Download ECE 441 lecture 15

Validating Computer System and
Network Trustworthiness
Prof. William H. Sanders
Department of Electrical and Computer Engineering and
Coordinated Science Laboratory
University of Illinois at Urbana-Champaign
[email protected]
www.mobius.uiuc.edu
www.perform.csl.uiuc.edu
www.iti.uiuc.edu
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 1
Course Outline
• Issues in Model-Based Validation of High-Availability
Computer Systems/Networks
• Combinatorial Modeling
• Stochastic Activity Network Concepts
• Analytic/Numerical State-Based Modeling
• Case Study: Embedded Fault-Tolerant Multiprocessor System
• Solution by Simulation
• Symbolic State-space Exploration and Numerical Analysis of
State-sharing Composed Models
• Case Study: Security Evaluation of a Publish and Subscribe
System
• The Art of System Trust Evaluation /Conclusions
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 2
What is Validated? -- Dependability
•
•
Dependability is the ability of a system to deliver a specified service.
System service is classified as proper if it is delivered as specified; otherwise it
is improper.
• System failure is a transition from proper to improper service.
• System restoration is a transition from improper to proper service.
failure
improper
service
proper
service
restoration
 The “properness” of service depends on the user’s viewpoint!
Reference: J.C. Laprie (ed.), Dependability: Basic Concepts and Terminology,
Springer-Verlag, 1992.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 5
Basic Validation Terms
• Measures -- What you want to know about a system. Used to determine if a
realization meets a specification
• Models -- Abstraction of the system at an appropriate level of abstraction and/or
details to determine the desired measures about a realization.
• Dependability Model Solution Methods -- Method by which one determines
measures from a model. Models can be solved by a variety of techniques:
– Combinatorial Methods -- Structure of the model is used to obtain a simple
arithmetic solution.
– Analytical/Numerical Methods -- A system of linear differential equations or
linear equations is constructed, which is solved to obtain the desired
measures
– Simulation -- The realization of the system is executed, and estimates of the
measures are calculated based on the resulting executions (known also as
sample paths or trajectories.)
 Möbius supports performance/reliability/availability validation by
analytical/numerical and simulation-based methods.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 6
Dependability Measures: Availability
Availability - quantifies the alternation between deliveries of proper and improper
service.
– A(t) is 1 if service is proper at time t, 0 otherwise.
– E[A(t)] (Expected value of A(t)) is the probability that service is proper at
time t.
– A(0,t) is the fraction of time the system delivers proper service during [0,t].
– E[A(0,t)] is the expected fraction of time service is proper during [0,t].
– P[A(0,t) > t*] (0  t*  1) is the probability that service is proper more than
100t*% of the time during [0,t].
– A(0,t)t is the fraction of time that service is proper in steady state.
– E[A(0,t)t], P[A(0,t)t > t*] as above.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 7
Other Dependability Measures
• Reliability - a measure of the continuous delivery of service
– R(t) is the probability that a system delivers proper service throughout [0,t].
• Safety - a measure of the time to catastrophic failure
– S(t) is the probability that no catastrophic failures occur during [0,t].
– Analogous to reliability, but concerned with catastrophic failures.
• Time to Failure - measure of the time to failure from last restoration. (Expected
value of this measure is referred to as MTTF - Mean time to failure.)
• Maintainability - measure of the time to restoration from last experienced
failure. (Expected value of this measure is referred to as MTTR - Mean time to
repair.)
• Coverage - the probability that, given a fault, the system can tolerate the fault
and continue to deliver proper service.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 8
How is Validation Done?
Validation
Measurement
Modeling
Passive
Active
(no fault (Fault Injection
injection) on Prototype)
Without
Contact
Simulation
With
Contact
HardwareImplemented
Möbius supports
model-based validation
of italicized (red) items.
Continuous
State
Discrete
Event
(state)
Analysis/
Numerical
Deterministic
Non-Deterministic
Probabilistic
SoftwareImplemented
Non-Probabilistic
Sequential Parallel
Stand-alone
Systems
Networks/
Distributed
Systems
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Non-State-space-based
State-space-based (Combinatorial)
Slide 9
Integrated Validation Procedure
R
S
Requirement
Decomposition
Q
P
Functional Model of the Relevant Subset of the System
…
ModuleB
ModuleA
AA1
M1
M2
L1
L2
AA2
AA3
M4
M3
ModuleZ
AP1
AP2
M5
M6
L3
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Functional Model
of the System
(Probabilistic or
Logical)
Assumptions
Supporting Logical
Arguments and
Experimentation
Slide 10
Probability Review: Exponential Random Variables
An exponential random variable X with parameter l has the CDF
P[X  t] = Fx(t) =
{
0
1-e-lt
The density function is given by f x (t ) 
fx(t) =
{
0
le-lt
t0
t>0
t0
t>0 .
d
Fx (t );
dt
1
1
Its mean is and its variance is 2 .
l
l
The exponential random variable is the only continuous random variable that is
“memoryless.”
To see this, let X be an exponential random variable representing the time that an
event occurs (e.g., a fault arrival).
Important Fact 1: P X  t  s X  s   P X  t  (memoryless property)!
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 12
Probability Review: Exponential Event Rate
• The fact that the exponential random variable has the memoryless property
indicates that the “rate” at which events occur is constant, i.e., it does not change
over time.
• Often, the event associated with a random variable X is a failure, so the “event
rate” is often called the failure rate or the hazard rate.
• The event rate of X is defined as the probability that the event associated with X
occurs within the small interval [t, t + Dt], given that the event has not occurred
by time t, per the interval size Dt:
Pt  X  t  Dt X  t 
.
Dt
• This can be thought of as looking at X at time t, observing that the event has not
occurred, and measuring the number of events (probability of the event) that
occur per unit of time at time t.
Important Fact 2: The exponential random variable has a constant failure rate!
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 13
Probability Review: Minimum of Two Independent Exponentials
Another interesting property of exponential random variables is that the minimum
of two independent exponential random variables is also an exponential random
variable.
Let A and B be independent exponential random variables with rates a and b
respectively. Let us define X = min{A,B}. What is FX(t)?
FX(t) = P[X  t]
= P[min{A,B}  t]
= P[A  t OR B  t]
= 1 - P[A > t AND B > t]
= 1 - P[A > t] P[B > t]
= 1 - (1 - P[A  t])(1 - P[B  t])
= 1 - (1 - FA(t))(1 - FB(t))
= 1 - (1 - [1 - e-at])(1 - [1 - e-bt])
= 1 - e-ate-bt
= 1 - e-(a + b)t
Important Fact 3: The minimum of two independent exponential random variables
is itself exponential with rate the sum of the two rates!
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 14
Probability Review: Competition of Two Independent Exponentials
If A and B are independent and exponential with rate a and b respectively, and A and B
are competing, then we know that one will “win” with an exponentially distributed
time (with rate a + b). But what is the probability that A wins?
P A  B    PA  B A  x  P A  x  dx

0
  P A  B A  x  f A  x dx

0
  P A  B A  x  ae - ax dx

0

  Px  B  ae - ax dx
0

  1 - PB  x  ae - ax dx
0


  1 - 1 - e -bx  ae - ax dx

0

  e -bx ae - ax dx
0

 a  e - a  b  x dx 
0
a
ab
Important Fact 4: If A and B are independent, competing exponentials, with rates
a and b respectively, the probability that A occurs before B is a/a + b!
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 15
Course Outline
• Issues in Model-Based Validation of High-Availability
Computer Systems/Networks
• Combinatorial Modeling
• Stochastic Activity Network Concepts
• Analytic/Numerical State-Based Modeling
• Case Study: Embedded Fault-Tolerant Multiprocessor System
• Solution by Simulation
• Symbolic State-space Exploration and Numerical Analysis of
State-sharing Composed Models
• Case Study: Security Evaluation of a Publish and Subscribe
System
• The Art of System Trust Evaluation /Conclusions
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 16
Combinatorial Methods
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 17
Introduction to Combinatorial Methods
• Combinatorial validation methods are the simplest kind of
analytical/numerical techniques and can be used for reliability and
availability modeling under certain assumptions.
• Assumptions are that component failures are independent, and for
availability, repairs are independent.
• When these assumptions hold, simple formulas for reliability and
availability exist.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 18
Lecture Outline
• Review definition of reliability
• Failure rate
• System reliability
– Maximum
– Minimum
– k of N
• Reliability formalisms
– Reliability block diagrams
– Fault trees
– Reliability graphs
• Reliability modeling process
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 19
Reliability
• One key to building highly available systems is the use of reliable components
and systems.
• Reliability: The reliability of a system at time t (R(t)) is the probability that the
system operation is proper throughout the interval [0,t].
• Probability theory and combinatorics can be directly applied to reliability
models.
• Let X be a random variable representing the time to failure of a component. The
reliability of the component at time t is given by
RX(t) = P[X > t] = 1 - P[X  t] = 1 - FX(t).
• Similarly, we can define unreliability at time t by
UX(t) = P[X  t] = FX(t).
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 20
Failure Rate
What is the rate that a component fails at time t? This is the probability that a
component that has not yet failed fails in the interval (t, t + Dt), as Dt  0.
Note that we are not looking at P[X  (t, t + Dt)] = fX(t). Rather, we are seeking
P[X  (t, t + Dt)| X > t].
P[ X  (t , t  Dt ), X  t ]
P[ X  t ]
P X  t , t  Dt 

1 - FX t 
P[ X  (t , t  Dt ) | X  t ] 

f X (t )
 rX (t )
1 - FX (t )
rX(t) is called the failure rate or hazard rate.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 21
Typical Failure Rate
Break in
Normal operation
Wear out
rX(t)
time
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 22
System Reliability
While FX can give the reliability of a component, how do you compute the
reliability of a system?
System failure can occur when one, all, or some of the components fail. If one
makes the independent failure assumption, system failure can be computed quite
simply. The independent failure assumption states that all component failures of a
system are independent, i.e., the failure of one component does not cause another
component to be more or less likely to fail.
Given this assumption, one can determine:
1) Minimum failure time of a set of components
2) Maximum failure time of a set of components
3) Probability that k of N components have failed at a particular time t.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 23
Maximum of n Independent Failure Times
Let X1, . . . , Xn be independent component failure times. Suppose the system fails
at time S if all the components fail.
Thus, S = max{X1, . . . , Xn}
What is Fs(t)?
Fs(t) = P[S  t]
= P[X1  t AND X2  t AND . . . AND Xn  t]
= P[X1  t] P[X2  t] . . . P[Xn  t]
By independence
= FX1 (t ) FX 2 (t )...FX n (t )
By definition
n
=
F
i 1
Xi
(t )
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 24
Minimum of n Independent Component Failure Times
Let X1, . . . , Xn be independent component failure times. A system fails at time
S if any of the components fail.
Thus, S = min{X1, . . . , Xn}.
What is FS(t)?
FS(t) = P[S  t] = P[X1  t OR X2  t OR . . . OR Xn  t]
Trick : If Ai is an event, and Ai is the set complement
such that Ai  Ai   and Ai  Ai  , then
P[ A1 OR A2 OR . . . OR An ]
 1 - P[ A1 AND A2 AND . . . AND An ]
This is an application of the law of total probability (LOTP).
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
A2
A1

A3
Slide 25
Minimum cont.
Fs(t) = P[X1  t OR X2  t OR . . . OR Xn  t]
= 1 - P[X1 > t AND X2 > t AND . . . AND Xn > t]
= 1 - P[X1 > t] P[X2 > t] . . . P[Xn > t]
= 1 - (1 - P[X1  t])(1 - P[X2  t]) . . . (1 - P[Xn  t])
By trick
By independence
By LOTP
n
= 1 -  (1 - FX (t ))
i 1
i
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 26
k of N
Let X1, . . . , Xn be component failure times that have identical distributions (i.e.,
FX (t )  FX (t ) = . . .). The system fails at time S if k of the N components fail.
1
2
FS(t) = P[at least k components failed by time t]
= P[k failed OR k + 1 failed OR . . . OR N failed]
= P[k failed] + P[k + 1 failed] + . . . + P[N failed]
- by independence
and axiom of
probability.
What is P[exactly k failed]?
= P[k failed and (N - k) have not]
N
k
N -k
=   FX (t ) (1 - FX (t ))
k
where FX(t) is the failure distribution of each component.
Thus,
N
FS (t )     FX (t ) i (1 - FX (t )) N -i
i k  i 
N
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 27
k of N in General
For non-identical failure distributions, we must sum over all combinations of at
least k failures.
Let Gk be the set of all subsets of {X1, . . . , XN} such that each element in Gk is a set
of size at least k, i.e.,
Gk = {gi  {X1, . . . , XN} : |gi|  k}.
The set Gk represents all the possible failure scenarios.
Now FS is given by








FS (t )     FX (t )    1 - FX (t ) 
gGk  X g
  X g

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 28
Component Building Blocks
Complex systems can be analyzed hierarchically.
Example: A computer fails if both power supplies fail or both memories fail or the
CPU fails.
FS(t) = 1 - (1 - FP1(t)FP2(t))(1- FM1(t)FM2(t))(1 - FC(t))
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 29
Summary
A system comprises N components, where the component failure times are given
by the random variables X1, . . . , XN. The system fails at time S with distribution
FS if:
Condition:
Distribution:
N
all components fail
FS (t )   FX (t )
one component fails
FS (t )  1 -  1 - FX (t ) 
i 1
i
N
i 1
i
k components fail,
identical distributions
N
N -i
FS (t )     FX (t ) i 1 - FX (t ) 
i k  i 
k components fail,
general case






FS (t )     FX (t )    1 - FX (t ) 
gGk  X g
  X g

N
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 30
Reliability Formalisms
There are several popular graphical formalisms to express system reliability. The
core of the solvers is the methods we have just examined. In particular, we will
examine
• Reliability Block Diagrams
• Fault Trees
• Reliability Graphs
There is nothing particularly special about these formalisms except their popularity.
It is easy to implement these formalisms, or design your own, in a spreadsheet, for
example.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 31
Reliability Block Diagrams
• Blocks represent components.
• A system failure occurs if there is no path from source to sink.
Series:
System fails if any component fails.
Parallel:
System fails if all components fail.
source
C1
C2
C3
sink
C1
source
C2
sink
C3
k of N:
System fails if at least k of N
components fail.
C1
source
C2
sink
C3
2 of 3
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 32
Example
A NASA satellite architecture under study is designed for high reliability. The
major computer system components include the CPU system, the high-speed
network for data collection and transmission, and the low-speed network for
engineering and control. The satellite fails if any of the major systems fail.
There are 3 computers, and the computer system fails if 2 or more of the computers
fail. Failure distribution of a computer is given by FC.
There is a redundant (2) high-speed network, and the high-speed network system
fails if both networks fail. The distribution of a high-speed network failure is given
by FH.
The low-speed network is arranged similarly, with a failure distribution of FL.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 33
RBG Example
computer
LSN
HSN
source
computer
sink
HSN
LSN
computer
2 of 3
  3  3 i
3- i 
FS (t )  1 - 1 -     FC (t )1 - FC t  

  i 2  i 
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

 1 -  FH t 2 1 -  FL t 2





Slide 34
Fault Trees
•
•
•
•
Components are leaves in the tree
A component fails = logical value of true, otherwise false.
The nodes in the tree are boolean AND, OR, and k of N gates.
The system fails if the root is true.
AND gates
true if all the components are true (fail).
AND
C1 C2 C3
OR
OR gates
true if any of the components are true (fail).
C1
k of N gates
true if at least k of the components are true (fail).
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
C2
C3
2 of 3
C1 C2 C3
Slide 35
Fault Tree Example
OR
2 of 3
C1
C2
AND
C3
H1
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
AND
H2
L1
L2
Slide 36
Combinatorial Methods: Review
A system comprises N components, where the component failure times are given
by the random variables X1, . . . , XN. The system fails at time S with distribution
FS if:
Condition:
Distribution:
N
all components fail
FS (t )   FX (t )
one component fails
FS (t )  1 -  1 - FX (t ) 
i 1
i
N
i 1
i
k components fail,
identical distributions
N
N -i
FS (t )     FX (t ) i 1 - FX (t ) 
i k  i 
k components fail,
general case






FS (t )     FX (t )    1 - FX (t ) 
gGk  X g
  X g

N
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 37
Reliability Formalisms
There are several popular graphical formalisms to express system reliability. The
core of the solvers is the methods we have just examined. In particular, we will
examine
• Reliability Block Diagrams
• Fault Trees
• Reliability Graphs
There is nothing particularly special about these formalisms except their popularity.
It is easy to implement these formalisms, or design your own, in a spreadsheet, for
example.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 38
Reliability Block Diagrams
• Blocks represent components.
• A system failure occurs if there is no path from source to sink.
Series:
System fails if any component fails.
Parallel:
System fails if all components fail.
source
C1
C2
C3
sink
C1
source
C2
sink
C3
k of N:
System fails if at least k of N
components fail.
C1
source
C2
sink
C3
2 of 3
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 39
Example
A NASA satellite architecture under study is designed for high reliability. The
major computer system components include the CPU system, the high-speed
network for data collection and transmission, and the low-speed network for
engineering and control. The satellite fails if any of the major systems fail.
There are 3 computers, and the computer system fails if 2 or more of the computers
fail. Failure distribution of a computer is given by FC.
There is a redundant (2) high-speed network, and the high-speed network system
fails if both networks fail. The distribution of a high-speed network failure is given
by FH.
The low-speed network is arranged similarly, with a failure distribution of FL.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 40
RBG Example
computer
LSN
HSN
source
computer
sink
HSN
LSN
computer
2 of 3
  3  3 i
3- i 
FS (t )  1 - 1 -     FC (t )1 - FC t  

  i 2  i 
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

 1 -  FH t 2 1 -  FL t 2





Slide 41
Fault Trees
•
•
•
•
Components are leaves in the tree
A component fails = logical value of true, otherwise false.
The nodes in the tree are boolean AND, OR, and k of N gates.
The system fails if the root is true.
AND gates
true if all the components are true (fail).
AND
C1 C2 C3
OR
OR gates
true if any of the components are true (fail).
C1
k of N gates
true if at least k of the components are true (fail).
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
C2
C3
2 of 3
C1 C2 C3
Slide 42
Fault Tree Example
OR
2 of 3
C1
C2
AND
C3
H1
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
AND
H2
L1
L2
Slide 43
Reliability Graphs
• The arcs represent components and have failure distributions.
• A failure occurs if there is no path from source to sink.
Can implement series:
source
1
FC1
2
FC2
3
sink
FC1
Can implement parallel:
source
1
FC2
2
sink
FC3
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 44
Reliability Graph Example
Reliability graphs can implement more complex interactions.
For example, a telephone network “fails” if there is no path from source to sink.
2
A
source
1
D
B
How do we solve this?
4
C
sink
E
3
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 45
Solving by Conditioning
P[ E  F ]
P[ F ]
If F and F are complementary events, i.e.,
Recall that P[ E | F ] 
F  F   and F  F  
then thereis a trick :
P[ E ]  P[ E  F ]  P[ E  F ]
P[ E ]  P[ E | F ]P[ F ]  P[ E | F ]P[ F ]
E
F

If you can solve P[ E | F ], P[ F ], P[ E | F ], and P[ F ], then you can solve P[ E ].
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 46
A
source
1
2
D
4
C
B
3
sink
E
First, condition the system on link C being failed.
Then the system becomes the series AD in parallel with the series BE.
A
source
1
2
D
4
B
sink
E
3
FS |C Fail (t )  P[ S  t | C  t ]  1 - 1 - FA (t ) 1 - FD (t ) 1 - 1 - FB (t ) 1 - FE (t ) 
and P[C  t ]  FC (t )
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 47
Second, condition the system on link C being up.
A
source
1
D
4
2,3
B
sink
E
FS |C up (t )  P[ S  t | C  t ]  1 - 1 - FA (t ) FB (t ) 1 - FD (t ) FE (t ) ,
and P[C  t ]  1 - P[C  t ]  1 - FC (t )
Thus, FS (t )  FS |C Fail (t ) FC (t )  FS |C up (t )1 - FC (t ) .
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 48
Conditioning Fault Trees
It is also possible to use conditioning to solve more complex fault trees. If the same
component appears more than once in a fault tree, it violates the independent failure
assumption. However, a conditioned fault tree can be solved.
Example: A component C appears multiple times in the fault tree.
FS t   FS C Fail t FC (t )  FS C Up t 1 - FC t 
Where S C Fail is the system given that C has failed
and S C Up is the system given that C has not failed.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 49
Reliability/Availability Point Estimates
• Frequently, the desired measure of a reliability model is the reliability at some
time t. Thus, the distribution of the system reliability is superfluous; R(t) is the
only thing of interest.
• This condition simplifies computation because all that is necessary for solution
is the reliability of the components at time t. Solution then becomes a
straightforward computation.
• If a system is described in terms of the availability of components at time t, then
we may compute the system availability in the same way that reliability is
computed. The restriction is that all component behaviors must be independent
of one another.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 50
Reliability/Availability Tables
A system comprises N components. Reliability of component i at time t is given by
RXi(t), and the availability of component i at time t is given by AXi(t).
Condition
system fails if all
components fail
System Reliability
n
AS t   1 -  1 - AXi t 
n
RS t   1 -  1 - RXi t 
i 1
i 1
system fails if
one component fails
n
n
AS t    AXi t 
RS t    RXi t 
i 1
i 1
system fails if at
N
N
i
N -i
least k components
RS t     1 - RXi t  RX t 
i k  i 
fail, identical distribution
system fails if at least
k components fail,
general case
System Availability
RS t  













1
R
t
R
t

X

  X 
gG  X g
 X g

k
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
N
i
N -i
AS t     1 - AX t  AX t 
i k  i 
N
AS t  











1
A
t
A
t




X
 X 
gG  X G
 X g

k
Slide 51
Estimating Component Reliability
• For hardware, MIL-HDBK-217 is widely used.
– Not always current with modern components.
– Lacks distributions; it only contains failure rates.
– While not perfect, it seems to be the best source that exists. However,
numbers from MIL-HDBK-217 should be used with caution.
• Due to the nature of software, no accepted mechanism exists to predict software
reliability before the software is built.
– Best guess is the reliability of previously built similar software.
• In all cases, numbers should be used with caution and adjusted based on
observation and experience.
• No substitute for empirical observation and experience!
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 52
Modeling Process
• Reliability models are built only after proper service is specified.
• Reliability models are built to answer the question “What subsystem or
components must be proper for the system to be proper?”
• Build models hierarchically out of subsystems.
• Estimation and guesses are acceptable, but state them explicitly.
• If unsure, do sensitivity analysis to see how much it matters.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 53
Reliability Modeling Process
• Realistic systems result in large RBDs and must be managed hierarchically.
RBD Process(system)
Define the system
Define “proper service”
Create RBD out of components
for each component
if component is simple
obtain reliability data of component
else
Do RBD Process(component)
end if
Compute reliability of system
Do results meet specification?
Modify design and repeat as necessary
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 54
Summary
– Reliability: review of definition
– Failure rate
– System reliability
• Independent failure assumption
• Minimum, maximum, k of N
• Reliability block diagrams, fault trees, reliability graphs
– Reliability modeling process
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 55
Stochastic Activity Network Concepts
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 56
Introduction
Stochastic activity networks, or SANs, are a convenient, graphical, high-level
language for describing system behavior. SANs are useful in capturing the
stochastic (or random) behavior of a system.
Examples:
– The amount of time a program takes to execute can be computed precisely if
all factors are known, but this is nearly impossible and sometimes useless.
At a more abstract level, we can approximate the running time by a random
variable.
– Fault arrivals almost always must be modeled by a random process.
We begin by describing a subset of SANs: stochastic Petri nets.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 58
Stochastic Petri Net Review
One of the simplest high-level modeling formalisms is called stochastic Petri nets.
A stochastic Petri net is composed of the following components:
• Places:
which contain tokens, and are like variables
• tokens:
which are the “value” or “state” of a place
• transitions:
which change the number of tokens in places
• input arcs:
which connect places to transitions
• output arcs:
which connect transitions to places
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 59
Firing Rules for SPNs
A stochastic Petri net (SPN) executes according to the following rules:
• A transition is said to be enabled if for each place connected by input arcs,
the number of tokens in the place is  the number of input arcs connecting
the place and the transition.
Example:
P1
P2
t1
Transition t1 is enabled.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 60
Firing Rules, cont.
•
•
A transition may fire if it is enabled. (More about this later.)
If a transition fires, for each input arc, a token is removed from the
corresponding place, and for each output arc, a token is added to the
corresponding place.
Example:
P1
P3
t1 fires
t1
P2
P4
Note: tokens are not necessarily conserved when a transition fires.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 61
Specification of Stochastic Behavior of an SPN
•
A stochastic Petri net is made from a Petri net by
– Assigning an exponentially distributed time to all transitions.
– Time represents the “delay” between enabling and firing of a transition.
– Transitions “execute” in parallel with independent delay distributions.
•
Since the minimum of multiple independent exponentials is itself exponential,
time between transition firings is exponential.
•
If a transition t becomes enabled, and before t fires, some other transition fires
and changes the state of the SPN such that t is no longer enabled, then t aborts,
that is, t will not fire.
•
Since the exponential distribution is memoryless, one can say that transitions
that remain enabled continue or restart, as is convenient, without changing the
behavior of the network.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 62
Notes on SPNs
• SPNs are much easier to read, write, modify, and debug than Markov chains.
• SPN to Markov chain conversion can be automated to afford numerical
solutions to Markov chains.
• Most SPN formalisms include a special type of arc called an inhibitor arc,
which enables the SPN if there are zero tokens in the associated place, and the
identity (do nothing) function. Example: modify SPN to give writes priority.
• Limited in their expressive power: may only perform +, -, >, and test-for-zero
operations.
• These very limited operations make it very difficult to model complex
interactions.
• Simplicity allows for certain analysis, e.g., a network protocol modeled by an
SPN may detect deadlock (if inhibitor arcs are not used).
• More general and flexible formalisms are needed to represent real systems.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 65
Stochastic Activity Networks
The need for more expressive modeling languages has led to several extensions to
stochastic Petri nets. One extension that we will examine is called stochastic
activity networks. Because there are a number of subtle distinctions relative to
SPNs, stochastic activity networks use different words to describe ideas similar to
those of SPNs.
Stochastic activity networks have the following properties:
•
•
•
•
•
•
A general way to specify that an activity (transition) is enabled
A general way to specify a completion (firing) rule
A way to represent zero-timed events
A way to represent probabilistic choices upon activity completion
State-dependent parameter values
General delay distributions on activities
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 66
SAN Symbols
Stochastic activity networks (hereafter SANs) have four new symbols in addition to
those of SPNs:
– Input gate:
– Output gate:
– Cases:
used to define complex enabling predicates and completion
functions
used to define complex completion functions
(small circles on activities) used to specify probabilistic choices
– Instantaneous activities:
used to specify zero-timed events
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 67
SAN Terms
1. activation - time at which an activity begins
2. completion - time at which activity completes
3. abort - time, after activation but before completion, when activity is no longer
enabled
4. active - the time after an activity has been activated but before it completes or
aborts.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 73
Illustration of SAN Terms
activation
completion
activity time
t
activation
aborted
enabled
activity time
activation
completion
and activation
activity
time
t
completion
activity
time
enabled
t
enabled
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 74
Completion Rules
When an activity completes, the following events take place (in the order listed),
possibly changing the marking of the network:
1. If the activity has cases, a case is (probabilistically) chosen.
2. The functions of all the connected input gates are executed (in an
unspecified order).
3. Tokens are removed from places connected by input arcs.
4. The functions of all the output gates connected to the chosen case are
executed (in an unspecified order).
5. Tokens are added to places connected by output arcs connected to the
chosen case.
Ordering is important, since effect of actions can be marking-dependent.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 75
General Delay Distributions
• SANs (and their implementation in Möbius) support many activity time
distributions, including:
•
•
•
•
•
•
Exponential
Hyperexponential
Deterministic
Weibull
Conditional Weibull
Normal
•
•
•
•
•
•
Erlang
Gamma
Beta
Uniform
Binomial
Negative Binomial
• All distribution parameters can be marking-dependent
• The obvious implication of general delay distributions is that there is no
conversion to a CTMC. Hence, no solutions to CTMCs are applicable.
However, simulation is still possible.
• Analytical/numerical solution is possible for certain mixes of exponential and
deterministic activities. See the Möbius manual for details.
• See [Kececioglu 91], for example, for appropriate use of some of these
distributions.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 80
Fault-Tolerant Computer Failure Model Example
A fault-tolerant computer system is made up of two redundant computers. Each
computer is composed of three redundant CPU boards. A computer is operational if
at least 1 CPU board is operational, and the system is operational if at least 1
computer is operational.
CPU boards fail at a rate of 1/106 hours, and there is a 0.5% chance that a board
failure will cause a computer failure, and a 0.8% chance that a board will fail in a
way that causes a catastrophic system failure.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 81
SAN computer for Computer Failure Model
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 82
Activity Case Probabilities and Input Gate Definition
Activity
CPUfail1
Case
1
2
3
Probability
0.987
0.005
0.008
Gate
Definition
Enabled1 Predicate
MARK(CPUboards1 > 0) && MARK(NumComp) > 0
Function
MARK(CPUboards1)--;
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 83
Output Gate Definitions
Definition
Function
if (MARK(CPUboards1) == 0)
MARK(NumComp)--;
Function
Uncovered1
MARK(CPUboards1) = 0;
MARK(NumComp)--;
Catastrophic1 Function
MARK(CPUboards1) = 0;
MARK(NumComp) = 0;
Gate
Covered1
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 84
Reward Variables
Reward variables are a way of measuring performance- or dependability-related
characteristics about a model.
Examples:
– Expected time until service
– System availability
– Number of misrouted packets in an interval of time
– Processor utilization
– Length of downtime
– Operational cost
– Module or system reliability
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 85
Reward Structures
Reward may be “accumulated” two different ways:
– A model may be in a certain state or states for some period of time, for
example, “CPU idle” states. This is called a rate reward.
– An activity may complete. This is called an impulse reward.
The reward variable is the sum of the rate reward and the impulse reward structures.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 86
Reward Structure Example
A web server failure model is used to predict profits. When the web server is fully
operational, profits accumulate at $N/hour. In a degraded mode, profits accumulate
at 16 N/hour. Repairs cost $K.
N

R m    16 N
0

m is a fully functioning marking
m is a degraded-mode marking
otherwise
- K
C a   
0
a is an activity representing repair
otherwise
By carefully integrating the reward structure from 0 to t, we get the profit at time t.
This is an example of an “interval-of-time” variable.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 87
Reward Variables
A reward variable is the sum of the impulse and rate reward structures over a
certain time.
Let [t, t + l] be the interval of time defined for a reward variable:
– If l is 0, then the reward variable is called an instant-of-time reward variable.
– If l > 0, then the reward variable is called an interval-of-time reward
variable.
– If l > 0, then dividing an interval-of-time reward variable by l gives a timeaveraged interval-of-time reward variable.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 88
Reward Variable Specification
Reward Structure
Interval-of-Time
Instant-of-Time
Time-Average Interval-of-Time
[t, t + l]
t
lim as t
goes to
infinity
[t, t + l]
[t, t + l] lim as l
[t, t + l]
[t, t + l]
[t, t + l]
lim as t
goes to
infinity
lim as l
goes to
infinity
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
lim as t goes to
goes to infinity
infinity
Slide 89
Reward Variables for Computer Failure Model
Reliability
Rate rewards
Subnet = computer
Predicate:
MARK(NumComp) > 0
Function:
1
Impulse reward
none
NumBoardFailures
Rate reward
none
Impulse reward
Subnet = computer
activity = CPUfail1, value = 1
activity = CPUfail2, value = 1
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 92
Reward Variables for Computer Failure Model
Performability
Rate rewards
Subnet = computer
Predicate:
1
Function:
MARK(NumComp)
Impulse reward
none
NumBoards
Rate reward
Subnet = computer
Predicate:
1
Function:
MARK(CPUBboards1) + MARK(CPUboards2)
Impulse reward
none
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 93
Model Composition
A composed model is a way of connecting different SANs together to form a larger
model.
Model composition has two operations:
– Replicate: Combine 2 or more identical SANs and reward structures
together, holding certain places common among the replicas.
– Join: Combine 2 or more different SANs and reward structures together,
combining certain places to permit communication.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 94
Composed Model Specification
• Join two or more
submodels
together
• Replicate
submodel a
certain number of
times
• Certain places in
different
submodels can be
made common
• Hold certain
places common to
all replicas
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 95
Rationale
There are many good reasons for using composed models.
– Building highly reliable systems usually involves redundancy. The
replicate operation models redundancy in a natural way.
– Systems are usually built in a modular way. Replicates and Joins are
usually good for connecting together similar and different modules.
– Tools can take advantage of something called the Strong Lumping Theorem
that allows a tool to generate a Markov process with a smaller state space
(to be described in Session 7).
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 96
Computer Failure Model Revisited: Single computer Model
(Note initial marking of NumComp is two since there will be two computers
in the composed model.)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 98
Composed Model for Computer Failure Model
Node
Rep1
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Reps
2
Common Places
NumComp
Slide 99
Composed Model
How does adding an additional computer affect reliability?
– In the composed model, change number of replications to 3 and change
various reward variables - easy (Use a global variable if you think suspect
you may want to do this.)
– In “flat” model, add another computer - hard
In composed model, the number of states in the underlying Markov chain is much
smaller, especially for large numbers of replications. (Details will be given in
Session 7.)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 102
Analytic/Numerical State-Based Modeling
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 103
Session Outline
• Review of Markov process theory and fundamentals
• Methods for constructing state-level models from SANs
• Analytic/numerical solution techniques
– Transient solution
• Standard uniformization (instant-of-time variables)
• Adaptive uniformization (instant-of-time variables)
• Interval-of-time uniformization (interval-of-time variables)
– Steady-state solution (steady-state instant-of-time variables)
• Direct solution
• Iterative solution
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 104
Weaknesses of Simulation
• Simulation relies on good pseudo-random number generation, sufficient
observations, and good statistical techniques to produce an approximate solution
• Increasing accuracy by a factor of n requires on the order of n2 more work,
which can be prohibitively expensive.
For example, a 5-Nines system reliability model will require approximately
100,000 observations to observe one failure. One digit of accuracy can easily
require over 1,000,000 observations!
(For many models, 1,000,000 observations can be generated quickly, but as
system failure becomes even rarer, standard simulation quickly becomes
infeasible.)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 105
The Case for Analytical/Numerical Techniques
If you can model using exponential delays and your model is sufficiently small,
continuous time Markov chains (CTMCs) offer some advantages. These include:
– Typically faster solution time for systems with rare events
– Typically takes less time to get more accurate answers
– Typically more confidence in the solution
In order to understand when we get these advantages, we must better understand the
methods of obtaining solutions to CTMCs.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 106
Random Variable Review
It is often convenient to assign a (real) number to every element in . This
assignment, or rule, or function, is called a random variable.
w

-1
0
1
X:
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 107
Random Process Review
Random processes are useful for characterizing the behavior of real systems.
A random process is a collection of random variables indexed by time.
Example: X(t) is a random process. Let X(1) be the result of tossing a die. Let X(2)
be the result of tossing a die plus X(1), and so on. Notice that time (T) = {1,2,3,
. . .}.
One can ask: P X 2  12  361
PX 3  14 X 1  2  361
E X n   3.5n
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 108
Describing a Random Process
Recall that for a random variable X, we can use the cumulative distribution FX to
describe the random variable.
In general, no such simple description exists for a random process.
However, a random process can often be described succinctly in various different
ways. For example, if Y is a random variable representing the roll of a die, and X(t)
is the sum after t rolls, then we can describe X(t) by
X(t) - X(t - 1) = Y,
P[X(t) = i|X(t - 1) = j] = P[Y = i - j],
or X(t) = Y1 + Y2 + . . . + Yt, where the Yi’s are independent.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 110
Classifying Random Processes: Characteristics of T
If the number of time points defined for a random process, i.e., |T|, is finite or
countable (e.g., integers), then the random process is said to be a discrete-time
random process.
If |T| is uncountable (e.g., real numbers) then the random process is said to be a
continuous-time random process.
Example: Let X(t) be the number of fault arrivals in a system up to time t. Since t 
T is a real number, X(t) is a continuous-time random process.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 111
Classifying Random Processes: State Space Type
Let X be a random process. The state space of a random process is the set of all
possible values that the process can take on, i.e.,
S = {y: X(t) = y, for some t  T}.
If X is a random process that models a system, then the state space of X can
represent the set of all possible configurations that the system could be in.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 112
Random Process State Spaces
If the state space S of a random process X is finite or countable
(e.g., S = {1,2,3, . . .}), then X is said to be a discrete-state random process.
Example: Let X be a random process that represents the number of bad
packets received over a network. X is a discrete-state random process.
If the state space S of a random process X is infinite and uncountable (e.g., S = ),
then X is said to be a continuous-state random process.
Example: Let X be a random process that represents the voltage on a
telephone line. X is a continuous-state random process.
We examine only discrete-state processes in this lecture.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 113
Stochastic-Process Classification Examples
Time
Continuous
Discrete
Analog signal
A to D converter
Computer
availability
model
round-based
network
protocol
model
State
Continuous
Discrete
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 114
Markov Process
A special type of random process that we will examine in detail is called the
Markov process. A Markov process can be informally defined as follows.
Given the state (value) of a Markov process X at time t (X(t)), the future
behavior of X can be described completely in terms of X(t).
Markov processes have the very useful property that their future behavior is
independent of past values.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 115
Markov Chains
A Markov chain is a Markov process with a discrete state space.
We will always make the assumption that a Markov chain has a state space in
{1,2, . . .} and that it is time-homogeneous.
A Markov chain is time-homogeneous if its future behavior does not depend on
what time it is, only on the current state (i.e., the current value).
We make this concrete by looking at a discrete-time Markov chain (hereafter
DTMC). A DTMC X has the following property:
P X t  k   j X t   i, X t - 1  nt -1 , X t - 2  nt -2 ,..., X O   nO 
 P X t  k   j X t   i 
(1)
 Pij k 
(2)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 116
DTMCs
Notice that given i, j, and k, Pij k  is a number!
Pij k  can be interpreted as the probability that if X has value i, then after k time-steps,
X will have value j.
Frequently, we write Pij to mean Pij1 .
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 117
Markov Chains
A Markov chain is a Markov process with a discrete state space.
We will always make the assumption that a Markov chain has a state space in
{1,2, . . .} and that it is time-homogeneous.
A Markov chain is time-homogeneous if its future behavior does not depend on
what time it is, only on the current state (i.e., the current value).
We make this concrete by looking at a discrete-time Markov chain (hereafter
DTMC). A DTMC X has the following property:
P X t  k   j X t   i, X t - 1  nt -1 , X t - 2  nt -2 ,..., X O   nO 
 P X t  k   j X t   i 
(1)
 Pij k 
(2)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 118
DTMCs
Notice that given i, j, and k, Pij k  is a number!
Pij k  can be interpreted as the probability that if X has value i, then after k time-steps,
X will have value j.
Frequently, we write Pij to mean Pij1 .
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 119
State Occupancy Probability Vector
Let p be a row vector. We denote pi to be the i-th element of the vector. If p is a
state occupancy probability vector, then pi(k) is the probability that a DTMC has
value i (or is in state i) at time-step k.
Assume that a DTMC X has a state-space size of n, i.e., S = {1, 2, . . . , n}. We say
formally
pi(k) = P[X(k) = i]
n
Note that  p i k   1 for all times k.
i 1
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 120
Computing State Occupancy Vectors: A Single Step Forward in Time
If we are given p(0) (the initial probability vector), and Pij for i, j = 1, . . . , n,
how do we compute p(1)?
Recall the definition of Pij.
Pij = P[X(k+1) = j | X(k) = i]
= P[X(1) = j | X(0) = i]
n
Since  pi 0  1,
i 1
p j 1  P X 1  j 
 P X 1  j X 0   1P X 0   1  ...  P X 1  j X 0   nP X 0   n
n
  P X (1)  j X 0   i P X 0   i 
i 1
n
  Pij pi 0 
i 1
n
  pi 0 Pij
i 1
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Slide 121
Transition Probability Matrix
n
We have p j 1   pi 0Pij , which holds for all j.
i 1
Notice that this resembles vector-matrix multiplication.
In fact, if we arrange the matrix P = {Pij}, that is, if
P=
p11
p1n
pn1
pnn
,
then pij = Pij, and p(1) = p(0)P, where p(0) and p(1) are row vectors, and p(0)P is a
vector-matrix multiplication.
The important consequence of this is that we can easily specify a DTMC in terms of
an occupancy probability vector p and a transition probability matrix P.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 122
Transient Behavior of Discrete-Time Markov Chains
Given p(0) and P, how can we compute p(k)?
We can generalize from earlier that
p(k) = p(k - 1)P.
Also, we can write p(k - 1) = p(k - 2)P, and so
p(k) = [p(k - 2)P]P
= p(k - 2)P2
Similarly, p(k - 2) = p(k - 3)P, and so
p(k) = [p(k - 3)P]P2
= p(k - 3)P3
By repeating this, it should be easy to see that
p(k) = p(0)Pk
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Slide 123
A Simple Example
Suppose the weather at Urbana-Champaign, Illinois can be modeled the following
way:
• If it’s sunny today, there’s a 60% chance of being sunny tomorrow, a
30% chance of being cloudy, and a 10% chance of being rainy.
• If it’s cloudy today, there’s a 40% chance of being sunny tomorrow, a
45% chance of being cloudy, and a 15% chance of being rainy.
• If it’s rainy today, there’s a 15% chance of being sunny tomorrow, a 60%
chance of being cloudy, and a 25% chance of being rainy.
If it’s rainy on Friday, what is the forecast for Monday?
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 124
Simple Example, cont.
Clearly, the weather model is a DTMC.
1) Future behavior depends on the current state only
2) Discrete time, discrete state
3) Time homogeneous
The DTMC has 3 states. Let us assign 1 to sunny, 2 to cloudy, and 3 to rainy. Let
time 0 be Friday.
p0   0,0,1
 .6 .3 .1 


P   .4 .45 .15
 .15 .6 .25


©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 125
Simple Example Solution
The weather on Saturday p(1) is
 .6 .3 .1 


p1  p0 P  0,0,1  .4 .45 .15  .15,.6,.25,
 .15 .6 .25


that is, 15% chance sunny, 60% chance cloudy, 25% chance rainy.
The weather on Sunday p(2) is
 .6 .3 .1 


p2   p1P  .15,.6,.25  .4 .45 .15  .3675,.465,.1675.
 .15 .6 .25


The weather on Monday p(3) is
p(3) = p(2)P = (.4316, .42, .1484),
that is, 43% chance sunny, 42% chance cloudy, and 15% chance rainy.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 126
Solution, cont.
Alternatively, we could compute P3 since we found
p(3) = p(0)P3.
Working out solutions by hand can be tedious and error-prone, especially for
“larger” models (i.e., models with many states). Software packages are used
extensively for this sort of analysis.
Software packages compute p(k) by (. . . ((p(0)P)P)P. . .)P rather than computing
Pk, since computing the latter results in a large “fill-in.”
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 127
Graphical Representation
It is frequently useful to represent the DTMC as a directed graph. Nodes represent
states, and edges are labeled with probabilities. For example, our weather
prediction model would look like this:
.45
2
.15
.3
1 = Sunny Day
2 = Cloudy Day
3 = Rainy Day
.6
.4
.1
.6
1
.15
3
.25
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 128
“Simple Computer” Example
Parr
Pbusy
Pidle
1
Pcom
2
Pfi
Pfb
Pr
X=1
X=2
X=3
computer idle
computer working
computer failed
3
Pff
 Pidle

P   Pcom
 Pr
Parr
Pbusy
0
Pfi 

Pfb 
Pff 
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 129
Limiting Behavior of DTMCs
It is sometimes useful to know the time-limiting behavior of a DTMC. This
translates into the “long term,” where the system has settled into some steady-state
behavior.
Formally, we are looking for lim pn .
n
To compute this, what we want is lim p0P n .
n 
There are various ways to compute this. The simplest is to calculate p(n) for
increasingly large n, and when p(n + 1)  p(n), we can believe that p(n) is a good
approximation to steady-state. This can be rather inefficient if n needs to be large.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 130
Classifications
It is much easier to solve for the steady-state behavior of some DTMC’s than
others. To determine if a DTMC is “easy” to solve, we need to introduce some
definitions.
Definition: A state j is said to be accessible from state i if there exists an n  0 such
that Pij( n )  0. We write i  j.
Note: recall that Pij( n )  P X (n)  j X (0)  i 
If one thinks of accessibility in terms of the graphical representation, a state j is
accessible from state i if there exists a path of non-zero edges (arcs) from node i to
node j.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 131
State Classification in DTMCs
Definition: A DTMC is said to be irreducible if every state is accessible from every
other state.
Formally, a DTMC is irreducible if
ij
for all i,j  S.
A DTMC is said to be reducible if it is not irreducible.
It turns out that irreducible DTMC’s are simpler to solve. One need only solve one
linear equation:
p = pP.
We will see why this is so, but first there is one more issue we must confront.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 132
Periodicity
Consider the following DTMC:
1
1
Does lim pn  exist? No!
1
p0   1,0 
2
n
n
 pi 
i 1
lim
However, n  n
does exist; it is called the time-averaged steady-state
distribution, and is denoted by p*.
Definition: A state i is said to be periodic with period d if Pij( n )  0 only when n is
some multiple of d. If d = 1, then i is said to be aperiodic.
A steady-state solution for an irreducible DTMC exists if all the states are aperiodic.
A time-averaged steady-state solution for an irreducible DTMC always exists.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 133
Steady-State Solution of DTMCs
The steady-state behavior can be computed by solving the linear equation
p = pP, with the constraint that
n
p
i 1
i
 1. For irreducible DTMC’s, it can be
shown that this solution is unique. If the DTMC is periodic, then this solution
yields p*.
One can understand the equation p = pP in two different ways.
• In steady-state, the probability distribution p(n + 1) = p(n)P, and by
definition p(n + 1) = p(n) in steady-state.
• “Flow” equations.
Flow equations require some visualization. Imagine a DTMC graph, where the
nodes are assigned the occupancy probability, or the probability that the DTMC has
the value of the node.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 134
i
Probability must be conserved, i.e.,
p i  1.
...
...
Flow Equations
Let piPij be the “probability mass” that moves from state j to state i in one time-step.
Since probability must be conserved, the probability mass entering a state must
equal the probability mass leaving a state.
Prob. mass in = Prob. mass out
n
n
p j Pji   p i Pij

j 1
j 1
n
 p i  Pij
j 1
Written in matrix form, p = pP.
 pi
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 135
Continuous Time Markov Chains (CTMCs)
For most systems of interest, events may occur at any point in time. This leads us
to consider continuous time Markov chains. A continuous time Markov chain
(CTMC) has the following property:
P X t     j X (t )  i, X (t - t1 )  k1 , X (t - t 2 )  k 2 ,..., X t - t n   k n 
 P X (t  )  j X (t )  i  ,
 Pij ()
for all   0, 0  t1  t 2  ...  t n
A CTMC is completely described by the initial probability distribution p(0) and the
transition probability matrix P(t) = [pij(t)]. Then we can compute p(t) = p0P(t).
The problem is that pij(t) is generally very difficult to compute.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 136
CTMC Properties
This definition of a CTMC is not very useful until we understand some of the
properties.
First, notice that pij() is independent of how long the CTMC has previously been in
state i, that is,
P X t    j X (u )  i for u  0, t 
 P X (t  )  j X (t )  i 
 pij ()
There is only one random variable that has this property: the exponential random
variable. This indicates that CTMCs have something to do with exponential
random variables. First, we examine the exponential r.v. in some detail.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 137
Exponential Random Variables
Recall the property of the exponential random variable. An exponential random
variable X with parameter l has the CDF
P[X  t] = Fx(t) =
{
0
1-e-lt
t0
t>0 .
The distribution function is given by f x (t ) 
fx(t) =
{
0
le-lt
t0
t>0
d
Fx (t );
dt
The exponential random variable is the only random variable that is “memoryless.”
To see this, let X be an exponential random variable representing the time that an
event occurs (e.g., a fault arrival).
We will show that PX  t  s X  s   P X  t  .
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 138
Memoryless Property
Proof of the memoryless property:
P X  t  s X  s  
P X  t  s , X  s 
P X  s 
P X  t  s 
P X  s 
1 - FX t  s 

1 - FX ( s )

e -l t  s 
 -ls
e
e -lt e -ls
 -ls
e
 e -lt
 P X  t 
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 139
Event Rate
The fact that the exponential random variable has the memoryless property
indicates that the “rate” at which events occur is constant, i.e., it does not change
over time.
Often, the event associated with a random variable X is a failure, so the “event rate”
is often called the failure rate or the hazard rate.
The event rate of X is defined as the probability that the event associated with X
occurs within the small interval [t, t + Dt], given that the event has not occurred by
time t, per the interval size Dt:
Pt  X  t  Dt X  t 
.
Dt
This can be thought of as looking at X at time t, observing that the event has not
occurred, and measuring the number of events (probability of the event) that occur
per unit of time at time t.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 140
Observe that:
Pt  X  t  Dt X  t  Pt  X  t  Dt , X  t 

Dt
P X  t   Dt
Pt  X  t  Dt 

P X  t   Dt
FX t  Dt  - FX t 

1 - FX t Dt

FX t  Dt  - FX (t )
1

Dt
1 - FX (t )

f X (t )
1 - FX (t )
in general.
In the exponential case,
f X (t )
le -lt
le -lt

 -lt  l.
-lt
1 - FX (t ) 1 - 1 - e  e
This is why we often say a random variable X is “exponential with rate l.”
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 141
Minimum of Two Independent Exponentials
Another interesting property of exponential random variables is that the minimum
of two independent exponential random variables is also an exponential random
variable.
Let A and B be independent exponential random variables with rates a and b
respectively. Let us define X = min{A,B}. What is FX(t)?
FX(t) = P[X  t]
= P[min{A,B}  t]
= P[A  t OR B  t]
= 1 - P[A > t AND B > t]
- see comb. methods section
= 1 - P[A > t] P[B > t]
= 1 - (1 - P[A  t])(1 - P[B  t])
= 1 - (1 - FA(t))(1 - FB(t))
= 1 - (1 - [1 - e-at])(1 - [1 - e-bt])
= 1 - e-ate-bt
= 1 - e-(a + b)t
Thus, X is exponential with rate a + b.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 142
Competition of Two Independent Exponentials
If A and B are independent and exponential with rate a and b respectively, and A
and B are competing, then we know that one will “win” with an exponentially
distributed time (with rate a + b). But what is the probability that A wins?
P A  B   PA  B A  x P A  x dx

0
  P A  B A  x  f A  x dx

0
  P A  B A  x  ae - ax dx

0

  Px  B  ae - ax dx
0

  1 - PB  x  ae - ax dx
0


  1 - 1 - e -bx  ae - ax dx

0

  e -bx ae - ax dx
0

 a  e - a b  x dx 
0
a
ab
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 143
Competing Exponentials in CTMCs
a
2
1
b
X(0) = 1
P[X(0) = 1] = 1
3
Imagine a random process X with state space S = {1,2,3}. X(0) = 1. X goes to state
2 (takes on a value of 2) with an exponentially distributed time with parameter a.
Independently, X goes to state 3 with an exponentially distributed time with
parameter b. These state transitions are like competing random variables.
We say that from state 1, X goes to state 2 with rate a and to state 3 with rate b.
X remains in state 1 for an exponentially distributed time with rate a + b. This is
1
called the holding time in state 1. Thus, the expected holding time in state 1 is a b .
The probability that X goes to state 2 is
a
a b
. The probability X goes to state 3 is
b
a b
.
This is a simple continuous-time Markov chain.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 144
Competing Exponentials vs. a Single
Exponential With Choice
Consider the following two scenarios:
1. Event A will occur after an exponentially distributed time with rate a. Event
B will occur after an independent exponential time with rate b.
2. After waiting an exponential time with rate a + b, event A occurs with
b
probability aab , and event B occurs with probability a b .
These two scenarios are indistinguishable. In fact, we frequently interchange the
two scenarios rather freely when analyzing a system modeled as a CTMC.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 145
State-Transition-Rate Matrix
A CTMC can be completely described by an initial distribution p(0) and a statetransition-rate matrix. A state-transition-rate matrix Q = [qij] is defined as follows:
qij =
rate of going from
state i to state j
-  qik
k i
i  j,
i = j.
Example: A computer is idle, working, or failed. When the computer is idle, jobs
arrive with rate a, and they are completed with rate b. When the computer is
working, it fails with rate lw, and with rate li when it is idle.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 146
“Simple Computer” CTMC
a
1
li
b
2
lw
3
Let X = 1 represent “the system is idle,” X = 2 “the system is working,” and X = 3 a
failure.
a
li 
- a  l i 
Q
b
- b  l w  l w 


0
0
0 

If the computer is repaired with rate m, the new CTMC looks like
a
a
li 
- a  l i 
2
1
b
Q
b
- b  l w  l w 


li
l
m
w
m
0
- m 

3
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 147
Analysis of “Simple Computer” Model
Some questions that this model can be used to answer:
– What is the availability at time t?
– What is the steady-state availability?
– What is the expected time to failure?
– What is the expected number of jobs lost due to failure in [0,t]?
– What is the expected number of jobs served before failure?
– What is the throughput of the system (jobs per unit time), taking into
account failures and repairs?
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 148
State-Space Generation from SANs
If the activity delays are exponential, it is straightforward to convert a SAN to a
CTMC. We first look at the simple case, where there is no composed model.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 149
State Space (Generated by Möbius)
State No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
CPUboards1
3
2
0
0
3
3
3
1
2
2
2
0
0
0
3
1
1
1
2
0
0
1
CPUboards2
3
3
3
3
2
0
0
3
2
0
0
2
0
2
1
2
0
0
1
1
1
1
NumComp
2
2
1
0
2
1
0
2
2
1
0
1
0
0
2
2
1
0
2
1
0
2
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
(Next State, Rate)
(2,.p1l),(3,p2l),(4,P3l),(5,p1l),(6,p2l,),(7,p3l)
(8,p1l),(3,p2l),(4,p3l),(9,p1l),(10,p2l),(11,p3l)
(12,p1l),(13,(p2+p3) l)
(9,p1l),(12,p2l),(14,p3l),(15,p1l),(6,p2l),(7,p3l)
(10,p1l),(13,(p2+p3) l)
(3,(p1+p2) l),(4,p3l),(16,p1l),(17,p2l),(18,p3l)
(16,p1l),(12,p2l),(14,p3l),(19,p1l),(10,p2l),(11,p3l)
(17,p1l),(13,(p2+p3) l)
(20,p1l),(13,(p2+p3) l)
(19,p1l),(20,p2l),(21,p3l),(6,(p1+p2) l),(7,p3l)
(12,(p1+p2) l),(14,p3l),(22,p1l),(17,p2l),(18,p3l)
(13, l)
(22,p1l),(20,p2l),(21,p3l),(10,(p1+p2l),(11,p3l)
(13, l)
(20,(p1+p2) l),(21,p3l),(17,(p1+p2) l),(18,p3l)
Slide 150
Underlying Markov Model (State Transition Rates Not Shown)
3
2
4
10
1
5
6
12
7
14
15
20
19
9
8
11
13
16
17
18
22
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
21
Slide 151
Reduced Base Model Construction
• “Reduced Base Model” construction techniques make use of composed model
structure to reduce the number of states generated.
• A state in the reduced base model is composed of a state tree and an impulse
reward.
• During reduced base model construction, the use of state trees permits an
algorithm to automatically determine valid lumpings based on symmetries in the
composed model.
• The reduced base model is constructed by finding all possible (state tree,
impulse reward) combinations and computing the transition rates between states.
• Generation of the detailed base model is avoided.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 152
Example Reduced Base Model State Generation
Composed Model
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
computer
Slide 153
Example Reduced Base Model States and Transitions
R (NumComp = 2)
(state 1)
2
computer (CPUboards = 3)
covered
catastrophic
uncovered
R (NumComp = 2)
1
1
computer
(CPUboards = 3)
computer
(CPUboards = 2)
(state 2)
R (NumComp = 1)
1
1
computer
(CPUboards = 3)
computer
(CPUboards = 0)
(state 3)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
R (NumComp = 0)
1
1
computer
(CPUboards = 3)
computer
(CPUboards = 0)
(state 4)
Slide 154
Markov Chain of Reduced Base Model
(State Transition Rates not Shown)
1
2
3
4
5
6
8
7
9
10
11
12
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
13
Slide 155
State-Space Generation in Möbius
(For generating random process representations of models with all
exponential or exponential/deterministic timed activities)
Print out states
and reward
variables
Print out absorbing
states. Useful to
detect problems
when attempting a
steady-state
solution.
Place comments, as
specified by edit
comments, in file.
State-space generation must be done before all analytic/numerical solutions are done.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 156
Numerical/Analytical Solution Techniques
1) Transient Solution
– Standard Uniformization (instant-of-time variables)
– Adaptive Uniformization (instant-of-time variables)
– Interval-of-time Uniformization (expected value, interval-of-time variables)
2) Steady-state Solution
– Direct Solution (instant-of-time steady-state variables)
– Iterative Solution (instant-of-time steady-state variables)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 157
CTMC Transient Solution
We have seen that it is easy to specify a CTMC in terms of the initial probability
distribution p(0) and the state-transition-rate matrix.
Earlier, we saw that the transient solution of a CTMC is given by p(t) = p(0)P(t),
and we noted that P(t) was difficult to define.
Due to the complexity of the math, we omit the derivation and show the
relationship
d
P(t )  QP(t )  P(t )Q, where Q is the state transition rate matrix of
dt
the Markov chain.
Solving this differential equation in some form is difficult but necessary to compute
a transient solution.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 158
Transient Solution Techniques
Solutions to dtd P(t )  QP(t ) can be done in many (dubious) ways*:
– Direct: If the CTMC has N states, one can write N2 PDEs with N2 initial
conditions and solve N2 linear equations.
– Laplace transforms: Unstable with multiple “poles”
– Nth order differential equations: Uses determinants and hence is numerically
unstable
n

(
Qt
)
Qt
.
– Matrix exponentiation: P(t) = eQt, where e  I  
n!
n 1
Qt by performing
Matrix
exponentiation
has
some
potential.
Directly
computing
e

(Qt ) n
I 
can be expensive and prone to instability.
n!
n 1
If the CTMC is irreducible, it is possible to take advantage of the fact that
Q = ADA-1, where D is a diagonal matrix. Computing eQt becomes AeDtA-1, where
e Dt  diag e d t , e d t ,..., e d t .
11
22
nn
* See C. Moler and C. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” SIAM
Review, vol. 20, no. 4, pp. 801-836, October 1978.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 159
Standard Uniformization
Starting with CTMC state transition rate matrix (Q) construct
1. Poisson process : rate l, l  q i ,i 
2. DTMC : P  I 
Q
l
Probability of k transitions
in time t
Then :
k

l t  - lt k
e P .
pt   p0 

k 0
k-step state transition probability
k!
In actual computatio n :
Ns
pt   
with pk  1  pk P .
k 0
lt k e -lt pk ,
k!
Choose truncation point to
obtain desired accuracy
Compute p(k) iteratively,
to avoid fill-in
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 160
Error Bound in Uniformization
• Answer computed is a lower bound, since each term in summation is positive,
and summation is truncated.
• Number of iterations to achieve a desired accuracy bound can be computed
easily.
Ns
lt k
k 0
k!
Error for each state  1 - 
e - lt
 Choose error bound, then compute Ns on-the-fly, as uniformization is done.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 161
Transient Uniformization Solver
(for transient solution of instant-of-time variables)
Instant-of-time variable time
points of interest. Multiple time
points may be specified,
separated by spaces.
Number of digits of accuracy in
the solution. Solution reported is
a lower bound.
Volume of intermediate results
reported. “1” gives the greatest
volume, greater numbers less.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 163
Accumulated Reward Solver (ars)
(solves for expected values of interval-of-time and time-averaged intervalof-time variables on intervals [t0, t1] when both t0 and t1 are finite)
Number of digits of
accuracy in the
solution. Solution
reported is a lower
bound.
Volume of
intermediate results
reported. “1” gives the
greatest volume, greater
numbers less.
Series of time
intervals for which
solution is desired.
Intervals are
separated by spaces.
Each interval can be
specified as t1:t2.
The accumulated reward solver is based on uniformization,
so the hints given for the transient solver apply here as well.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 167
Steady-State Behavior of CTMCs, cont.
This yields the elegant equation p*Q = 0, where p*  lim pt , the steady-state
t 
probability distribution. If the CTMC is irreducible, then p* can be computed with
n
the constraint that
p
i 1
*
i
 1.
Definition: A CTMC is irreducible if every state in the CTMC is reachable from
every other state.
If the CTMC is not irreducible, then more complex solution methods are required.
Notice that for irreducible CTMCs, the steady-state distribution is independent of
the initial-state distribution.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 170
Direct Steady-State Solution
One steady-state solver in Möbius is the direct steady-state solver. This solver
~
solves the augmented matrix p* Q  eiT using a form of Gaussian elimination.
Pros:
Can get a very accurate solution in a fixed amount of time;
“stiffness” (described later) does not affect solution time.
Cons: Solution complexity is O(n3), so does not scale well to large models;
memory requirements are high due to fill-in and are not known a
priori.
Recommendation: Use for small CTMCs (tens of states) or medium-sized and stiff
CTMCs (hundreds to a few thousands), or when high accuracy is required.
Reminder: High accuracy in solution does not mean high accuracy in prediction.
Use accuracy to do relative comparisons.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 171
Direct Steady-State Solver (dss)
(for steady-state solution of instant-of-time variables)
Stopping criterion used in
iterative refinement
phase, after direct
solution is done.
Volume of
intermediate
results reported.
“1” gives the
greatest volume,
greater numbers
less.
Number of rows to search
for the “best” pivot when
performing LU
decomposition
“Grace” factor by which
elements may become
pivots
Value that, when multiplied by smallest matrix element, is
threshold at which elements may be dropped in LU
decomposition.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 172
Iterative Solution Methods
The simplest iterative solution methods are called stationary iterative methods, and
they can be expressed as
p(k + 1) = p(k)M,
where M is a constant (stationary) matrix. Computing p(k + 1) from p(k) requires one
vector-matrix multiplication, or one iteration, which on modern workstations is
extremely fast.
The simplest stationary iterative method for CTMCs is called the power method.
Recall p*Q = 0. Let M = Q + I.
p(M - I) = 0
pM - p = 0
pM = p
p(k + 1) = p(k)(Q + I)
The power method typically converges (gets close to the answer) slowly.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 174
Iterative Solution Characteristics
Stationary iterative solution methods have the following characteristics:
– Low memory usage (no fill-in); predictable memory usage
– Low time per iteration, proportional to the number of non-zero entries
– Fast solution time for non-stiff matrices (tens or hundreds of iterations)
– Stop when sufficiently accurate
– Slow solution time for stiff matrices
– Difficult to quantify accuracy, especially for stiff matrices
– Easy to implement
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 175
Gauss-Seidel
One of the most widely used stationary iterative methods is called Gauss-Seidel.
The algorithm appears as follows:
for k  1 to convergence
for i  1 to n
 k 1
pi
1
qii
 i -1  k 1
  p j q ji 

 j 1

p j q ji 

j  i 1

n
k 
end for
end for
An intuitive explanation for this algorithm:
 k 1
- qii pi
i -1
 p j
flow out of node i
j 1
 k 1
q ji 
n
k 
p
 j q ji
j  i 1
flow into node i
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 178
SOR
There is an extension to Gauss-Seidel called successive over-relaxation, or SOR,
that sometimes gives better performance.
Let Dxi  xi k 1 - xi k  , where x  k  and x  k 1 are the kth and k  1th Gauss - Seidel
iterate. The k  1th SOR iterate, ~
x  k 1 , is computed as
i
~
xi k 1  xi k   wDx ,
where 0  w  2.
Choosing w is a hard problem in general. Automatic techniques for choosing w
exist but are not implemented in Möbius.
Note: w = 1 is the same as Gauss-Seidel.
Recommendation: Leave w = 1 unless you are solving a similar system many times
and the matrix is stiff.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 179
Iterative Steady-State Solver (iss)
(for steady-state solution of instant-of-time variables)
Stopping criterion,
expressed as 10-x, where x is
given. The criterion used is
the infinity difference norm.
SOR weight factor.
Values < 1 guarantee
convergence, but slow it.
Values >= 1 speed
convergence, but may not
converge.
Maximum number of
iterations allowed.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 180
Möbius Analytical Solvers
Model Class
All activities
exponential
Exponential and
Deterministic
activities
Analytic Solvers (for reward variables only)
SteadyInstant-of-time
Mean,
state or
or
Variance, or
Transient
Interval-of-time
Distribution
a
SteadyInstant-of-time
Mean,
state
Variance, and
Distribution
Transient
Instant-of-time
Mean,
Variance, and
Distribution
Interval-of-time
Mean
Steadystate
Instant-of-timeb
Mean,
Variance, and
Distribution
Applicable
Analytic
Solver
dss and iss
trs and atrs
ars
diss and
adiss
a
if only rate rewards are used, the time-averaged interval-of-time steady-state measure is
identical to the instant-of-time steady-state measure (if both exist).
b provided the instant-of-time steady-state distribution is well-defined. Otherwise, the timeaveraged interval-of-time steady-state variable is computed and only results for rate
rewards should be derived.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 183
Case Study: Fault-Tolerant Embedded
Multiprocessor System
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 184
Problem Origin
• This problem was originally posed in 1992 as a reliability model of a large,
embedded fault-tolerant computer, presumably for space-borne applications. It
was posed as a hierarchical model with non-perfect coverage at each level, with
the purpose of showing the inadequacy of existing techniques.
– Combinatorial methods were incapable of including coverage at all levels of
the hierarchy, thus grossly overstating the reliability.
– Markov- or SPN-based methods create far too many states to solve.
– Monte-Carlo simulation works, but provides only an estimate (which is
often not good enough).
– A specialized tool was developed to do numerical integration of a semiMarkov process to solve this and similar problems.
• In Möbius, we solve a smaller version of the same architecture “exactly” using
Markov models generated by SANs. This is made possible by automatic state
lumping using composed models.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 186
Problem Description
• System consists of 2 computers
• Each computer consists of
– 3 memory modules (2 must be operational)
– 3 CPU units (2 must be operational)
– 2 I/O ports (1 must be operational)
– 2 error-handling chips (non-redundant)
• Each memory module consists of
– 41 RAM chips (39 must be operational)
– 2 interface chips (non-redundant)
• A CPU consists of 6 non-redundant chips
• An I/O port consists of 6 non-redundant chips
• 10 to 20 year operational life
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 187
Diagram of Fault-Tolerant Multiprocessor System
..
..
..
41 RAMs
41 RAMs
41 RAMs
2 int. ch.
2 int. ch.
2 int. ch.
memory module
memory module
2 ch.
memory module
errorhandlers
interface bus
..
..
..
..
..
6 CPU
chips
6 CPU
chips
6 CPU
chips
6 I/O
chips
6 I/O
chips
CPU module
CPU module
CPU module
I/O port
I/O port
...
computer
computer
computer
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 188
Definition of “Operational”
• The system is operational if at least one computer is operational
• A computer is operational if all the modules are operational
– A memory module is operational if at least 39 RAM chips and both interface
chips are operational.
– A CPU unit is operational if all 6 CPU chips are operational
– An I/O port is operational if all 6 I/O chips are operational
– The error-handling unit is operational if both error-handling chips are
operational
• Failure rate per chip is 100 failures per 1 billion hours
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 189
Coverage
• This system could be modeled using combinatorial methods if we did not take
coverage into account. Coverage is the chance that the failure of a chip will not
cause the larger system to fail even if sufficient redundancy exists. I.e.,
coverage is the probability that the fault is contained.
The coverage probabilities are given in the following table:
Redundant Component
RAM Chip
Memory Module
CPU Unit
I/O Port
Computer
Fault Coverage Probability
0.998
0.95
0.995
0.99
0.95
• For example, if a RAM chip fails, there is a 0.2% chance the memory module
will fail even if sufficient redundancy exists. If the memory module fails, there
is a 5% chance the computer will fail. If a computer fails, there is a 5% chance
the system will fail.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 190
Outline of Solution: List of SANs
• The model is composed of four SANs:
1. memory_module
2. cpu_module
3. errorhandlers
4. io_port_module
• Each SAN models the behavior of the module in the event of a module
component failure.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 191
List of Places
• Seven places represent the state of the system:
1. cpus – the number of operational CPU modules
2. ioports – the number of operational I/O modules
3. errorhandlers – whether the two error-handler chips are operational
4. computer_failed – the number of failed computers
5. memory_failed – the number of failed memory modules
6. memory_chips – number of operational RAM chips
7. interface_chips – number of operational interface chips
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 192
List of Activities
• Five activities represent failures in the system
1. cpu_failure – the failure of any CPU chip
2. ioport_failure – the failure of any I/O chip
3. errorhandling_chip_failure – the failure of either error-handler chip
4. memory_chip_failure – the failure of a RAM chip
5. interface_chip_failure – the failure of a memory interface chip
Cases on these activities represent behavior based on coverage or non-coverage.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 193
Tricks of the Trade
Since we intend to solve this model analytically, we want the fewest number of
states possible.
• We don’t care which component failed or what particular failed state the
model is in. Therefore, we lump all failure states into the same state.
• We don’t care which computer or which module is in what state. Therefore,
we make use of replication to further reduce the number of states.
• We use marking-dependent rates to model RAM chip failure, making use of
the fact that the minimum of independent exponentials is an exponential.
• We use cases to denote coverage probabilities, and adjust the probabilities
depending on the state of the system.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 194
Composed Model
Node
Join1
Node
Rep1
Reps
3
Rep2
2
Common Places
computer_failed
memory_failed
computer_failed
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Common Places
Subtree 1

computer_failed

memory_failed
cpus
errorhandlers
ioports
2





3





4





Slide 195
cpu_modules SAN
Place
cpus
ioports
errorhandlers
memory_failed
computer_failed
Marking
3
2
2
0
0
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 196
cpu_modules SAN, cont.
cpu_modules input gate predicates and functions:
Gate
IG1
Enabling Predicate
(MARK(cpus) > 1) &&
(MARK(memory_failed) < 2) &&
(MARK(computer_failed) < 2)
Function
identity
cpu_modules activity time distributions:
Activity
cpu_failure
Distribution
expon(0.0052596 * MARK(cpus))
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 197
cpu_modules SAN, cont.
cpu_modules case probabilities for activities:
Case
1
2
3
Probability
module_cpu_failure
if (MARK(cpus) == 3)
return(0.995);
else
return(0.0);
if (MARK(cpus) == 3)
return(0.00475);
else
return(0.95);
if (MARK(cpus) == 3)
return (0.00025);
else
return(0.05);
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
• case 1: chip failure
covered
• case 2: chip failure
causes computer failure
• case 3: chip failure
causes system
(catastrophic) failure
Slide 198
cpu_modules SAN, cont.
cpu_modules output gate functions:
Gate
OG1
OG2
OG3
Function
if (MARK(cpus) == 3)
MARK(cpus) - -;
MARK(cpus) = 0;
MARK(ioports) = 0;
MARK(errorhandlers) = 0;
MARK(memory_failed) = 2;
MARK(computer_failed) ++;
MARK(cpus) = 0;
MARK(ioports) = 0;
MARK(errorhandlers) = 0;
MARK(memory_failed) = 2;
MARK(computer_failed) = 2;
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 199
Model Solution
The modeled two-computer system with non-perfect coverage at all levels (i.e., the
model as described), the state space contains 10,114 states. The 10 year mission
reliability was computed to be .995579.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 213
Impact of Coverage
• Coverage can have a large impact on reliability and state-space size. Various
coverage schemes were evaluated with the following results.
Design description
100% coverage at all levels
Nonperfect coverage considered at all levels
Nonperfect coverage considered at all levels,
no spare memory module
Nonperfect coverage considered at all levels,
no spare CPU module
Nonperfect coverage considered at all levels,
no spare IO port
Nonperfect coverage considered at all levels,
no spare memory module, CPU module, or
IO port
100% coverage at all levels, no spare
memory module, CPU module, IO port, or
RAM chips
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
4278
10114
1335
Reliability
(10-year
mission time)
0.999539
0.995579
0.987646
3299
0.973325
3299
0.985419
511
0.935152
6
0.702240
State-space size
Slide 214
Solution by Simulation
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 216
Motivation
• High-level formalisms (like SANs) make it easy to specify realistic systems, but
they also make it easy to specify systems that have unreasonably large state
spaces.
• State-of-the-art tools (like Mobius) can handle state-level models with a few
ten’s of million states, but not more.
• When state spaces become too large, discrete event simulation is often a viable
alternative.
• Discrete-event simulation can be used to solve models with arbitrarily large state
spaces, as long as the desired measure is not based on a “rare event.”
• When “rare events” are present, variance reduction techniques can sometimes be
used.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 218
Advantages of Simulation
• Simulation can be applied to any SAN model. The most
prominent difference, compared with analytic solvers, is that
generally distributed activities can be used.
• Simulation does not require the generation of a state space and
therefore does not require a finite state space. Therefore, much
more detailed models can be solved.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 219
Disadvantages of Simulation
• Simulation only provides an estimate of the desired measure. An
approximate confidence interval is constructed that contains the actual
result with some user-specified probability.
• Higher desired accuracy dramatically increases the necessary simulation
time. As a rule, to make the confidence interval n times narrower, the
simulation has to be run n2 times as long.
• The “rare event problem” may arise. If simulation is used to estimate a
small probability, such as the reliability of a highly-reliable system,
extremely long simulations may have to be performed to encounter the
particular event often enough.
• Complicated models can require long simulation times, even if the rare
event problem is not an issue. The simulators in Möbius perform the
necessary event scheduling very efficiently, but it should be realized that
simulation is not a panacea.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 220
Simulation as Model Experimentation
• State-based methods (such as Markov chains) work by enumerating all possible
states a system can be in, and then invoking a numerical solution method on the
generated state space.
• Simulation, on the other hand, generates one or more trajectories (possible
behaviors from the high-level model), and collects statistics from these
trajectories to estimate the desired performance/dependability measures.
• Just how this trajectory is generated depends on the:
– nature of the notion of state (continuous or discrete)
– type of stochastic process (e.g., ergodic, reducible)
– nature of the measure desired (transient or steady-state)
– types of delay distributions considered (exponential or general)
• We will consider each of these issues in this module, as well as the simulation of
systems with rare events.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 221
Types of Simulation
Continuous-state simulation is applicable to systems where the notion of state is
continuous and typically involves solving (numerically) systems of differential
equations. Circuit-level simulators are an example of continuous-state simulation.
Discrete-event simulation is applicable to systems in which the state of the system
changes at discrete instants of time, with a finite number of changes occurring in
any finite interval of time.
Since we will focus on validating end-to-end systems, rather than circuits, we will
focus on discrete-event simulation.
There are two types of discrete-event simulation execution algorithms:
– Fixed-time-stamp advance
– Variable-time-stamp advance
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 222
Fixed-Time-Stamp Advance Simulation
• Simulation clock is incremented a fixed time Dt at each step of the simulation.
• After each time increment, each event type (e.g., activity in a SAN) is checked
to see if it should have completed during the time of the last increment.
• All event types that should have completed are completed and a new state of the
model is generated.
• Rules must be given to determine the ordering of events that occur in each
interval of time.
• Example:
0
e1
e2
Dt
e3
2Dt
3Dt
e4
e5
4Dt
e6
5Dt
• Good for all models where most events happen at fixed increments of time (e.g.,
gate-level simulations).
• Has the advantage that no “future event list” needs to be maintained.
• Can be inefficient if events occur in a bursty manner, relative to time-step used.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 223
Variable-Time Step Advance Simulation
• Simulation clock advanced a variable amount of time each step of the
simulation, to time of next event.
• If all event times are exponentially distributed, the next event to complete and
time of next event can be determined using the equation for the minimum of n
exponentials (since memoryless), and no “future event list” is needed.
• If event times are general (have memory) then “future event list” is needed.
• Has the advantage (over fixed-time-stamp increment) that periods of inactivity
are skipped over, and models with a bursty occurrence of events are not
inefficient.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 224
Basic Variable-Time-Step Advance Simulation
Loop for SANs
A) Set list_of_active_activities to null.
B) Set current_marking to initial_marking.
C) Generate potential_completion_time for each activity that may complete in the
current_marking and add to list_of_active_activities.
D) While list_of_active_activities  null:
1) Set current_activity to activity with earliest potential_completion_time.
2) Remove current_activity from list_of_active_activities.
3) Compute new_marking by selecting a case of current_activity, and executing
appropriate input and output gates.
4) Remove all activities from list_of_active_activities that are not enabled in
new_marking.
5) Remove all activities from list_of_active_activities for which new_marking is a
reactivation marking.
6) Select a potential_completion_time for all activities that are enabled in
new_marking but not on list_of_active_activities and add them to
list_of_active_activities.
E) End While.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 225
Types of Discrete-Event Simulation
• Basic simulation loop specifies how the trajectory is generated, but does not
specify how measures are collected, or how long the loop is executed.
• How measures are collected, and how long (and how many times) the loop is
executed depends on type of measures to be estimated.
• Two types of discrete-event simulation exist, depending on what type of measures
are to be estimated.
– Terminating - Measures to be estimated are measured at fixed instants of
time or intervals of time with fixed finite point and length. This may also
include random but finite (in some sense) times, such as a time to failure.
– Steady-state - Measures to be estimated depend on instants of time or
intervals whose starting points are taken to be t  .
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 226
Issues in Discrete-Event Simulation
1) How to generate potential completion times for events
2) How to estimate dependability measures from generated trajectories
– Transient measures
– Steady-state measures
3) How to implement the basic simulation loop
– Sequential or parallel
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 227
Generation of Potential Completion Times
1) Generation of uniform [0,1] random variates
– Used as a basis for all random variate samples
– Types
• Linear congruential generators
• Tausworthe generators
• Other types of generators
– Tests of uniform [0,1] generators
2) Generation of non-uniform random variates
– Inverse transform technique
– Convolution technique
– Composition technique
– Acceptance-rejection technique
– Technique for discrete random variates
3) Recommendations/Issues
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 228
Generation of Uniform [0,1] Random Number Samples
Goal: Generate sequence of numbers that appears to have come from uniform [0,1]
random variable.
Importance: Can be used as a basis for all random variates.
Issues:
1) Goal isn’t to be random (non-reproducible), but to appear to be random.
2) Many methods to do this (historically), many of them bad (picking
numbers out of phone books, computing p to a million digits, counting
gamma rays, etc.).
3) Generator should be fast, and not need much storage.
4) Should be reproducible (hence the appearance of randomness, not the
reality).
5) Should be able to generate multiple sequences or streams of random
numbers.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 229
Linear Congruential Generators (LCGs)
• Introduced by D. H. Lehmer (1951). He obtained
xn = an mod m
xn = (axn - 1) mod m
• Today, LCGs take the following form:
xn = (axn - 1 + b) mod m, where
xn are integers between 0 and m - 1
a, b, m non-negative integers
• If a, b, m chosen correctly, sequence of numbers can appear to be uniform and
have large period (up to m).
• LCGs can be implemented efficiently, using only integer arithmetic.
• LCGs have been studied extensively; good choices of a, b, and m are known.
See, e.g., Law and Kelton (1991), Jain (1991).
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 230
Tausworthe Generators
• Proposed by Tausworthe (1965), and are related to cryptographic methods.
• Operate on a sequence of binary digits (0,1). Numbers are formed by selecting
bits from the generated sequence to form an integer or fraction.
• A Tausworthe generator has the following form:
bn = cq - 1bn - 1  cq - 2bn - 2  . . .  c0bn - q
where bn is the nth bit, and ci (i = 0 to q - 1) are binary coefficients.
• As with LCGs, analysis has been done to determine good choices of the ci.
• Less popular than LCGs, but fairly well accepted.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 231
Generation of Non-Uniform Random Variates
• Suppose you have a uniform [0,1] random variable, and you wish to have a
random variable X with CDF FX. How do we do this?
• All other random variates can be generated from uniform [0,1] random variates.
• Methods to generate non-uniform random variates include:
– Inverse Transform - Direct computation from single uniform [0,1] variable
based on observation about distribution.
– Convolution - Used for random variables that can be expressed as sum of
other random variables.
– Composition - Used when the distribution of the desired random variable
can be expressed as a weighted sum of the distributions of other random
variables.
– Acceptance-Rejection - Uses multiple uniform [0,1] variables and a function
that “majorizes” the density of the random variate to be generated.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 232
Inverse Transform Technique
Suppose we have a uniform [0,1] random variable U.
If we define X = F-1(U), then X is a random variable with CDF FX = F.
To see this,
FX(a) = P[X  a]
= P[F-1(U)  a]
= P[U  F(a)]
= F(a)
Thus, by starting with a uniform random variable, we can generate virtually any
type of random variable.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 233
Example of Inverse Transform
Let X be an exponentially distributed random variable with parameter l. Let U be a
uniform [0,1] random variable generated by a pseudo-random number generator.
FX a   1 - e -la
1
X  FX-1 U   - ln1 - U 
l
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 234
Convolution Technique
• Technique can be used for all random variables X that can be expressed as the
sum of n random variables
X = Y1 + Y2 + Y3 + . . . + Yn
• In this case, one can generate a random variate X by generating n random
variates, one from each of the Yi, and summing them.
• Examples of random variables:
– Sum of n Bernoulli random variables is a binomial random variable.
– Sum of n exponential random variables is an n-Erlang random variable.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 235
Composition Technique
• Technique can be used when the distribution of a desired random variable can be
expressed as a weighted sum of other distributions.
• In this case F(x) can be expressed as

F  x    pi Fi  x 
i 0
where pi  0,

pi  1.

i 0
• The composition technique is as follows:
1) Generate random variate i such that P[I = i] = pi for i = 0, 1, . . .
(This can be done as discussed for discrete random variables.)
2) Return x as random variate from distribution Fi(x), where i is as chosen
above.
• A variant of composition can also be used if the density function of the desired
random variable can be expressed as weighted sum of other density functions.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 236
Acceptance-Rejection Technique
• Indirect method for generating random variates that should be used when other
methods fail or are inefficient.
• Must find a function m(x) that “majorizes” the density function f(x) of the
desired distribution. m(x) majorizes f(x) if m(x)  f(x) for all x.
• Note:


c   m x dx   f  x dx  1, so m x  is not necessarily a density function,
-
-
m( x )
is a density function.
c
If random variates for m(x) can be easily computed, then random variates for f(x)
can be found as follows:
1) Generate y with density m(x)
2) Generate u with uniform [0,1] distribution
but m x  
•
3) If u 
f ( y)
, return y, else goto 1.
m( y )
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 237
Generating Discrete Random Variates
• Useful for generating any discrete distribution, e.g., case probabilities in a SAN.
• More efficient algorithms exist for special cases; we will review most general
case.
• Suppose random variable has probability distribution p(0), p(1), p(2), . . . on
non-negative integers. Then a random variate for this random variable can be
generated using the inverse transform method:
1) Generate u with distribution uniform [0,1]
2) Return j satisfying
j -1
j
pi   u   pi 

i 0
i 0
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 238
Recommendations/Issues in Random Variate Generation
• Use standard/well-tested uniform [0,1] generators. Don’t assume that because a
method is complicated, it produces good random variates.
• Make sure the uniform [0,1] generator that is used has a long enough period.
Modern simulators can consume random variates very quickly (multiple per
state change!).
• Use separate random number streams for different activities in a model system.
Regular division of a single stream can cause unwanted correlation.
• Consider multiple random variate generation techniques when generating nonuniform random variates. Different techniques have very different efficiencies.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 239
Estimating Dependability Measures: Estimators and
Confidence Intervals
• An execution of the basic simulation loop produces a single trajectory (one
possible behavior of the system).
• Common mistake is to run the basic simulation loop a single time, and presume
observations generated are “the answer.”
• Many trajectories and/or observations are needed to understand a system’s
behavior.
• Need concept of estimators and confidence intervals from statistics:
– Estimators provide an estimate of some characteristic (e.g., mean or
variance) of the measure.
– Confidence intervals provide an estimate of how “accurate” an estimator is.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 240
Typical Estimators of a Simulation Measure
• Can be:
– Instant-of-time, at a fixed t, or in steady-state
– Interval-of-time, for fixed interval, or in steady-state
– Time-averaged interval-of-time, for fixed interval, or in steady-state
• Estimators on these measures include:
– Mean
– Variance
– Interval - Probability that the measure lies in some interval [x,y]
• Don’t confuse with an interval-of-time measure.
• Can be used to estimate density and distribution function.
– Percentile - 100bth percentile is the smallest value of estimator x such that
F(x)  b.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 241
Different Types of Processes and Measures Require
Different Statistical Techniques
• Transient measures (terminating simulation):
– Multiple trajectories are generated by running basic simulation loop multiple
times using different random number streams. Called Independent
Replications.
– Each trajectory used to generate one observation of each measure.
• Steady-State measures (steady-state simulation):
– Initial transient must be discarded before observations are collected.
– If the system is ergodic (irreducible, recurrent non-null, aperiodic), a single
long trajectory can be used to generate multiple observations of each
measure.
– For all other systems, multiple trajectories are needed.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 242
Confidence Interval Generation: Terminating Simulation
Approach:
– Generate multiple independent observations of each measure, one
observation of each measure per trajectory of the simulation.
– Observations of each measure will be independent of one another if different
random number streams are used for each trajectory.
– From a practical point of view, new stream is obtained by continuing to
draw numbers from old stream (without resetting stream seed).
Notation (for subsequent slides):
– Let F(x) = P[X  x] be measure to be estimated.
– Define m = E[X], s2 = E[(X - m)2].
– Define xi as the ith observation value of X (ith replication, for terminating
simulation).
Issue: How many trajectories are necessary to obtain a good estimate?
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 243
Terminating Simulation: Estimating the Mean of a
Measure I
• Wish to estimate m = E[X].
• Standard point estimator of m is the sample mean
1 N
mˆ   xn
N n 1
( mˆ is unbiased, i.e., Emˆ   m , and Varmˆ  
s2
N
, where s 2  Var X )
• To compute confidence interval, we need to compute sample variance:
1 N
1 N 2
N
2
2




s 
x
m

x
m
ˆ
ˆ
 n
 n N -1
N - 1 n 1
N - 1 n 1
2
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 244
Terminating Simulation: Estimating the Mean of a
Measure II
• Then, the (1 - a) confidence interval about x can be expressed as:
t N -1 1 - a2 s
t N -1 1 - a2 s
mˆ  m  mˆ 
N
N
Where
– t N -1 1 - a2  is the1001 - a2 th percentile of the student's t distribution with
N - 1 degrees of freedom (values of this distribution can be found in tables).
– s  s 2 is the sample standard deviation.
– N is the number of observations.
• Equation assumes xn are distributed normally (good assumption for large
number of xi).
• The interpretation of the equation is that with (1 - a) probability the real value
(m) lies within the given interval.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 245
Terminating Simulation: Estimating the Variance of a
Measure I
• Computation of estimator and confidence interval for variance could be done
like that done for mean, but result is sensitive to deviations from the normal
assumption.
• So, use a technique called jackknifing developed by Miller (1974).
• Define
sˆ i 
1
N -1
2
2


x
m
ˆ
 n N -2 i
N - 2 ni
Where
mˆ i 
1
xn

N - 1 n i
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 246
Terminating Simulation: Estimating the Variance of a
Measure II
• Now define
1 N
Z i  Ns -  N - 1 sˆ and Z   Z i , for i  1,2,..., N
N i 1
(where s2 is the sample variance as defined for the mean)
2
2
i
• And
1 N
2


s 
Z
Z
 i
N - 1 i 1
2
Z
• Then
t N -1 1 - a2 s Z
t N -1 1 - a2 s Z
2
Zs Z 
N
N
is a (1 - a) confidence interval about s2.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 247
Terminating Simulation: Estimating the Percentile of an
Interval About an Estimator
• Computed in a manner similar to that for mean and variance.
• Formulation can be found in Lavenberg, ed., Computer Performance Modeling
Handbook, Academic Press, 1983.
• Such estimators are very important, since mean and variance are not enough to
plan from when simulating a single system.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 248
Confidence Interval Generation: Steady-State
Simulation
• Informally speaking, steady-state simulation is used to estimate measures that
depend on the “long run” behavior of a system.
• Note that the notion of “steady-state” is with respect to a measure (which has
some initial transient behavior), not a model.
• Different measures in a model will converge to steady state at different rates.
• Simulation trajectory can be thought of as having two phases: the transient phase
and the steady-state phase (with respect to a measure).
• Multiple approaches to collect observations and generate confidence intervals:
– Replication/Deletion
– Batch Means
– Regenerative Method
– Spectral Method
• Which method to use depends on characteristics of the system being simulated.
• Before discussing these methods, we need to discuss how the initial transient is
estimated.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 249
Estimating the Length of the Transient Phase
Problem: Observations of measures are different during so-called “transient phase,”
and should be discarded when computing an estimator for steady-state behavior.
Need: A method to estimate transient phase, to determine when we should begin to
collect observations.
Approaches:
– Let the user decide: not sophisticated, but a practical solution.
– Look at long-term trends: take a moving average and measure differences.
– Use more sophisticated statistical measures, e.g., standardized time series
(Schruben 1982).
Recommendation:
– Let the user decide, since automated methods can fail.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 250
Methods of Steady-State Measure Estimation:
Replication/Deletion
• Statistics similar to those for terminating simulation, but observations collected
only on steady-state portion of trajectory.
• One or more observations collected per trajectory:
O11
O12
O21
O31
O32
O13
O14
O22
O23 O24
O33
O34
trajectory 1
trajectory 2
...
transient
phase
trajectory n
• Compute
Mi
xi 
O
j 1
ij
th
M i as i observation, where Mi is the number of observations in
trajectory i.
• xi are considered to be independent, and confidence intervals are generated.
• Useful for a wide range of models/measures (the system need not be ergodic),
but slower than other methods, since transient phase must be repeated multiple
times.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 251
Methods of Steady-State Measure Estimation: Batch
Means
• Similar to Replication/Deletion, but constructs observations from a single
trajectory by breaking it into multiple batches.
• Example
O11 O12
O13 ... O1n
O21
O22
initial transient
1
O23 ... O2n
2
O31
O32 ... O3n3
...
• Observations from each batch are combined to construct a single observation;
these observations are assumed to be independent and are used to construct the
point estimator and confidence interval.
• Issues:
– How to choose batch size?
– Only applicable to ergodic systems (i.e., those for which a single trajectory
has the same statistics as multiple trajectories).
– Initial transient only computed once.
• In summary, a good method, often used in practice.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 252
Other Steady-State Measure Estimation Methods I
• Regenerative Method (Crane and Iglehart 1974, Fishman 1974)
– Uses “renewal points” in processes to divide “batches.”
– Results in batches that are independent, so approach used earlier to generate
confidence intervals applies.
– However, usually no guarantee that renewal points will occur at all, or that
they will occur often enough to efficiently obtain an estimator of the
measure.
• Autoregressive Method (Fishman 1971, 1978)
– Uses (as do the two following methods) the autocorrelation structure of
process to estimate variance of measure.
– Assumes process is covariance stationary and can be represented by an
autoregressive model.
– Above assumption often questionable.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 253
Other Steady-State Measure Estimation Methods II
• Spectral Method (Heidelberger and Welch 1981)
– Assumes process is covariance stationary, but does not make further
assumptions (as previous method does).
– Efficient method, if certain parameters chosen correctly, but choice requires
sophistication on part of user.
• Standardized Time Series (Schruben 1983)
– Assumes process is strictly stationary and “phi-mixing.”
– Phi-mixing means that Oi and Oi + j become uncorrelated if j is large.
– As with spectral method, has parameters whose values must be chosen
carefully.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 254
Summary: Measure Estimation and Confidence Interval
Generation
1) Only use the mean as an estimator if it has meaning for the situation being
studied. Often a percentile gives more information. This is a common mistake!
2) Use some confidence interval generation method! Even if the results rely on
assumptions that may not always be completely valid, the methods give an
indication of how long a simulation should be run.
3) Pick a confidence interval generation method that is suited to the system that
you are studying. In particular, be aware of whether the system being studied is
ergodic.
4) If batch means is used, be sure that batch size is large enough that batches are
practically uncorrelated. Otherwise the simulation can terminate prematurely
with an incorrect result.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 255
Summary/Conclusions: Simulation-Based Validation
Techniques
1) Know how random variates are generated in the simulator you use. Make sure:
– A good uniform [0,1] generator is used
– Independent streams are used when appropriate
– Non-uniform random variates are generated in a proper way.
2) Compute and use confidence intervals to estimate the accuracy of your
measures.
– Choose correct confidence interval computation method based on the nature
of your measures and process
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 256
Simulator Editor
Maximum and Minimum
Number of Replications to Run
Number of Batches between
each calculation of the variance
Trace-Level for Debugging
File Name of Output File
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 261
Batch and Replication Outputs (Variable Output Option)
Typical batch output:
Variable Name
Batch Number
Simulation Time
Time (CPU seconds)
Batch Mean
Mean
Variance
:
:
:
:
:
:
:
utilization
10
1.100000e + 04
41
8.467695e - 01
8.447065e - 01 + / - 1.516121e - 03
4.417886e - 02 + / - 5.035103e - 04
:
:
:
:
:
:
:
:
utilization
2400
1.000000e + 02
1498
1.000000e + 00
8.466667e - 01 + / - 8.196275e - 03
4.196934e - 02
4.196934e - 02 + / - 2.588252e - 03
Typical replication output:
Variable Name
Replication Number
Simulation Time
Time (CPU seconds)
Current Value
Mean
Sample Variance
Variance
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 262
Möbius Simulation Techniques
Simulation Characteristics
Steady-state Instant-of-time or
Mean,
Variable
Applicable
or Transient Interval-of-time Variance, or
Simulator
Distribution
Transient
Instant-of-time
Mean,
Reward Variable tsim and itsim
and
Variance,
Activity
tsim
Interval-of-time
and
Variable
Distribution
Steady-state Instant-of-time
Mean,
Reward Variable
ssim
Variance,
and Activity
and
Variable
Distribution
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 263
Symbolic State-space Exploration and Numerical
Analysis of State-sharing Composed Models
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 265
Motivation
• State-space (SS) explosion or largeness problem in
discrete-state systems
– Costly generation and representation of SS (space and
time)
– Costly representation of CTMC (space)
– Costly representation of solution vector (space) and
costly iteration/solution time (time)
• Typical solutions:
– Largeness avoidance, e.g., using lumping techniques
• CTMC level
• Model level
– Largeness tolerance using BDD, MDD, MTBDD,
Kronecker, or Matrix Diagrams (MD)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 266
What Is New?
• Our approach combines
– Model-level lumping induced by structural symmetries
• Number of states   solution vector size 
• Number of states   iteration time 
– MDD and Matrix Diagram (MD) data structures
• Enables us to represent lumped CTMCs not possible using
sparse matrix
• An order of magnitude faster than unlumped sparse
representation although it induces slowdown in solution time
compared to lumped sparse representation
• State-sharing composed models as opposed to action-synchronization
– Maintain almost the same generality
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 267
State-sharing Composed Models
• Join and Replicate operators
Join
SV1
M1
SV1
M2
Join
M1
M2
M1
M1
Rep (3)
M1

M1
• Any atomic model formalism that can share state variables
– E.g., SAN, PEPAk, and Buckets and Balls
• Replicate induces symmetry
• Global and local actions
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 268
Introduction to MDD
• Represents function
where
• Special case : n = 1, f represents
a set of vectors
0 1 2
0 1
0 1
0 1 2
0
0 1
0 1 2
1
{(0,0,1), (0,0,2), (0,1,1), (0,1,2),
(1,0,1), (1,0,2), (1,1,0), (1,1,1),
(2,0,0), (2,0,1), (2,1,1), (2,1,2)}
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 269
Introduction to MDD
• Represents function
where
• Special case : n = 1, f represents
a set of vectors
• Representation of a set of states
of a discrete-state model
– Partition set of SVs
– Assign index to unique value
assignment of variables of
each block
– Vector of indices represents
a state
0 1 2
0 1
0 1
0 1 2
0
0 1
0 1 2
1
{(0,0,1), (0,0,2), (0,1,1), (0,1,2),
(1,0,1), (1,0,2), (1,1,0), (1,1,1),
(2,0,0), (2,0,1), (2,1,1), (2,1,2)}
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 270
Introduction to MDD
• Represents function
where
0
• Special case : n = 1, f represents
a set of vectors
– Partition set of SVs
– Assign index to unique value
assignment of variables of
each block
– Vector of indices represents
a state
•
Augment by state offsets
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
8
12
0 1 2
0
• Representation of a set of states
of a discrete-state model
4
2
4
0
0 1
0
2
4
0 1
0
1
0 1 2
0
2
4
0 1
0
2
0
1
2
2
0 1 2
1
{(0,0,1), (0,0,2), (0,1,1), (0,1,2),
(1,0,1), (1,0,2), (1,1,0), (1,1,1),
(2,0,0), (2,0,1), (2,1,1), (2,1,2)}
Slide 271
MDD data structure by example
• Partitioning SVs based on
composition structure
– Maximizing efficiency of local
SS exploration
– Simplifying global
SS exploration
• Dependability model
for multicomputer system
Rep2 (N)
Join
Rep1 (M)
cpu
error handler
IO port
memory
MDD level
assignment
Rep2
0
Join
1
Rep1
2
mem
3 outer replicate
mem
2+M
inner replicate
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 272
Algorithm Overview
1.
2.
3.
4.
Generate MDD representation of unlumped SS
Build MD representation of CTMC
Convert unlumped SS to lumped SS
Solve CTMC by iterating through MD data structure
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 273
Symbolic Generation of Unlumped SS
•
•
•
•
•
•
•
set of visited states
set of unexplored states
expands using
sequences of firings of local
actions
expands
using single action firing of
global actions
Never generate potential or unreachable states
Creating necessary matrices and data structures to
construct MD of the CTMC at a later stage
No consideration of lumping properties
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 274
Symbolic SSG (Local Actions)
• Restriction: immediate actions
are local
i
j
• On-the-fly elimination of
A
B local
vanishing states
transition
 i to j
• Local SS expansion in levels
corresponding to atomic models.
i
j
No assumption of knowing the
A A B
local state space in advance 
– Online computation of transitive closure based on Ibaraki
and Katoh’s algoritm
• Avoids costly computation of tr. closure from scratch
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 275
Symbolic SSG (Global Actions)
• Global action a in component c affects more than one level
• No “product-form”-like restriction 
Effect of a on each level need not be determined locally
• More difficult to handle than synchronizing actions
• Expensive operation
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 276
Lumping
• Redundant states (paths)
Rep
x
AM
1
2
AM
1
2
AM
2
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
1
1
Slide 277
Lumping
• Redundant states (paths)
• Rep node c implies
equivalence relation Rc
Rep
x
x
1
AM
1
2
AM
1
2
AM
2
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
1
1
1
2
Slide 278
Lumping
• Redundant states (paths)
• Rep node c implies
equivalence relation Rc
Rep
x
x
1
AM
1
2
AM
1
2
AM
2
1
1
1
2
• Overall equivalence relation
• Canonical representative state in each class min(v)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 279
Lumping
• Redundant states (paths)
• Rep node c implies
equivalence relation Rc
Rep
x
x
1
AM
1
2
AM
1
2
AM
2
1
1
1
2
• Overall equivalence relation
• Canonical representative state in each class min(v)
•
may become exponentially large  break it up into
many extremely smaller MDDs  faster computation of
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 280
Lumping
•
where
is the set of all states v where
min(v) =v
• may become huge  break up
MDDs
into extremely smaller
–
•
is often less structured than
larger in number of nodes
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
and therefore
Slide 281
SSG and Lumping Performance
• Worst case example: No local behavior
• Drastic decrease in number of states in the lumped SS (up to 6 orders of
magnitude)
• Increase in number of nodes in the lumped state space but still small compared
to other entities
• Very small unlumped and lumped SS representation
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 282
CTMC Generation and Enumeration
• Use Matrix Diagrams (MD) (Ciardo/Miner)
– CTMC of largest example has <40000 nodes and takes <3MB of
memory
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 283
CTMC Generation and Enumeration
• Use Matrix Diagrams (MD) (Ciardo/Miner)
– CTMC of largest example has <30000 nodes and takes <5MB of
memory
• Projection of the MD on the lumped SS? Problem: some needed
transitions are deleted
wrong
correct
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 284
CTMC Generation and Enumeration
• Use Matrix Diagrams (MD) (Ciardo/Miner)
– CTMC of largest example has <40000 nodes and takes <3MB of
memory and at most a few seconds to build
• Projection of the MD on the lumped SS? Problem: some needed
transitions are deleted
• Project rows on lumped SS
wrong
and columns on
unlumped SS
correct
• Redirect transitions on-the-fly
• DFS-based enumeration of
MD using “sorting” MDD
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 285
CTMC Enumeration Performance
• Fairly fast iteration: less than 6 times slower than lumped sparse matrix
• Solving larger CTMCs
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 286
Integration into Möbius
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 287
Case Study: Survivability Evaluation
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 288
Defending Against a Wide Variety of Attacks
Nation-states,
Terrorists,
Multinationals
Economic intelligence
Information terrorism
Military spying
HIGH
Disciplined strategic
cyber attack
INNOVATION
Selling secrets
Civil disobedience
Serious hackers
Harassment
Embarrassing organizations
Stealing credit cards
Collecting trophies
Script kiddies
PLANNING
STEALTH
COORDINATION
Copy-cat attacks
Curiosity
Thrill-seeking
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
LOW
Slide 289
Intrusion Tolerance: A New Paradigm for Security
Prevent Intrusions
(Access Controls, Cryptography,
Trusted Computing Base)
But intrusions will occur
Trusted Computing
Base
Access Control &
Physical Security
Cryptography Multiple Security Levels
1st Generation: Protection
Detect Intrusions, Limit Damage
(Firewalls, Intrusion Detection Systems,
Virtual Private Networks, PKI)
Boundary
Controllers
Firewalls
Intrusion
Detection
Systems
VPNs
PKI
2nd Generation: Detection
But some attacks will succeed
Tolerate Attacks
(Redundancy, Diversity, Deception,
Wrappers, Proof-Carrying Code,
Proactive Secret Sharing)
Intrusion
Tolerance
Big Board View of
Attacks
Real-Time Situation
Awareness
& Response
Graceful
Degradation
Hardened
Operating
System
3rd Generation: Tolerance
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 290
Validation of Computer System/Network Survivability
• Security is no longer absolute
• Trustworthy computer systems/networks must operated through
attacks, providing proper service in spite of possible partially
successful attacks
• Intrusion tolerance claims to provide proper operation under such
conditions
• Validation of security/survivability must be done:
– During all phases of the design process, to make design
choices
– During testing, deployment, operation, and maintenance,
to gain confidence that the “amount” of intrusion
tolerance provided is as advertised.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 291
Validating Computer System Security: Research Goal
CONTEXT: Create robust software and hardware that are faulttolerant, attack resilient, and easily adaptable to changes in
functionality and performance over time.
GOAL: Create an underlying scientific foundation,
methodologies, and tools that will:
– Enable clear and concise specifications,
– Quantify the effectiveness of novel solutions,
– Test and evaluate systems in an objective manner, and
– Predict system assurance with confidence.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 292
Existing Security/Survivability Validation Approaches
• Most traditional approaches to security validation have focus on
avoiding intrusions (non-circumventability), or have not been
quantitative, instead focusing on and specifying procedures that
should be followed during the design of a system (e.g., the Security
Evaluation Criteria [DOD85, ISO99]).
• When quantitative methods have been used, they have typically either
been based on formal methods (e.g., [Lan81]), aiming to prove that
certain security properties hold given a specified set of assumptions,
or been quite informal, using a team of experts (often called a “red
team,” e.g. [Low01]) to try to compromise a system.
• Both of these approaches have been valuable in identifying system
vulnerabilities, but probabilistic techniques are also needed.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 293
Example Probabilistic Validation Study
• Evaluation of DPASA-DV Project design
– Designing Protection and Adaptation into a
Survivability Architecture: Demonstration and
Validation
• Design of a Joint Battlespace Infosphere
– Publish, Subscribe and Query features (PSQ)
– Ability to fulfill its mission in the presence of
attacks, failures, or accidents
• Uses Multiple, synergistic validation techniques
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 294
JBI Design
Overview
JBI Management Staff
Quad 1
JBI Core
Quad 2
Quad 3
Quad 4
Executive
Zone
Operations
Zone
Crumple
Zone
Network
Access Proxy
(Isolated Process Domains in SE-Linux)
Domain6
Local Controller First Restart Domains Eventually Restart Host
Protection
Domains
Isolation among
selected functions on
individual core hosts
and on clients
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Domain1
Domain2
Forward/
Ratelimit
PS
TCP
Domain3
Domain4
Domain5
Proxy Logic
Inspect / Forward / Rate Limit
Sensor
Rpts
DC
PSQImpl
Eascii
IIOP
RMI
IIOP
UDP
TCP
TCP
STCP
PSQImpl
Slide 295
Survivability/Security Validation Goal
• Provide convincing evidence that the design, when
implemented, will provide satisfactory mission support
under real use scenarios and in the face of cyber-attacks.
• More specifically, determine whether the design, when
implemented will meet the project goals:
• This assurance case is supported by:
– Rigorous logical arguments
– Experimental evaluation
– A detailed executable model of the design
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 296
Goal: Design a Publish and Subscribe Mechanism that …

Provides 100% of critical functionality when under sustained attack
by a “Class-A” red team with 3 months of planning.

Detects 95% of large scale attacks within 10 mins. of attack
initiation and 99% of attacks within 4 hours with less than 1% false
alarm rate.

Displays meaningful attack state alarms. Prevent 95% of attacks
from achieving attacker objectives for 12 hours.

Reduces low-level alerts by a factor of 1000 and display meaningful
attack state alarms.

Shows survivability versus cost/performance trade-offs.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 297
Integrated Survivability Validation Procedure
R
S
Requirement
Decomposition
Q
P
Functional Model of the Relevant Subset of the System
Model for
Access Proxy
Model for
Client
AA1
M1
M2
AA2
…
AA3
M4
M3
Model for
PSQ Server
AP1
AP2
M5
M6
(Network Domains)
L1
L2
L3
Functional Model
of the System
(Probabilistic or
Logical)
Assumptions
Supporting Logical
Arguments and
Experimentation
(ADF)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 298
Integrated Survivability Validation Procedure
Steps
R
S
1. A precise statement of
the requirements
Q
P
Functional Model of the Relevant Subset of the System
Model for
Access Proxy
Model for
Client
AA1
M1
M2
AA2
…
AA3
M4
M3
Model for
PSQ Server
AP1
AP2
M5
M6
(Network Domains)
L1
L2
2. High-level functional
model description:
a) Data and alerts
flows for the
processes related
to the
requirements,
b) Assumed attacks
and attack effects
[Threat/vulnerability analysis;
whiteboarding]
L3
(ADF)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 299
Integrated Survivability Validation Procedure
Steps
R
S
Q
P
Functional Model of the Relevant Subset of the System
Model for
Access Proxy
Model for
Client
AA1
M1
M2
AA2
…
AA3
M4
M3
Model for
PSQ Server
AP1
AP2
M5
M6
(Network Domains)
L1
L2
3. Detailed descriptions
of model component
behaviors representing
2a and 2b, along with
statements of
underlying
assumptions made for
each component.
[Probabilistic modeling
or logical
argumentation,
depending on
requirement]
L3
(ADF)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 300
Integrated Survivability Validation Procedure
Steps
R
S
Q
P
Functional Model of the Relevant Subset of the System
Model for
Access Proxy
Model for
Client
AA1
M1
M2
AA2
…
AA3
M4
M3
Model for
PSQ Server
AP1
AP2
M5
M6
(Network Domains)
L1
L2
L3
(ADF)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
4. Construct executable
functional model
[Probabilistic
modeling, if model
constructed in 3 is
probabilistic]
In Parallel
5. a) Verification of the
modeling assumptions
of Step 3 [Logical
argumentation] and,
b) where possible,
justification of model
parameter values
chosen in Step 4.
[Experimentation]
Slide 301
Integrated Survivability Validation Procedure
Steps
R
S
Q
P
Functional Model of the Relevant Subset of the System
Model for
Access Proxy
Model for
Client
AA1
M1
M2
AA2
…
AA3
M4
M3
Model for
PSQ Server
AP1
AP2
M5
M6
6. Run the executable
model for the
measures that
correspond to the
requirements of Step
1. [Probabilistic
modeling]
(Network Domains)
L1
L2
L3
(ADF)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 302
Integrated Survivability Validation Procedure
Steps
R
?
S
Q
P
Functional Model of the Relevant Subset of the System
Model for
Access Proxy
Model for
Client
AA1
M1
M2
AA2
…
AP1
AA3
M4
M3
Model for
PSQ Server
M5
(Network Domains)
L1
L2
L3
(ADF)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
7. Comparison of results
obtained in Step 6,
noting in particular
the configurations
and parameter values
for which the
requirements of Step
1 are satisfied.
AP2
M6
Note that if the
requirement being
addressed is not
quantitative, steps
4 and 6 are
skipped.
Slide 303
Step 1: Requirement Specification
• Expressed in an argument graph:
JBI critical mission
objectives
JBI critical
functionality
Initialized JBI
provides
essential services
Authorized
publish
processed
successfully
Authorized
subscribe
processed
successfully
JBI mission
Detection / Correlation
Requirements
JBI properly
initialized
Authorized
query
processed
successfully
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
IDS objectives
Authorized
join/leave
processed
successfully
Unauthorized
activity
properly
rejected
Confidential
info not
exposed
Slide 304
Argument Graph for the Design
Requirements decomposition
PIP requirements 1 – 4
JBI survivability
requirements
Executable model
JBI intrusion detection
requirements
Initialized JBI provides
essential services
Model assumptions
Supporting arguments
Authorized publish is
processed successfully
Authorized subscribe is
processed successfully
Dataflow
Timeliness
Integrity
Authorized query is
processed
successfully
Authorized join/leave
is processed
successfully
IDS / Correlation
requirements
JBI is properly
initialized
Unauthorized activity
is properly rejected
Confidential info is
not exposed
Confidentiality
(from functional
model execution)
Confidentiality of
Application-layer
Messages
Functional model
faithful to design
IO Confidentiality
(end-to-end)
IO Confidentiality
in Transit
Functional Model
Assumptions Hold
IO Confidentiality
in Storage
IConfidentiality of
Network
Communications
Design Team Review
Attack Model
Assumptions Hold
Component Model
Assumptions Hold
No
Compromise
or Failure of
QIS
QA1: QIS
Incorruptibility
Hard-wired
Configuration
Attack Model
Parameter
Selection
CERT
Vulnerability
DB Analysis
DoS Causes
Processing
Delays
DoS Does
Not Corrupt
Other
Components
DoS Attacks
Do Not
Propagate from
Clients to Core
Type Enforcement
Hardened Kernel
Solaris
No Data
Attacks
Outside the
Platform
SA3: IO
Authenticity
Initial Targets
of
Infrastructure
Attacks
Attacks
Originate
Outside the
Platform
SELinux
Physically
Protected
Correctness of
Rate Control
Mechanisms
Variation over
Anticipated
Ranges
Isolation of
Intruded
Process
Domains
Platform Mechanisms
Targets for
Loss of IO
Confidentiality
Electrically
Isolated
Infrastructure
Attack
Propagation
PA2: Alternate
Path
Availability
SA1: IO
Integrity in
PSQ Server
SA2: Client
Confidentiality
in PSQ Server
AA2: AP
Applicationlayer Integrity
AA3: AP
Application-layer
Confidentiality
PA1: ClientCore
Communication
I&C
Data Attack
Propagation
DA1: DC
Communications
QA2: QIS
Communication
Cutoff
QA3: QIS
Input
Integrity
QA4: QIS
Function
Correctness
Connectivity
Physical
Integrity
Correctness of
Reattachment
Protocol
Correctness of
Registration
Protocol
SA4: Networklayer I & C
MA1: SM Byzantine
Agreement
AA1: AP
Function
Correctness
SeA1: Sensor
False Alarm
Rate
SeA2: Sensor
Detection Delay
SeA3: Sensor
Detection
Probability
Correctness of Modified
ITUA Protocols
Electrical
Integrity
Gate
Configuration and
Truth Table
Proxy Protocol
Configuration
CoA1:
Corrleator
False Alarm
Rate
Can Identify
Malformed Traffic
IDS Experimental
Evaluation
Process Domain
Policies
Windows
Design
Faithfully
Implemented
Absence of
Insider Threat
PsA1: ADF
Policy Server
Input
Correctness
System Connectivity
PsA2: ADF
Policy Server
Synchronization
No Cryptography
in Access Proxy
IKENA StormWatch
No Tunneling Attacks
Restricted Routing
Network Topology
Not
Preconfigured
Correctness of
Certificate
Exchange
Correctness of
Managed Switch
ADF NIC Firmware
Initialization
ADF Key Initialization
ADF Agent
Initialization
ADF Host
Independence
ADF Protocol
Correctness
ADF Agent
Correctness
Policy Server
Integrity
VPG Integrity
VPG
Confidentiality
ADF Policy
Correctness
DoD Common
Access Card (CAC)
PKCS #11
ADF NIC
services
protected
No Unauthorized
Indirect Access
No Unauthorized
Direct Access
ADF Correctness
Keys Protected
from Theft
ADF NIC Physical
Security
Not
Reconfigurable
Private Key
Confidentiality
Physical Topology
Algorithmic
Framework
Keys Not Guessable
Key Length
Physical Protection
of CAC device
Protection of CAC
Authentication Data
No Compromise of
Authorized Process
Accessing CAC
Key Lifetime
Tamperproof
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 305
Step 2: System and Attack Assumption Definition
Example
High level description
…
Steps 4-5
Access proxy verifies if
the client is in valid
session by sending the
session key
accompanying the IO to
the Downstream
Controller for verification
Step 6
Access Proxy forwards
the IO to the PSQ
Server in its quadrant.
....
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 306
Attack Model Description
• Definitions
– Intrusion, prevented intrusion, tolerated intrusion
– New vulnerabilities
• Assumptions
– Outside attackers only
– Attacker(s) with unlimited resources
– Consider successful (and harmful) attacks only
– No patches applied for vulnerabilities found during the
mission/scenario execution
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 307
Attack Model Description
• Attack propagation
– MTTD: mean time to discovery of a vulnerability
– MTTE: mean time to exploitation of a vulnerability
• 3 types of vulnerabilities:
– Infrastructure-Level Vulnerabilities  attacks in depth
• OS vulnerability
• Non-JBI-specific application-level vulnerability
• pcommon : common-mode failure
– Data-Level Vulnerabilities  attacks in breadth
• Using the application data of JBI software
– Across process domains
• flaw in protection domains
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 308
Attack Model Description
• Attack effects
– Compromise
• Launching pad for further attacks
• Malicious behavior
– Crash
• Attack propagation stopped
– (DoS)
– Distinction between OSes with and without protection domains
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 309
Attack Model Description
• Intrusion Detection
– pdetect=0 if the sensors are compromised
– pdetect > 0 otherwise.
• Attack Responses
– Restart Processes
– Secure Reboot
– Permanent Isolation
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 310
Infrastructure Attacks Example
Access Proxy, Quad 1, OS 1
AP Hb
Se
AP
Alert
Se
Ac
ADF NIC
DC
Policy Server, Quad 1, OS 1
ADF NIC
Outside
DC, Quad 1, OS 1
AP IO
ADF NIC
T=85 min.:
discovery of a
vulnerability on
the Main PD,
OS1
PS
all quad
components
Quadrant
1
LC
Ac
Outside
LC
PSQ Server, Quad 1, OS 1
Guardian, Quad 1, OS 1
PSQ
Gu
Ac
LC
ADF NIC
Se
Ac
LC
SM, Quad 1, OS 1
ADF NIC
SD
Se
ADF NIC
Ac
ADF NIC
Se
Publishing Client, OS1
SM
Correlator, Quad 1, OS 1
Co
Crumple Zone
Access Proxy, Quad 2, OS 2
Access Proxy, Quad 3, OS 3
AP IO
Access Proxy, Quad 4, OS 4
ADF NIC
LC
Operations Zone
Executive Zone
PSQ Server, Quad 2, OS 2
SM, Quad 1, OS 2
SM, Quad 1, OS 3
SM, Quad 1, OS 4
ADF NIC
ADF NIC
ADF NIC
ADF NIC
ADF NIC
ADF NIC
ADF NIC
ADF NIC
ADF NIC
PSQ Server, Quad 3, OS 3
AP IO
PSQ
AP Hb
PSQ Server, Quad 4, OS 4
AP IO
PSQ
AP Hb
Se PSQ
APAP Hb
Se
Alert
AP
Ac
Se
Se
Alert
AP
Ac
Se
LC Ac
Ac Alert
Se
LC
Ac
Outside
LC Ac
LC
LC
©2005 William H. Sanders. All rights reserved. Do LC
not duplicate without permission of the author.
SM
SM
SM
Slide 311
Model of Access Proxy
4.4 Access Proxy
4.4.1 Model Description
AM1: If a process domain in the DJM proxy is not corrupted, it forwards the traffic it is designated to handle from the Quadrant
isolation switch to core quadrant elements and vice versa. All traffic being forwarded is well-formed (if the proxy is correct).
The following kinds of traffic are handled:
1. IOs (together with tokens) sent from publishing clients to the core (we do not distinguish between IOs sent via different
protocols such as RMI or SOAP/HTTP).
….
AM2: Attacks on access proxy: attacks on an access proxy are enabled if either/both
1. Quadrant isolation switch is ON, and one or more clients are corrupted, leading to:
a) Direct attacks: can cause the corruption of the process domain corresponding to the domain of the attacking process on
the compromised client.
….
AM3: If an attack occurs on the access proxy, it can have the following effects:
1. Direct attacks leading to process corruption:
a) Enable corruption of other process domains on the host.
…..
4.4.2 Facts and Simplifications
AF1: Each access proxy runs on a dedicated host machine.
AF2: DoS attacks result in increased delays.
….
Assumptions
Step 3: Detailed descriptions of model component behaviors
and Assumptions (Access Proxy)
4.4.3 Assumptions
AA1: Only well-formed traffic is forwarded by a correct access proxy.
AA2: The access proxy cannot access cryptographic keys used to sign messages that pass through it.
AA3: Access proxy cannot access the contents on an IO if application-level end-to-end encryption is being used.
AA4: Attacks on an access proxy can only be launched from compromised clients, or from corrupted core elements that interact
with the access proxy during the normal course of a mission.
….
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 312
Step 4: Construct Executable Functional Model
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 313
Step 5: Supporting Logical Arguments
JBI critical mission
objectives
JBI mission awareness
JBI critical functionality
Initialized JBI provides
essential services
Authorized publish
processed successfully
Authorized subscribe
processed successfully
Authorized query
processed
successfully
JBI properly initialized
Authorized join/leave
processed
successfully
Unauthorized activity
properly rejected
Confidential info
not exposed
IDS objectives
Dataflow
Timeliness
Integrity
Confidentiality
(from functional
model execution)
IO Confidentiality
(end-to-end)
Notification
Confidentiality
Functional model
assumptions hold
Functional model
faithful to design
Design Team Review
CA1: Origin of
Attacks on
Clients
CA2: Attack
Propagation
from Clients
AA4: Origin of
Attacks on
Access Proxy
AA5: Attacks
from AP
DA1: DC
Communications
DA2: Origin of
Attacks on DC
GA2: Attacks
from Guardian
SA1: Origin of
Attacks on
PSQ Server
SA2: Attacks
from PSQ
Server
SeA1: Attacks
from IDS Sensor
AcA2: Attacks
from Actuator
LA2: Attacks
from Local
Controller
CoA2: Origin
of Attacks on
Correlator
CoA3: Attacks
from Correlator
MA2: Origin
of Attacks on
SM
MA3: Attacks
from SM
PA1: ClientCore
Communication
I&C
SA6: Networklayer I & C
PsA1: ADF
Policy Server
Input
Correctness
PsA1: ADF
Policy Server
Synchronization
AA6: DoS from
Compromised
Core
AA1: AP
Function
Correctness
Bidirectional
Flow Control
AA8: DoS
Prevention by
Access Proxy
Correctness of
Flow Control
Mechanisms
QA1: QIS
Incorruptibility
Hard-wired
Configuration
QA2: QIS
Communication
Cutoff
Electrically
Isolated
Physically
Protected
QA3: QIS
Input
Integrity
Connectivity Physical
Integrity
QA4: QIS
Function
Correctness
Electrical
Integrity
Gate
Configuration and
Truth Table
Correctness of
Registration
Protocol
SA3: IO
Integrity in
PSQ Server
Proxy Protocol
Configuration
Restricted Routing
SA4: Client
Confidentiality
in PSQ Server
AA2: AP
Applicationlayer Integrity
AA3: AP
Application-layer
Confidentiality
Correctness of
Reattachment
Protocol
Correctness of
Certificate
Exchange
System Connectivity
Network Topology
PA2: Alternate
Path
Availability
SA5: IO
Authenticity
No Cryptography
in Access Proxy
Private Key
Confidentiality
MA1: SM Byzantine
Agreement
CA3: Client
Process
Corruption
AA7: AP
Process
Corruption
DA3:
Process
Corruption
on DC
GA1: Process
Corruption on
Guardian
SA7: Process
Corruption in
PSQ Server
SeA5:
Process
Corruption in
Sensor
Not
Reconfigurable
ScA1: Process
Corruption in
Subscribed
Client
Correctness of Modified
ITUA Protocols
SeA2: Sensor
False Alarm
Rate
SeA3: Sensor
Detection Delay
SeA4: Sensor
Detection
Probability
CoA1:
Corrleator
False Alarm
Rate
CoA4: Alert
Integrity
IDS Experimental
Evaluation
ADF NIC
services
protected
Platform Mechanisms
No Unauthorized
Direct Access
LA1: Process
Corruption in
Local
Controller
Process Isolation
Not
Preconfigured
Can Identify
Malformed Traffic
No Tunneling Attacks
AcA1: Process
Corruption in
Actuator
Component-specific
policy
No Unauthorized
Indirect Access
SELinux
Trusted Solaris
Windows 2000
Physical Topology
Keys Protected
from Theft
ADF Correctness
DoD Common
Access Card (CAC)
ADF NIC Physical
Security
ADF NIC Firmware
Initialization
ADF Agent
Initialization
ADF Protocol
Correctness
Policy Server
Integrity
PKCS #11
ADF Key Initialization
ADF Host
Independence
ADF Agent
Correctness
VPG Integrity
Algorithmic
Framework
Keys Not Guessable
Key Length
Physical Protection Protection of CAC
of CAC device
Authentication Data
No Compromise of
Authorized Process
Accessing CAC
Type Enforcement
Hardened Kernel
Hardened Kernel
Kernel Loadable
Wrappers
VMWare over SELinux
Key Lifetime
ADF Policy
Correctness
Tamperproof
VPG
Confidentiality
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 314
Logical Argument Sample
PSQ Server
Model
Functional
Model
Model
Assumptions
SA3: IO
Integrity in
PSQ Server
SA4: Client
Confidentiality
in PSQ Server
No Unauthorized
Direct Access
DoD Common
Access Card
(CAC)
PKCS #11
Compliance
AA2: AP
Applicationlayer Integrity
Private Key
Confidentiality
Supporting
Arguments
Keys Protected
from Theft
Access Proxy
Model
No Cryptography
in Access Proxy
No Unauthorized
Indirect Access
Keys Not
Guessable
Algorithmic
Framework
Key
Length
AA3: AP
Application-layer
Confidentiality
Not
Preconfigured
Physical
Protection of
CAC device
Not
Reconfigurable
Protection of
CAC
Authentication
Data
ADF NIC
services
protected
No Compromise
of Authorized
Process
Accessing CAC
Key
Lifetime
Tamperproof
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 315
Steps 6 and 7: Measures and Results
• Assumptions: CPUB is the conjunction of
– C1PUB= the publishing client is successfully registered with the
core
– C2PUB= the publishing client's mission application interacts
with the client as intended
• Definition of a successful publish: EPUB is the conjunction of
– E1PUB = the data flow for the IO is correct
– E2PUB = the time required for the publish operation is less than
tmax
– E3PUB = the content of the IO received by the subscriber has
the same essential content as that assembled by the publisher
• Measure: P[EPUB|CPUB]
– Fraction of successful publishes in a 12 hour period
– Between clients that cannot be compromised
• Objective
– P[EPUB|CPUB] ≥ pPUB for a 12-hour mission
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 316
Vulnerability Discovery Rate Study
Fraction of successful publishes
versus MTTD
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Number of successful intrusions
versus MTTD
Slide 317
Varying the number of OS and OS w/ process domains
With data attacks
Without data attacks
4 OS, 4 pd,
3 OS, 3 pd,
AP OS<>core AP OS<>core
4 OS total
4 p.d
3 p.d
2 p.d
2 OS total
3 OS total
1 p.d
0 p.d
3 p.d
2 p.d
1 p.d
0 p.d
2 p.d
1 p.d
1 OS total
0 p.d
1 p.d
0 p.d
1.00
0.94
0.90
0.93
0.89
0.85
0.84
0.83
0.82
0.84
0.84
0.81
0.78
0.80
Fraction of successful Publishes
0.87
0.83
0.76
0.75
0.72
0.70
0.70
0.81
0.78
0.76
0.76
0.72
0.70
0.66
0.64
0.63
0.61
0.76
0.71
0.59
0.60
0.57
0.52
0.50
0.40
0.30
0.20
0.10
0.00
1.1
6.1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Experiment
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 318
Autonomic Distributed Firewall (ADF) NIC policies
Fraction of successful publishes
Total number of intrusions
140
1
Per process domain
120
0.96
Total Number of Intrusions
Fraction of Successful Publishes
0.98
0.94
0.92
0.9
0.88
Per component
No restriction
100
80
60
40
0.86
Per process domain
0.84
Per component
0.82
No restriction
20
0
0.8
100
100
1000
1000
MTTD
MTTD (min)
•
•
Per-pd policies considerably increase the performance (10% unavailability vs. 1.5%
at MTTD=100 minutes)
ADF NICs can handle per-port policies => should take advantage of this feature,
implying to set the communication ports in advance
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 319
Design and Implementation Oriented Validation of Survivable Systems
A. Agbaria, T. Courtney, M. Ihde, W. H. Sanders, M. Seri, and S. Singh
Design Phase Validation
• A study of the design reveals that
integrity and confidentiality can be
regarded as probability-1 events.
• We obtain the following logical
decomposition:
• PUB1: P[E1  E2| E3  E4  C] ≥ p
• PUB2: P[E3| C] = 1
• PUB3: P[E4| C] = 1
• It can be shown that:
(PUB1  PUB2  PUB3)  PUB
Model
Assumptions
AA2: AP
Applicationlayer Integrity
Supporting
Arguments
DoD
(CAC)
PKCS #11
Compliance
Sub-requirements
Step 2: If R is
logically
decomposable,
decompose it
iteratively.
Logical
Decomposition
AA3: AP
Application-layer
Confidentiality
No
Unauthorized
Indirect
Access
Key
Leng
th
Tamperproof
Key
Lifeti
me
No
Cryptography in
AP
Not
Preconfig
ured
Physical
Protection of
CAC device
Decomposable?
No
Not
Reconfigu
rable
Protection of
CAC
Authenticati
on Data
Step 3: For
every atomic
requirement Ra
Logical
Argumentation
• Let PUB be the requirement of “successfully
process a publish request”.
• Let C be the preconditions.
• Let E be the desired event, i.e., the
successful of a request to publish.
• E is a conjunction of:
• E1 = the data flow of the publish is
correct
• E2 = timeliness
• E3 = integrity
• E4 = confidentiality
• The requirement: PUB: P[E|C] ≥ p
Def eat conf identiality
of IO data
Attack Tree
Gate 1
Read data on client
Read data on transit
Gate 2
Gate 3
Build high-level description of System and
its operational environment
Step 4:
Detailed
description of
components
Read data on core
Gate 4
Defeat the firewall
access control
Compromise client
Escalate privileges
Read data
Defeat the firewall
and sniff off wire
Get in middle of
client/core traffic
Defeat the firewall
access control
Ev ent 1
Ev ent 2
Ev ent 3
Gate 5
Gate 6
Gate 7
Ev ent 1
Read from data file
Read from memory
Defeat the firewall
crypto
Steal key/certificate
Sniff packets
Defeat the firewall
access control
Defeat the firewall
crypto
Ev ent 4
Ev ent 5
Ev ent 6
Ev ent 7
Ev ent 8
Ev ent 1
Ev ent 6
Attack Graph
Yes
Tear down current
TCP connections
Read from AP
Gate 12.1
Re-route traffic at
both ends
Read data
Steal key/certificate
Compromise AP
Escalate privileges
Read IO as it passes
through
Gate 8
Gate 9
Ev ent 7
Ev ent 13
Ev ent 14
Ev ent 15
Ev ent 9
Perform ARP
spoofing
Modify network
routing
Steal key/certificate
Decrypt & read data
Ev ent 10
Ev ent 11
Ev ent 7
Ev ent 12
Automatic
construction
Data Flow
ADF NIC
services
protected
Keys Not
Guessable
Alg.
Framew
ork
Yes
Quantitative?
Private Key
Confidentiality
No Unauthorized
Direct Access
Keys
Protected
from Theft
Requirement
Access Proxy
Model
Functional
Model
Step 1:
Formulate a
precise
statement of
R.
Implementation Phase Validation
Not
valid
No Compromise of
Authorized Process
Accessing CAC
Step 5: Justify
the modeling
assumptions of
Step 4
Step 6:
Construct a
simulation
model
Verify assumptions
& parameter values
Probabilistic measures
Probabilistic model of the system and its operational environment
Infrastructure-level attacks
Survivable Publish Subscribe System
Management Staff
Step 7:
Evaluation and
comparing
System not valid
Executive Quad 1 Quad 2
Zone
Core
Compare with
requirement
Quad 3
Quad 4
Operations
Zone
Crumple
Zone
System valid w.r.t.
the requirement
Client Zone
Network
Access Proxy
(Isolated Process Domains in SE-Linux)
Domain6
Local Controller
First Restart Domains
Domain1
Domain2
Forward/Rate limit
PS
TCP
©2005 William H.
Sanders.
rights
Do not
IN
F O All
RM
Areserved.
TION
T duplicate
R U S Twithout
I Npermission
S T I T ofUthe
T Eauthor.
University of Illinois at Urbana-Champaign
Eventually Restart Host
Domain3 Domain4 Domain5
Proxy Logic Inspect / Forward / Rate Limit
Sensor Rpts
DC
PSQImpl
PSQImpl
Eascii
IIOP
RMI
IIOP
UDP
TCP
TCP
TCP
www.iti.uiuc.edu
Slide 320
The Art of Dependability Evaluation /
Conclusions
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 321
Course Outline Revisited
• Issues in Model-Based Validation of High-Availability
Computer Systems/Networks
• Stochastic Activity Network Concepts
• Analytic/Numerical State-Based Modeling
• Case Study: Embedded Fault-Tolerant Multiprocessor System
• Solution by Simulation
• The Art of System Dependability /Conclusions
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 322
Model Solution Issues
• In general:
– Use “tricks” from probability theory to reduce complexity of model
– Choose the right solution method
• Simulation:
– Result is just an estimator based on a statistical experiment
– Estimation of accuracy of estimate essential
– Use confidence Intervals!
• Analytic/Numerical model solution:
– Avoid state space explosion
• Limit model complexity
• Use structure of model (symmetries) to reduce state space size
– Understand accuracy/limitations of chose numerical method
• Transient Solution
• (Iterative or Direct) Steady-state solution
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 323
The “Art” of Performance and Dependability Validation
• Performance and dependability validation is an art because:
– There is no recipe for producing a good analysis,
– The key is knowing how to abstract away unimportant details, while
retaining important components and relationships,
– This intuition only comes from experience,
– Experience comes from making mistakes.
• There are many ways to make mistakes.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 324
Doing it Right: Model Construction
•
•
•
Understand the desired measure before you build the model.
The desired measure determines the type of model and the level of detail required. No
model is universal!
Steps in constructing a model:
1. Choose the desired measures:
• Choice of measures form a basis for comparison.
• It’s easy to choose wrong measure and see patterns where none exist.
• Measures should be refined during the design and validation process.
2. Choose the appropriate level of detail/abstraction for model components.
• Key is to represent model at the right level of detail for the chosen measures.
• It is almost never possible or practical to include all system aspects.
• Model the system at the highest level possible to obtain a good estimate of the
desired measures.
3. Build the model.
• Decide how to break up the model into modules, and how the modules will
interact with one another.
• Test the model as you build it, to ensure it executes as intended.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 325
Doing it Right: Model Solution
• Use the appropriate model solution technique:
– Just because you have a hammer doesn’t mean the world is a nail.
– There is no universal model solution technique (not even simulation!)
– The appropriate model solution technique depends on model characteristics.
• Use representative input values:
– The results of a model solution are only as good as the inputs.
– The inputs will never be perfect.
– Understand how uncertainty in inputs affects measures.
– Do sensitivity analysis.
• Include important points in the design/parameter space:
– Parameterize choices when design or input values are not fixed.
– A complete parametric study is usually not possible.
– Some parameters will have to be fixed at “nominal” values.
– Make sure you vary the important ones.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 326
Doing it Right: Model Interpretation/Documentation
• Make all your assumptions explicit:
– Results from models are only as good as the assumptions that were made in
obtaining them.
– It’s easy to forget assumptions if they are not recorded explicitly.
• Understand the meaning of the obtained measures:
– Numbers are not insights.
– Understand the accuracy of the obtained measures, e.g., confidence intervals
for simulation.
• Keep social aspects in mind:
– Performance and dependability analysts almost always bring bad news.
– Bearers of bad news are rarely welcomed.
– In presentations, concentrate on results, not the process.
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 327
Next Steps
• You have:
– Learned theory related to reliability, availability, and
performance validation using SANs and Möbius
– Learned about the advantages and disadvantages of various
(analytical/numerical and simulation-based) solution
algorithms.
• There are many places to go for further information:
– Möbius Software Web pages
(www.mobius.uiuc.edu)
– Performability Engineering Research Group Web pages
(www.perform.csl.uiuc.edu)
©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.
Slide 328