Download Continuous Random Variables and Reliability Analysis

In the Name of the Most High
Continuous Random Variables and Reliability
Analysis
Behzad Akbari
Spring 2009
Tarbiat Modares University
These slides are based on the slides of Prof. K.S. Trivedi (Duke University)
Definitions
• Distribution function:
• If FX(x) is a continuous function of x, then X is a
continuous random variable.
– FX(x): discrete in x  Discrete rv’s
•
Definitions
(Continued)
Equivalence:
• CDF (cumulative distribution function)
• PDF (probability distribution function)
• Distribution function
• FX(x) or FX(t) or F(t)
Probability Density Function (pdf)
•
•
X : continuous rv, then,
pdf properties:
1.
2.
Definitions
(Continued)
• Equivalence: pdf
– probability density function
– density function
– density
dF
– f(t) =
dt
F (t )  
t

f ( x)dx
t
  f ( x)dx ,
0
For a non-negative
random variable
Exponential Distribution
•
•
•
•
Arises commonly in reliability & queuing theory.
A non-negative random variable
It exhibits memoryless (Markov) property.
Related to (the discrete) Poisson distribution
– Interarrival time between two IP packets (or voice calls)
– Time to failure, time to repair etc.
• Mathematically (CDF and pdf, respectively):
CDF of exponentially distributed
random variable with  = 0.0001
F(t)
12500
25000
t
37500
50000
Exponential Density Function
(pdf)
f(t)
t
Memoryless property
• Assume X > t. We have observed that the
component has not failed until time t.
• Let Y = X - t , the remaining (residual) lifetime
• The distribution of the remaining life, Y, does not
depend on how long the component has been
operating. Distribution of Y is identical to that of X.
Memoryless property
• Assume X > t. We have observed that the
component has not failed until time t.
• Let Y = X - t , the remaining (residual)
lifetime
Gt ( y )  P(Y  y | X  t )
 P( X  y  t | X  t )
P(t  X  y  t )
 y

 1 e
P( X  t )
Memoryless property (Continued)
• Thus Gt(y) is independent of t and is identical to the original
exponential distribution of X.
• The distribution of the remaining life does not depend on
how long the component has been operating.
Reliability as a Function of Time
• Reliability R(t): failure occurs after time ‘t’. Let X
be the lifetime of a component subject to failures.
• Let N0: total no. of components (fixed); Ns(t):
surviving ones; Nf(t): failed one by time t.
Definitions
(Continued)
Equivalence:
• Reliability
• Complementary distribution function
• Survivor function
• R(t) = 1 -F(t)
Failure Rate or Hazard Rate
• Instantaneous failure rate: h(t) (#failures/10k hrs)
• Let the rv X be EXP( λ). Then,
• Using simple calculus the following applies to any rv,
Hazard Rate and the pdf
f (t )
f (t )
h (t ) 

R(t ) 1  F (t )
h(t) t = Conditional Prob. system will fail in
(t, t + t) given that it has survived until time t
f(t) t = Unconditional Prob. System will fail in
(t, t + t)
• Difference between:
– probability that someone will die between 90
and 91, given that he lives to 90
– probability that someone will die between 90
and 91
Weibull Distribution
• Frequently used to model fatigue failure, ball bearing failure
etc. (very long tails)
Rt   e
t 0
• Reliability:
• Weibull distribution is capable of modeling DFR (α < 1), CFR
(α = 1) and IFR (α >1) behavior.
 t
• α is called the shape parameter and  is the scale parameter
Failure rate of the weibull
distribution with various values
of  and  = 1
5.0
1.0
2.0
3.0
4.0
Infant Mortality Effects in System
Modeling
• Bathtub curves
– Early-life period
– Steady-state period
– Wear out period
• Failure rate models
Bathtub Curve
Failure Rate (t)
•Until now we assumed that failure rate of equipment is time (age)
independent. In real-life, variation as per the bathtub shape has been
observed
Infant Mortality
(Early Life Failures)
Steady State
Operating Time
Wear out
Early-life Period
• Also called infant mortality phase or reliability
growth phase
• Caused by undetected hardware/software defects
that are being fixed resulting in reliability growth
• Can cause significant prediction errors if steadystate failure rates are used
• Availability models can be constructed and solved
to include this effect
• Weibull Model can be used
Steady-state Period
• Failure rate much lower than in early-life
period
• Either constant (age independent) or slowly
varying failure rate
• Failures caused by environmental shocks
• Arrival process of environmental shocks can
be assumed to be a Poisson process
• Hence time between two shocks has the
exponential distribution
Wear out Period
• Failure rate increases rapidly with age
• Properly qualified electronic hardware do
not exhibit wear out failure during its
intended service life (Motorola)
• Applicable for mechanical and other
systems
• Weibull Failure Model can be used
Bathtub curve
DFR phase: Initial design, constant bug fixes
CFR phase: Normal operational phase
IFR phase: Aging behavior
h(t)
(burn-in-period)
(wear-out-phase)
CFR
(useful life)
DFR
IFR
t
Decreasing failure rate
Increasing fail. rate
Failure Rate Models
•We use a truncated Weibull Model
Failure-Rate Multiplier
7
6
5
4
3
2
1
0
0
2,190
4,380
6,570
8,760
10,950 13,140 15,330 17,520
Operating Times (hrs)
•Infant mortality phase modeled by DFR Weibull and the
steady-state phase by the exponential
Failure Rate Models (cont.)
• This model has the form:
1  t  8,760
W ( t )  C 1 t 
  SS
t  8,760
• where:
• C 1  W 1,  SS  steady-state failure rate
•  is the Weibull shape parameter
• Failure rate multiplier = W ( t)  SS
Failure Rate Models (cont.)
• There are several ways to incorporate time dependent failure rates in
availability models
• The easiest way is to approximate a continuous function by a
decreasing step function
Failure-Rate Multiplier
7
6
1
5
4
2
3
2
1
0
0
2,190
4,380
 SS
6,570 8,760 10,950 13,140 15,330 17,520
Operating Times (hrs)
Failure Rate Models (cont.)
•Here the discrete failure-rate model is
defined by:
0  t  4,380
W ( t )   1
 2
4,380  t  8,760
  ss
t  8,760
Uniform Random Variable
• U(a,b)  pdf constant over the (a,b) interval and
CDF is the ramp function
Uniform distribution
• The distribution function is given by:
0 ,
{
F(x)=
xa ,
ba
1 ,
x < a,
a <x<b
x > b.
HypoExponential
• HypoExp: multiple Exp stages in series.
• 2-stage HypoExp denoted as HYPO(λ1, λ2). The
density, distribution and hazard rate function are:
• HypoExp results in IFR: 0  min(λ1, λ2)
• Disk service time may be modeled as a 3-stage
Hypoexponential as the overall time is the sum of
the seek, the latency and the transfer time
Erlang Distribution
• Special case of HypoExp: All stages have
same rate.
Gamma Random Variable
• Gamma density function is,
• Gamma distribution can capture all three failure
modes, viz. DFR, CFR and IFR.
– α = 1: CFR
– α <1 : DFR
– α >1 : IFR
HyperExponential Distribution
• Hypo or Erlang  Sequential Exp( ) stages.
• Alternate Exp( ) stages  HyperExponential.
• CPU service time may be modeled as HyperExp
Gaussian (Normal) Distribution
• Bell shaped pdf
• Central Limit Theorem: mean of a large number of
mutually independent rv’s (having arbitrary
distributions) starts following Normal distribution
as n 
• μ: mean, σ: std. deviation, σ2: variance (N(μ, σ2))
• μ and σ completely describe the statistics. This is
significant in statistical estimation/signal
processing/communication theory etc.
Normal Distribution (contd.)
• N(0,1) is called normalized Guassian.
• N(0,1) is symmetric i.e.
– f(x)=f(-x)
– F(z) = 1-F(z).
• Failure rate h(t) follows IFR behavior.
– Hence, N( ) is suitable for modeling long-term wear or
aging related failure phenomena.
Order statistics: kofn, TMR
Order Statistics: KofN
X1 ,X2 ,..., Xn iid (independent and identically distributed)
random variables with a common distribution function
F().
Let Y1 ,Y2 ,...,Yn be random variables obtained by
permuting the set X1 ,X2 ,..., Xn so as to be in
increasing order.
To be specific:
Y1 = min{X1 ,X2 ,..., Xn} and
Yn = max{X1 ,X2 ,..., Xn}
Order Statistics: KofN
(Continued)
• The random variable Yk is called the k-th ORDER
STATISTIC.
• If Xi is the lifetime of the i-th component in a system of n
components. Then:
– Y1 will be the overall series system lifetime.
– Yn will denote the lifetime of a parallel system.
– Yn-k+1 will be the lifetime of an k-out-of-n system.
Order Statistics: KofN (Continued)
To derive the distribution function of Yk, we note that the
probability that exactly j of the Xi's lie in (- ,y] and (n-j)
lie in (y, ) is:
n j
n j
F
(
y
)
[
1

F
(
y
)]
 
 j
hence
n j
n j
FYk ( y )     F ( y ) [1  F ( y )]
j k  j 
n
Applications of order statistics
• Reliability of a k out of n system
n
Rkofn (t )   ( nj )[ R(t )] j [1  R(t )]n  j
j k
• Series system:
Rseries (t )  [ R(t )]
n
n
or  Ri (t )
i 1
• Parallel system: R parallel(t )  1  [1  R(t )]n
• Minimum of n EXP random variables is special case
of Y1 = min{X1,…,Xn} where Xi~EXP(i)
Y1~EXP( i)
• This is not true (that is EXP dist.) for the parallel case
Triple Modular Redundancy (TMR)
R(t)
R(t)
Voter
R(t)
• An interesting case of order statistics occurs when
we consider the Triple Modular Redundant (TMR)
system (n = 3 and k = 2). Y2 then denotes the time
until the second component fails. We get:
RTMR (t )  3R (t )  2R (t )
2
3
TMR (Continued)
• Assuming that the reliability of a single
component is given by,
e
 t
we get:
RTMR (t )  3e
2 t
 2e
3t
TMR
(Continued)
• In the following figure, we have plotted
RTMR(t) vs t as well as R(t) vs t.
TMR
(Continued)
Cold standby (dynamic redundancy)
X
Y
Lifetime of Lifetime of
Spare
Active
EXP()
EXP()
Total lifetime 2-Stage Erlang
R(t )  (1  t )e
 t
EXP()
Assumptions: Detection & Switching perfect; spare
does not fail
EXP()
Sum of RVs: Standby Redundancy
• Two independent components, X and Y
– Series system (Z=min(X,Y))
– Parallel System (Z=max(X,Y))
– Cold standby: the life time Z=X+Y
Sum of Random Variables
• Z = Φ(X, Y)  ((X, Y) may not be independent)
• For the special case, Z = X + Y
• The resulting pdf is (assuming independence),
• Convolution integral (modify for the non-negative
case)
Convolution (non-negative case)
Z = X + Y, X & Y are independent random
variables (in this case, non-negative)
t
f Z (t )   f X ( x) fY (t  x) dx
0
• The above integral is often called the
convolution of fX and fY. Thus the density of
the sum of two non-negative independent,
continuous random variables is the
convolution of the individual densities.
Cold standby derivation
• X and Y are both EXP() and independent.
• Then
t
f t (t )   e e
 x
 ( t  x )
dx
0
2  t
 e
t
 dx
0
 t
  te , t  0
2
Cold standby derivation
(Continued)
• Z is two-stage Erlang Distributed
t
FZ (t )   f Z ( z )dz  1  (1  t )e
0
R(t )  1  F (t )
 t
 (1  t )e , t  0
 t
Convolution: Erlang
Distribution
• The general case of r-stage Erlang Distribution
• When r sequential phases have independent
identical exponential distributions, then the
resulting density is known as r-stage (or r-phase)
Erlang and is given by:
Convolution: Erlang
EXP()
EXP()
(Continued)
EXP()
r r 1  t
t e
f (t ) 
(r  1)!
(  t ) k  t
F (t )  1  
e
k!
k 0
r 1
Warm standby
•With Warm spare, we have:
•Active unit time-to-failure: EXP()
•Spare unit time-to-failure: EXP()
EXP(+ )
EXP()
2-stage hypoexponential distribution
Warm standby derivation
• First event to occur is that either the active or the spare wil
fail. Time to this event is min{EXP(),EXP()} which is
EXP( + ).
• Then due to the memoryless property of the exponential,
remaining time is still EXP().
• Hence system lifetime has a two-stage hypoexponential
distribution with parameters
1 =  +  and 2 =  .
Warm standby derivation
(Continued)
• X is EXP(1) and Y is EXP(2) and are
independent
1 = 2
• Then fZ(t) is
t
f Z (t )   1e
 1 x
2e
 2 ( t  x )
dx
0
12  t 12  t

e 
e
1  2
2  1
2
1
Hot standby
•With hot spare, we have:
•Active unit time-to-failure: EXP()
•Spare unit time-to-failure: EXP()
EXP(2)
EXP()
2-stage hypoexponential
TMR and TMR/simplex
as hypoexponentials
TMR/Simplex
EXP(3)
EXP()
TMR
EXP(3)
EXP(2)
Hypoexponential: general case
r
• Z=
 X , where X
i 1
i
1
,X2 , … , Xr
are mutually independent and Xi is exponentially
distributed with parameter i
(i = j for i = j).
Then Z is a r-stage hypoexponentially
distributed random variable.
EXP(1)
EXP(2)
EXP(r)
Hypoexponential: general case
KofN system lifetime as a
hypoexponential
At least, k out of n units should be operational for the
system to be Up.
EXP(n)
Y1
EXP((n-1))
Y2
...
EXP(k)
Yn-k+1
EXP((k-1))
Yn-k+2
...
EXP()
Yn
KofN with warm spares
At least, k out of n + s units should be operational
for the system to be Up. Initially n units are active
and s units are warm spares.
EXP(n
s)
EXP(n
+(s-1) )
...
EXP(n
+ )
EXP(n)
...
EXP(k)
Sum of Normal Random Variables
• X1, X2, .., Xk are normal ‘iid’ rv’s, then, the rv
Z = (X1+ X2+ ..+Xk) is also normal with,
• X1, X2, .., Xk are normal. Then,
follows Gamma