Download PERFORMANCE ANALYSIS ON A SHOESTRING Michael Tortorella Rutgers University

PERFORMANCE ANALYSIS
ON A SHOESTRING
Michael Tortorella
Rutgers University
Piscataway, NJ 08854 USA
SOME ALTERNATE
TITLES
™Crossing a non-Jackson network, with or
without a map
™Latency in Markov-routed networks with
node and link delays
™Path-additive functionals on networks
with Markovian routing
™Markovian routing approximations to
address-based routing
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 2
OVERVIEW
™Problem summary and basics
™Review of the traffic equation
™Path-additive functionals
™Node-to-node values
‰
Cross-network values
™Three models of address-based routing
™Conclusion
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 3
PROBLEM
NUTSHELLS
™Networks deliver services by transporting
certain objects from point to point
‰
Physical objects
+Packages
+Oil, gas, electricity
‰
Notional objects
+Packets
™Service quality includes metrics on how long it
takes to complete a trip across the network
‰
Cross-network delay
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 4
PROBLEM
NUTSHELLS
™Computing cross-network delay in a productform network is easy
‰
Everything is independent
™Many factors can cause the standard models
to be unsuitable
‰
‰
Packet size distribution not exponential
Delays on links
™How can we analyze cross-network delays
absent the standard framework?
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 5
BASICS
NETWORK
™A network is a graph G = (N, L)
N = set of nodes, |N| = n
‰ L = set of links ⊂ N × N
‰ O ⊂ N = set of originating nodes
‰ D ⊂ N = set of destination nodes
‰ For i ∈ O and j ∈ D, uij = exogenous demand =
‰
traffic to be carried from i to j
+Barrels per hour
+Terabits per second
+etc.
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 6
BASICS
ROUTING
™Let Xn represent the node at which a customer
is located at the nth step in its routing process
™ rij = P{Xn+1 = j | Xn = i, Xn–1, Xn–2, …, X1}
™Routing is homogeneous
‰
Doesn’t depend on n
™Routing is Markovian
‰
No dependence on Xn–1, Xn–2, …, X1
™ rij = P{Xn+1 = j | Xn = i } is routing matrix
‰
Usually rii = 0
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 7
OVERVIEW
™Problem summary and basics
™Review of the traffic equation
™Path-additive functionals
™Node-to-node values
‰
Cross-network values
™Three models of address-based routing
™Conclusion
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 8
TRAFFIC EQUATION
™Network in equilibrium
‰
Rate of flow into each node equals the rate of flow
out of the node
™λi = total arrival rate into node i
™ ui* = ui1 + ⋅ ⋅ ⋅ + uin = total arrival rate of
exogenous demand into node i
∗
i
™Then λ i = u +
n
∑λ r
j =1
j ji
, i = 1,… , n
™If R is convergent, then λ = u*(I – RT)–1
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 9
OVERVIEW
™Problem summary and basics
™Review of the traffic equation
™Path-additive functionals
™Node-to-node values
‰
Cross-network values
™Three models of address-based routing
™Conclusion
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 10
PATHS
™A path in G is a sequence of alternating
nodes and (single) links joining them,
beginning and ending with a node
‰
Set of all paths in G is P(G)
™A single node is a legitimate path
™So is a pair of nodes connected by a
single link
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 11
PATH-ADDITIVE
FUNCTIONALS
™a : P(G) → R is path-additive if for every
π ∈ P(G),
k −1
a (π) = ∑ [ a (ν m ) + a (ν m , ν m +1 ) ] + a (ν k )
m =1
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 12
CUSTOMER
SAMPLE PATHS
™Customer ω travels a path X0(ω) ( = i),
X1(ω), …, Xk(ω) ( = j) called π(ω)
™If a is path-additive, we define Aij(ω) by
Aij (ω) = a (i, j ) I { X 0 = i, X 1 = j} +
∞
+ ∑ a (π(ω)) I { X 0 = i, X 1 = ν1 , … , X k = ν k , X k +1 = j}
k =1
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 13
OVERVIEW
™Problem summary and basics
™Review of the traffic equation
™Path-additive functionals
™Node-to-node values
‰
Cross-network values
™Three models of address-based routing
™Conclusion
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 14
NODE-TO-NODE
VALUES
™Define skm = [a(k) + a(k, m)]I{(k, m) ∈ L}
‰
Assume these are random variables with
distributions Fkm
+Could be degenerate
‰
Independent of the routing process
™Let skm = Eskm and S be the matrix of the
skm
™Let Ā be the matrix of Eaij − Ea(j)
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 15
NODE-TO-NODE
VALUES
™Theorem: In an open network,
Ā = (I − R)−1⋅ S#R ⋅ (I − R)−1
‰
Single-link values independent of routing
process
™Example: let skm = 1 for every (k, m) ∈ L
Then Aij is the (random) number of links
traveled from i to j
‰ The expected number of links traveled
from i to j is (the (i, j) entry in) R(I − R)–2
‰
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 16
NODE-TO-NODE
VALUES
™Example:
2
p
1
4
1–p
DIMACS Workshop
13 October 2011
3
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 17
EXAMPLE
™R =
⎛ 0 p 1− p
⎜
0
⎜0 0
⎜0 0
0
⎜⎜
0
⎝0 0
™R(I − R)−2 =
DIMACS Workshop
13 October 2011
0⎞
⎟
1⎟
1⎟
⎟
0 ⎟⎠
⎛0
⎜
⎜0
⎜0
⎜⎜
⎝0
p 1− p 2⎞
⎟
0
0
1⎟
0
0
1⎟
⎟
0
0
0 ⎟⎠
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 18
EXAMPLE
™Single-link delays dkm
™(I − R)−1·S#R· (I − R)−1 =
⎛ 0 d12 p d13 (1 − p )
⎜
0
0
⎜0
⎜0
0
0
⎜⎜
0
0
⎝0
DIMACS Workshop
13 October 2011
p ( d12 + d 24 ) + (1 − p ) ( d13 + d 34 ) ⎞
⎟
d 24
⎟
⎟
d 34
⎟⎟
0
⎠
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 19
NODE-TO-NODE
VALUES
™Let wij = Esij2 and W = (wij )
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 20
NODE-TO-NODE
VALUES
™Theorem: In an open network in which
the skm are uncorrelated, E(Aij − a(j))2 is
given by the (i, j) entry of
(I − R)−1[W#R + 2S#R⋅(I − R)−1⋅S#R](I − R)−1
™Not a problem if single-element delays
are deterministic
‰
The delay on a single link counts the
processing delay at the link’s head node
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 21
CROSS-NETWORK
DELAY
™Expected value is latency
™Standard deviation is jitter
™Packet loss probability is P{delay = ∞}
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 22
DISTRIBUTIONS
™Let F and G be n×n matrices of CDFs
™Define matrix convolution of F and G by
n
( F ∗ G )ij ( x) = ∑ Fik ∗ Gkj ( x)
k =1
™Repeated convolutions denoted by
superscripts *2, *3, etc.
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 23
DISTRIBUTIONS
™Theorem: Let skm be nonnegative and
mutually stochastically independent
and
∞
let M(x) denote the matrix ∑ ( R # F )∗m ( x) .
m =1
Then, in an open network,
P{Aij − a(j) ≤ x} = Mij(x) and M satisfies
the matrix-integral equation
M = R#F + (R#F)∗M.
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 24
EXAMPLE
A JACKSON NETWORK
™Consider a network of K nodes in series
™ O = {1}, D ={K}
DIMACS Workshop
13 October 2011
⎛0
⎜
⎜0
⎜0
RK = ⎜
⎜
⎜0
⎜⎜
⎝0
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
0⎞
⎟
0⎟
0⎟
⎟
0⎟
1⎟
⎟⎟
0⎠
©2011 M. Tortorella
Slide 25
EXAMPLE
A JACKSON NETWORK
⎛1
⎜0
⎜
⎜0
−1
( I − RK ) = ⎜
⎜
⎜0
⎜⎜
⎝0
DIMACS Workshop
13 October 2011
1 1 1
1 1 1
0 1 1
0 0 0
0 0 0
1⎞
1 ⎟⎟
1⎟
⎟ and
⎟
1⎟
⎟⎟
1⎠
⎛1
⎜1
⎜
⎜1
T −1
( I − RK ) = ⎜
⎜
⎜1
⎜⎜
⎝1
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
0 0 0
1 0 0
1 1 0
1 1
1 1
1
1
0⎞
0 ⎟⎟
0⎟
⎟
⎟
0⎟
⎟⎟
1⎠
©2011 M. Tortorella
Slide 26
EXAMPLE
A JACKSON NETWORK
k
™The traffic equation gives λ k = ∑ uk , k = 1,…, K .
m =1
™Jackson’s theorem: if λk < μk for all k,
then nodes are independent M/M/1
queues with a(k) = (μk − λk)−1
™Put sjk = (μk − λk)−1δj,k−1 for j, k = 1, …, K
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 27
EXAMPLE
A JACKSON NETWORK
™Mean inter-node delays are
⎛1
⎜0
⎜
⎜0
⎜
⎜
⎜0
⎜⎜
⎝0
1 1 1
1 1 1
0 1 1
0 0 0
0 0 0
DIMACS Workshop
13 October 2011
1 ⎞ ⎛ 0 a1 0
1 ⎟⎟ ⎜⎜ 0 0 a2
1⎟ ⎜ 0 0 0
⎟⎜
⎟⎜
1⎟ ⎜ 0 0 0
⎟⎟ ⎜⎜
1⎠ ⎝ 0 0 0
0
0
a3
0
0
⎞⎛1
⎟⎜0
⎟⎜
⎟⎜0
⎟⎜
⎟⎜
aK −1 ⎟ ⎜ 0
⎟⎟ ⎜⎜
0 ⎠⎝0
0
0
0
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
1 1 1
1 1 1
0 1 1
0 0 0
0 0 0
1⎞
1 ⎟⎟
1⎟
⎟
⎟
1⎟
⎟⎟
1⎠
©2011 M. Tortorella
Slide 28
EXAMPLE
A JACKSON NETWORK
™Simplifies to
⎛ 0 a1 a1 + a2
⎜0 0
a2
⎜
⎜0 0
0
⎜
0
⎜0 0
⎜
⎜⎜
0
⎝0 0
DIMACS Workshop
13 October 2011
a1 + a2 + a3
a 2 + a3
a3
0
0
a1 +
+ a K −1 ⎞
a2 + + aK −1 ⎟⎟
a3 + + aK −1 ⎟
⎟
a4 + + aK −1 ⎟
⎟
⎟⎟
0
⎠
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 29
OVERVIEW
™Problem summary and basics
™Review of the traffic equation
™Path-additive functions
™Node-to-node values
‰
Cross-network values
™Three models of address-based routing
™Conclusion
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 30
ADDRESS-BASED
ROUTING
™Every arriving customer has a stated
destination
™Routers (nodes) contains lookup tables
that determine the next node given
The destination address and
‰ The current state of congestion in the
network
‰
™Incompatible with the Markovian routing
model
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 31
ADDRESS-BASED
ROUTING
™In address-based routing,
A customer leaves the network
immediately upon reaching destination
‰ A customer visits a given node at most
once
‰ Every origin has different routing tables
‰
™Can we mimic any of these
(approximately) in a Markovian routing
model?
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 32
CUSTOMER REPEATS
NO NODES
™Let tij(m) be the probability that a
customer reaches j from i in m steps
with no repeats
™Let Tij = tij(1) + tij(2) + ⋅⋅⋅ + tij(n)
™Let T be the matrix of the Tij
™Let B be the matrix whose (i, j) element
is rij − ∑ rik tkj( n )
k ≠i , j
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 33
CUSTOMER REPEATS
NO NODES
™Theorem: T = (I − R)−1B
™Note tij(n) = ∑ rσ σ rσ σ
σ∈Sn
1 2
2 3
rσn−1σn
™Let Mij(m) be the probability that a
customer starting from i reaches j for
the first time in m steps
™Of interest: |Mij(m) - tij(m)| and other
measures of how far M is from T
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 34
INDIVIDUAL ROUTING
MATRICES
™There is a separate routing matrix Ri for
each node i as an originating node
™Given X0 = i, the Markov chain is acyclic
™Requires treating each originating node
separately
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 35
CONCLUSION
™Generalized cross-network delay
Markovian routing
‰ Link delays
‰ Non-Poisson arrivals
‰ Non-exponential service times
‰
™Path-additive functionals
™Markovian approximations for addressbased routing
DIMACS Workshop
13 October 2011
PATH-ADDITIVE FUNCTIONALS
ON MARKOV-ROUTED NETWORKS
©2011 M. Tortorella
Slide 36