Chapter 6 Product-Form Queuing Network Models

Prof. Ali Movaghar

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Product-Form Queuing Networks Consider a modular system consisting of

multiple subsystems The more independent subsystems, the

easier overall system analysis For example, each subsystem can be modeled as

a queue and the overall system analysis can be obtained from the product of terms corresponding to each queue!

We call such queuing networks product-form networks.

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Product-Form Queuing Networks (Con.) There are several properties that lead to

product-form (PF) queuing networks Local balance Reversibility Quasi-reversibility Station balance MM

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Local Balance

The effective rate at which the system leaves state S due to the service completion (of a chain r customer) at station i equals the effective rate at which the system enters state S due to an arrival (of chain r customers) to station i

n n+1

Effective arrival rate

Effective service completion

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Reversible Markov Process

Consider a film we make from the behavior of an isolated service station and then run the film backward The departing customers in the real system will be

seen as arrival in backward run. Let X(t) denote the random process describing the

number of customers at the station Let Xb(t) denote the corresponding process in the

backward run of the film If X(t) and Xb(t) are statistically identical, we say

X(t) is reversible.

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Reversible Markov Process (Con.) A random process X(t) is reversible,

If for any finite sequence of time instants t1, …, tk and a parameter , the joint distribution of X(t1), …, X(tk) is the same as that of X(-t1),…, X(-tk)

M/M/c queue is a reversible system M/M/1 queue with batch arrival is not reversible

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Reversible Markov Process (Con.) A reversible process must be stationary

X(+t1),…, X(+tk) has the same distribution as

X(-t1),…, X(-tk) which has the same distribution as X(t1),…, X(tk).

Thus all joint distribution are independent of time shift as required for stationary

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Reversible Markov Process (Con.) Reversible Markov processes can be

characterized by an important property known as detailed balance.

Lemma : Let X(t) be a stationary, discrete parameter Markov chain with one-step transition matrix Q[q(i,j)]. If X(t) is reversible than it satisfy detailed balance equation :

P(i)q(i,j) = P(j) q(j,i)

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Reversible Markov Process (Con.) Similarly, a continuous Markov chain is

reversible if i,j. P(i)qi,j = P(j) qi,j where qi,j is the transition rates.

Conversely if we can find probabilities P(i)’s with P(i)=1 satisfying detailed balance property, then X(t) is reversible and P(i)’s form its stationary distribution.

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Reversible Markov Process (Con.) Lemma (Kolmogorove criterion)

A stationary Markov chain is reversible if and only if for every finite sequence of states i0, …, ik the transition probability (or rates in the continuous parameter case) satisfy the following equation:

q(i0, i1) q(i1, i2)…q(ik-1, ik) q(ik, i0)=

q(ik, ik-1) q(ik-1, ik-2)…q(i1, i0) q(i0, ik)

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Reversible Markov Process (Con.) Lemma: Let X(t) denote an ergodic Markov

chain with transition matrix Q. Then X(-t) has the same stationary distribution as X(t). Let Q*=[q*(i,j)] denote the transition matrix of X(-t), then :

q*(i,j)=P(j)q(j,i)/P(i)

In other words, for a reversible process Q=Q*

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Reversible Markov Process (Con.) Example : Consider M/M/c queue

Global balance results P(n)(λ+cμ)=P(n-1)λ+P(n+1)cμ Local balance results P(n)(λ)=P(n+1)cμ Local balance is contained in global balance

Thus Piqi,j = Pj qj,i and is reversible Number of leaves has Poisson distribution with rate λ

1

λ

0

μ

n

λ

c

cμ

n+1

λ

cμ

…

…

λ

2μ

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Reversible Markov Process (Con.) Reversibility of M/M/c results in following

lemma. The departure process of M/M/c system is

Poisson and the number in the queue at any time t is independent of the departure process prior to t.

Thus we can easily analyze feed-forward network of M/M systems

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Reversible Markov Process (Con.) Example : Find P(n1, n2)

P(n1) = (1-1)n1

P(n2) = (1-2)n2

The arrival to second queue (departure from first queue) is independent of number of customers at first queue (Previous Lemma). P(n1, n2) = (1-1) (1-2) n1 n2

M/M/cλ μ1

M/M/cμ2

Number of customers = n1 Number of customers = n2

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Reversible Markov Process (Con.) Example

P(n1, n2,n3) = (1-1) (1-2) (1-3) n1 n2 n3

Avg. Queue Length = 1/(1- 1)+ 2/(1- 2)+ 3/(1- 3) W = Avg. Queue Length /λ What if there is a feed-back from third queue to first queue:

Quasi-reversibility preserves product-form property

M/M/cλ μ1

M/M/cμ2

n1 n2

M/M/cμ3

n3

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Quasi-Reversible Queuing Systems Let X(t) denote the queue length process at a

queuing system. Then X(t) is quasi-reversible if for any time instant t0, X(t0) is independent of Arrival time of customers after t0 Departure times of customer prior to t0

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Open Single-Chain Product-Form Networks

qsi : Probability that an external arrival is directed to station i (similarly qid is defined)

(n) : Generation of customers at source with Poisson distribution .

i(n): External arrival to station i (qsi(n)= i(n)) μi : basic service rate of station i Ci(n) : capacity of station i at load n

i

j

src sink

Networkqs1

qs2

qsM

q1d

q2d

qMd

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Open Single-Chain Product-Form Networks (Con.)

vi : the visit ratio of station i relative to the source node λi(n) : the arrival rate of station i when there n customers in

the entire network (λi(n) = vi (n) ) Visit ratio to source and so destination should be 1 By flow balance we have :

M M

si i idi 1 i 1

M M

id ij i si j jij 1 j 1

q 1 and v q 1

q 1 q and v q v q 1

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Open Single-Chain Product-Form Networks (Con.)

n=(n1,…,nM) : state of network where ni is the number of customers at station i.

ei=(0,…,0,1,0,…,0) : 1 at position i The rate at which the system leaves state n due to

an arrival or completion at station i is qsi(n)+ μi(ni)qid+ μi(ni)qij or qsi(n)+μi(ni)

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Open Single-Chain Product-Form Networks (Con.)

The global balance equation results

n

n-ei+ej

n-ei n+ei

μj(nj+1)qij μi(ni)qij

(n-1)qsi

μi(ni)qid

(n)qsi

μi(ni+1)qid

M M M

si i i j j i j jii 1 i 1 j 1

M

i si i i i idi 1

P(n) [ (n)q (n )] (n 1)P(n e e )q

(n 1)P(n e )q (n 1)P(n e )q

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Open Single-Chain Product-Form Networks (Con.) Since all stations are individually qausi-

reversible, the entire network should be qausi-reversible :

μi(ni)P(n)= λi(n-1)P(n-ei) , i=1,..,M We can find P(n) for station i and then the

desired product-form solution for network: inn 1 Mi

k 0 i 1 i i

uP(n) (k) P(0)V (n )