Networks of queues Networks of queues reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity,

Networks of queues Networks of queues reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's theorem, sojourn times Richard J. Boucherie Stochastic Operations Research department of Applied Mathematics University of Twente

Networks of Queues: lecture 2
Nelson, sec Last time on NoQ … Reversibility, stationarity Time reversed process Truncation of reversible processes Output M/M/1 queue Tandem network of M/M/1 queues Jackson network of M/M/1 queues Partial balance Kelly/Whittle network Examples Summary / Exercises

Reversibility; stationarity
Stationary process: A stochastic process is stationary if for all t1,…,tn,τ Theorem: If the initial distribution of a Markov chain is a stationary distribution, then a Markov chain is stationary Reversible process: A stochastic process is reversible if for all t1,…,tn,τ

Reversibility; stationarity
Lemma: A reversible process is stationary. Theorem: A stationary Markov chain is reversible if and only if there exists a collection of positive numbers π(j), jεS, summing to unity that satisfy the detailed balance equations When there exists such a collection π(j), jεS, it is the equilibrium distribution. Global balance:

Kolmogorov’s criteria
Theorem: A stationary Markov chain is reversible iff for each finite sequence of states. Furthermore, for each sequence for which the denominator is positive.

Theorem: Kelly’s lemma (Proposition 10
Theorem: Kelly’s lemma (Proposition 10.2) Let X(t) be a stationary Markov process with transition rates q(j,k). If we can find a collection of numbers q’(j,k) such that q’(j)=q(j), jεS, and a collection of positive numbers π(j), jεS, summing to unity, such that then q’(j,k) are the transition rates of the time-reversed process, and π(j), jεS, is the equilibrium distribution of both processes. Note: X reversible iff q’(j,k)=q(j,k) for all j,k Time reversed process

Truncation of reversible processes
Theorem: If the transition rates of a reversible Markov process with state space S and equilibrium distribution are altered by changing q(j,k) to cq(j,k) for where c>0, then the resulting Markov process is reversible in equilibrium and has equilibrium distribution where B is the normalizing cst. If c=0 then the reversible Markov process is truncated to A and the resulting Markov process is reversible with equilibrium distribution A S\A

Example: two M/M/1 queues
M/M/1 queue is reversible Consider two M/M/1 queues, queue i with Poisson arrival process rate λi, service rate μi Independence: Now introduce a common capacity restriction Queues no longer independent, but 2

M/M/1 queue, Poisson(λ) arrivals, exponential(μ) service
Output M/M/1 queue M/M/1 queue, Poisson(λ) arrivals, exponential(μ) service M/M/1 queue is reversible due to detailed balance X(t) number of customers in M/M/1 queue: in equilibrium reversible Markov process. Forward process: upward jumps Poisson (λ) Reversed process X(-t): upward jumps Poisson (λ) = downward jump of forward process Downward jump process of X(t) Poisson (λ) process

Output M/M/1 queue (2) Let t0 fixed. Arrival process Poisson, thus arrival process after t0 independent of number in queue at t0. For reversed process X(-t): arrival process after –t0 independent of number in queue at –t0 Reversibility: joint distribution departure process up to t0 and number in queue at t0 for X(t) have same distribution as arrival process to X(-t) up to –t0 and number in queue at –t0. Burkes theorem: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0 Holds for each reversible Markov process with Poisson arrivals as long as an arrival causes the process to change state

Tandem network of M/M/1 queues
M/M/1 queue, Poisson(λ) arrivals, exponential(μ) service Equilibrium distribution Tandem of J M/M/1 queues, exp(λi) service queue i Xi(t) number in queue i at time t Queue 1 in isolation: M/M/1 queue. Departure process queue 1 Poisson, thus queue 2 in isolation: M/M/1 queue State X1(t0) independent departure process prior to t0, but this determines (X2(t0),…, XJ(t0)), hence X1(t0) independent (X2(t0),…, XJ(t0)). Similar Xj(t0) independent (Xj+1(t0),…, XJ(t0)). Thus X1(t0), X2(t0),…, XJ(t0) mutually independent, and

Example: feed forward network of M/M/1 queues
4 3 2 5

2 4 3 5

2 4 3 5 Houston, we have a problem

Jackson network : Definition
M/M/1 queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates

Jackson network : Definition
M/M/1 queues, exponential service queue j, j=1,…,J Traffic equations Irreducible, unique solution, interpretation, exercise Jackson network: open Gordon Newell network: closed

Jackson network : Global balance equations
Closed network: Open network:

closed network : equilibrium distribution
Theorem: The equilibrium distribution for the closed Jackson network containing N jobs is Where (traffic equations) Proof

Theorem: The equilibrium distribution for the closed Jackson network containing N jobs is and satisfies partial balance traffic equations

Open network : equilibrium distribution
Theorem: The equilibrium distribution for the open Jackson network is and satisfies partial balance Where (traffic equations) Proof

Partial balance Detailed balance: Prob flow between each two states matches Partial balance: prob flow out of state n due to departure from queue j is balanced by prob flow into state n due to arrival to queue j, for each queue j, j=0,…,J Global balance: total prob flow out of state n equals total prob flow into state n

Partial balance Theorem: a distribution that satisfies partial balance is the equilibrium distribution

Kelly Whittle network State space S Transition rates
Where  is non-negative function, and φ positive function notation

Kelly Whittle network Theorem: The equilibrium distribution for the Kelly Whittle network is where and π satisfies partial balance

Insert equilibrium distribution and rates in partial balance
This is the beauty of partial balance!

Examples Independent service, Poisson arrivals
equilibrium distribution

Examples Simple queue s-server queue Infinite server queue
Each station may have different service type

Summary / next / exercises:
Output reversible queue Tandem network of queues Jackson network Kelly Whittle network Partial balance All customers identical Quasi reversibility, customer types BCMP networks Insensitivity Exercises: 1,6,9,10,11,13,14,20,21,22,24,25,26

Networks of queues Networks of queues reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity,

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