Networks of queues

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 10.1-10.3.5 • 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 equationsWhen 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 ifffor each finite sequence of states. Furthermore, for each sequence for which the denominator is positive.

Time reversed process • 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 thatthen 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

S\A A Truncation of reversible processes Theorem: If the transition rates of a reversible Markov process with state space S and equilibrium distributionare 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

Example: two M/M/1 queues 1 • 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 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

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 Output M/M/1 queue (2)

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 Tandem network of M/M/1 queues

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

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

1 4 3 2 5 Example: feed forward network of M/M/1 queues Houston, we have a problem

M/M/1 queues, exponential service queue j, j=1,…,J statemovedepartarrive 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 : Definition

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

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

closed network : equilibrium distribution

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

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

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 πsatisfies partial balance

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

Independent service, Poisson arrivals equilibrium distribution Examples

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

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