Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Examples Summary / Next Exercises

Last time: Reversibility and stationarity; various properties Definition: Reversible process: A stochastic process is reversible if for all t1,…,tn, Theorem: A stationary Markov chain is reversible if and only if there exists a collection of positive numbers π(j), jS, summing to unity that satisfy the detailed balance equations When there exists such a collection π(j), jS, it is the equilibrium distribution Theorem 1.13: Kelly’s lemma Let X(t) be a stationary Markov processwith transition rates q(j,k). If we can find a collection of numbers q’(j,k) such that q’(j)=q(j), jS, and a collection of positive numbers (j), jS, summing to unity, such that then q’(j,k) are the transition rates of the time-reversed process, and (j), jS, is the equilibrium distribution of both processes.

PASTA: Poisson Arrivals See Time Averages fraction of time system in state n probability outside observer sees n customers at time t probability that arriving customer sees n customers at time t (just before arrival at time t there are n customers in the system) PASTA

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Output simple queue Simple queue, Poisson() arrivals, exponential() service X(t) number of customers in M/M/1 queue: in equilibrium reversible Markov process. Forward process: upward jumps Poisson () Reversed process X(-t): upward jumps Poisson () = downward jump of forward process Downward jump process of X(t) Poisson () process

Output simple 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. 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

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Tandem network of simple queues Simple queue, Poisson() arrivals, exponential() service Equilibrium distribution Tandem of J M/M/1 queues, exp(i) service queue i Xi(t) number in queue i at time t Queue 1 in isolation: simple queue. Departure process queue 1 Poisson, thus queue 2 in isolation: simple 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

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Jackson network : Definition Simple queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates Traffic equations Irreducible, unique solution, interpretation, exercise Jackson network: open Gordon Newell network: closed

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Jackson network : Equilibrium distribution Simple queues, Transition rates Traffic equations Closed network Open network Global balance equations: Closed network: Open network:

closed network : equilibrium distribution Transition rates Traffic equations Closed network Global balance equations: Theorem: The equilibrium distribution for the closed Jackson network containing N jobs is Proof

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Partial balance Global balance verified via partial balance Theorem: If distribution satisfies partial balance, then it is the equilibrium distribution. Interpretation partial balance

Jackson network : Equilibrium distribution Transition rates Traffic equations Open network Global balance equations: Theorem: The equilibrium distribution for the open Jackson network containing N jobs is, provided αj<1, j=1,…,J, Proof

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Kelly / Whittle network Transition rates for some functions :S[0,),  :S(0,) Traffic equations Open network Partial balance equations: Theorem: Assume that then satisfies partial balance, and is equilibrium distribution Kelly / Whittle network

Examples Independent service, Poisson arrivals Alternative

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

Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

Summary / next: Equilibrium distributions Reversibility Output reversible Markov process Tandem network Jackson network Partial balance Kelly-Whittle network NEXT: Sojourn times

Exercises [R+SN] 2.1.1, 2.1.2, 2.3.1, 2.3.4, 2.3.5, 2.3.6, 2.4.1, 2.4.2, 2.4.6, 2.4.7