1. Flows and Networks Plan for today (lecture 3): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

2. 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), 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

3. Poisson process Definition : Poisson process Let S1,S2,… be a sequence of independent exponential(  ) r.v. Let Tn=S1+…+Sn, T0=0 and N(s)=max{n,Tn≤s}. The counting process {N(s),s≥0} is called Poisson process. Theorem : If {N(s),s≥0} is a Poisson process, then (i) N(0)=0, (ii) N(t+s)-N(s)=Poisson(  t), and (iii) N(t) has independent increments. Conversely, if (i), (ii), (iii) hold, then {N(s),s≥0} is a Poisson process

4. 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) in general For Poisson arrivals PASTA

5. Flows and Networks Plan for today (lecture 3): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

6. Output simple queue Simple queue, Poisson(  ) arrivals, exponential(  ) service X(t) number of customers in M/M/1 queue Equilibrium distribution satisfies detailed balance X(t) 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

7. Output simple queue (2) Let t 0 fixed. Arrival process Poisson, thus arrival process after t 0 independent of number in queue at t 0. For reversed process X(-t): arrival process after –t 0 independent of number in queue at –t 0 Reversibility: joint distribution departure process up to t 0 and number in queue at t 0 for X(t) have same distribution as arrival process to X(-t) up to –t 0 and number in queue at –t 0. In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t 0 is independent of the departure process prior to t 0 Holds for each reversible Markov process with Poisson arrivals as long as an arrival causes the process to change state

8. Flows and Networks Plan for today (lecture 3): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

9. 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 X i (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 X 1 (t 0 ) independent departure process prior to t 0, but this determines (X 2 (t 0 ),…, X J (t 0 )), hence X 1 (t 0 ) independent (X 2 (t 0 ),…, X J (t 0 )). Similar X j (t 0 ) independent (X j+1 (t 0 ),…, X J (t 0 )). Thus X 1 (t 0 ), X 2 (t 0 ),…, X J (t 0 ) mutually independent, and

10. 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

11. 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

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

13. 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

14. 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

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

16. 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 α j <1, j=1,…,J, Proof

17. 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

18. 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

19. Examples Independent service, Poisson arrivals Alternative

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

21. 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

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

23. 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