1. 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

2. Flows and Networks Plan for today (lecture 3): Last time / Questions? Reversibility, stationarity Truncation Kolmogorov’s criteria Poisson process PASTA Output simple queue Tandem network Summary / Next Exercises

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. Flows and Networks Plan for today (lecture 3): Last time / Questions? Reversibility, stationarity Truncation Kolmogorov’s criteria Poisson process PASTA Output simple queue Tandem network Summary / Next Exercises

5. 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

6. PASTA: Poisson Arrivals See Time Averages Let C(t,t+h) event customer arrives in ( t,t+h) For Poisson arrivals q(n,n+1)=  so that Alternative explanation; PASTA holds in general! PASTA

7. Flows and Networks Plan for today (lecture 3): Last time / Questions? Reversibility, stationarity Truncation Kolmogorov’s criteria Poisson process PASTA Output simple queue Tandem network Summary / Next Exercises

8. 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

9. 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

10. Flows and Networks Plan for today (lecture 3): Last time / Questions? Reversibility, stationarity Truncation Kolmogorov’s criteria Poisson process PASTA Output simple queue Tandem network Summary / Next Exercises

11. 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 (t0) independent departure process prior to t0, but this determines (X 2 (t0),…, X J (t0)), hence X 1 (t0) independent (X 2 (t0),…, X J (t0)). Similar X j (t0) independent (X j+1 (t0),…, X J (t0)). Thus X 1 (t0), X 2 (t0),…, X J (t0) mutually independent, and

12. Flows and Networks Plan for today (lecture 5): Last time / Questions? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean sojourn time Summary / Next Exercises

13. Waiting time simple queue (1) Consider simple queue FCFS discipline – W : waiting time typical customer in M/M/1 (excludes service time) –N customers present upon arrival –S r (residual) service time of customers present PASTA Voor j=0,1,2,…

14. Waiting time simple queue (2) Thus is exponential (  -  )

15. Flows and Networks Plan for today (lecture 5): Last time / Questions? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean sojourn time Summary / Next Exercises

16. Little’s law (1) Let – A(t) : number of arrivals entering in (0,t] – D(t) : number of departure from system (0,t] – X(t) : number of jobs in system at time t Equilibrium for t  ∞ In equilibrium: average number of arrivals per time unit = average number of departures per time unit

17. Little’s law (2) F j sojourn time j-th departing job S(t) obtained sojourn times jobs in system at t

18. Assume following limits exist (ergodic theory, see SMOR) Then Little’s law Little’s law (3)

19. Little’s law (4) Intuition –Suppose each job pays 1 euro per time unit in system –Count at arrival epoch of jobs: job pays at arrival for entire duration in system, i.e., pays EF –Total average amount paid per time unit  EF –Count as cumulative over time: system receives on average per time unit amount equal to average number in system –Amount received per time unit EX Little’s law valid for general systems irrespective of order of service, service time distribution, arrival process, …

20. Flows and Networks Plan for today (lecture 5): Last time / Questions? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean sojourn time Summary / Next Exercises

21. Recall: 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 Theorem 2.2: If service discipline at each queue in tandem of J simple queues is FCFS, then in equilibrium the waiting times of a customer at each of the J queues are independent Proof: Kelly p. 38 Tandem M/M/s queues: overtaking Distribution sojourn time: Ex 2.2.2 Sojourn time tandem simple queues

22. 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

23. 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

24. 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

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

26. 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

27. 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

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

29. 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

30. 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

31. 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

32. Examples Independent service, Poisson arrivals Alternative

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

34. 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

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

36. 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