1. 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 Product form preserving blocking Summary / Next Exercises

2. Last time: output simple queue; partial balance 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 Global balance; partial balance;detailed balance, traffic equations

3. 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

4. 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

5. 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

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

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. Examples Independent service, Poisson arrivals Alternative

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

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. 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,…

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

18. 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

19. 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

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

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

22. 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, …

23. 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

24. 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

25. 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

26. Jackson network : Mean sojourn time Simple queues, FCFS, Transition rates Traffic equations Open network Global balance equations: Open network: Sojourn time in each queue: Sojourn time on path i,j,k:

27. 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 Product form preserving blocking Summary / Next Exercises

28. Blocking in tandem networks of simple queues (1) Simple queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates Traffic equations Solution

29. Blocking in tandem networks of simple queues (2) Simple queues, exponential service queue j, j=1,…,J Transition rates Traffic equations Solution Equilibrium distribution Partial balance PICTURE J=2

30. Blocking in tandem networks of simple queues (3) Simple queues, exponential service queue j, j=1,…,J Suppose queue 2 has capacity constraint: n2<N2 Transition rates Partial balance? PICTURE J=2 Stop protocol, repeat protocol, jump-over protocol

31. 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 Product form preserving blocking Summary / Next Exercises

32. Summary / next: Waiting times / sojourn times Distribution Little’s law Mean Blocking in Jackson network Partial balance Product form preserving blocking protocols NEXT: Optimization / applications

33. Exercises [R+SN] 1.3.3, 2.2.2, 2.2.4, 2.2.5, 2.2.6