1. 1 CS SOR: Polling models Vacation models Multi type branching processes Polling systems (cycle times, queue lengths, waiting times, conservation laws, service policies, visit orders) J.A.C. Resing. Polling systems and multitype branching processes, Queueing Systems, 13, p 409 – 426 Richard J. Boucherie department of Applied Mathematics University of Twente

2. 2 CSSOR (lecture 4): polling as multi type branching process; work decomposition S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper, Robert B. Operations Research, Sep/Oct 1985, Vol. 33 Issue 5, p1117-1129 J.A.C. Resing. Polling systems and multitype branching processes, Queueing Systems, 13, p 409 – 426 I. Adan: Queueing Systems, lecture notes Polling model Multi-type branching process Polling model as branching process Work decomposition

3. 3 Polling models N infinite buffer queues, Q 1, …, Q N Service time distribution at queue j: B j (.), mean β j, LST β j (.) Poisson arrivals to queue j at rate λ j Single server in cyclic order Switch over times: random variable S j, mean σ j, LST σ j (.)

4. 4 Polling : offspring While served at queue i, customer is replaced by random population with p.g.f. Exhaustive gated

5. 5 Polling : offspring Assumption If the server arrives at Q i to find k i customers, then during the course of the servers visit, each of these k i customers is effectively replaced in an iid manner by a random population with p.g.f. Examples: exhaustive, gated Not included: 1-limited, because all but first have pgf s i

6. 6 CSSOR (lecture 4): polling as multi type branching process; work decomposition S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper, Robert B. Operations Research, Sep/Oct 1985, Vol. 33 Issue 5, p1117-1129 J.A.C. Resing. Polling systems and multitype branching processes, Queueing Systems, 13, p 409 – 426 I. Adan: Queueing Systems, lecture notes Polling model Multi-type branching process Polling model as branching process Work decomposition

7. 7 Multitype branching processes Polling model is MTBP with immigration Switchover times: immigration in each state No switchover times: immigration in state zero

8. 8 MTBP with immigration in each state

9. 9 Recall result for M/G/1 gated vacation!

10. 10 CSSOR (lecture 4): polling as multi type branching process; work decomposition S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper, Robert B. Operations Research, Sep/Oct 1985, Vol. 33 Issue 5, p1117-1129 J.A.C. Resing. Polling systems and multitype branching processes, Queueing Systems, 13, p 409 – 426 I. Adan: Queueing Systems, lecture notes Polling model Multi-type branching process Polling model as branching process Work decomposition

11. 11 Polling model with switchover times as MTBP

12. 12 Proof: ancestral line Consider two successive times server at Q1: t(n) and t(n+1). Let C A customer be typical customer present Collection of cust present at t(n+1) consists of replacements of customers present at t(n) and the replacements of customers C B arriving during switching intervals. Poisson arrivals and all serv disciplines satisfy property 1 implies MTBP with immigration in each state. Compute p.g.f.’s g and f Polling model with switchover times as MTBP

13. 13 Polling model with switchover times as MTBP

14. 14 Polling model with switchover timesas MTBP

15. 15 CSSOR (lecture 4): polling as multi type branching process; work decomposition S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper, Robert B. Operations Research, Sep/Oct 1985, Vol. 33 Issue 5, p1117-1129 J.A.C. Resing. Polling systems and multitype branching processes, Queueing Systems, 13, p 409 – 426 I. Adan: Queueing Systems, lecture notes Polling model Multi-type branching process Polling model as branching process Work decomposition

16. 16 Work conservation: multi class M/G/1 Amount of work in system independent of service discipline E[V]=E[V FCFS ] for any work conserving discipline Multi class queue, i=1,…,N arrival rates λ i service times B i, meanβ i, second moment β i, (2) finite Traffic intensity ρ=Σ i λ i β i Stability ρ<1

17. 17 Work conservation: multi class M/G/1 Observe system as single class M/G/1 Arrival rate λ=Σ i λ i Service times B distributed as B i w.p. λ i/ λ In particular β=E[B]=Σ i β i λ i/ λ=ρ/λ, β (2) =E[B 2 ]=Σ i β i (2) λ i/ λ Pollaczek-Khintchine: E[V FCFS ]=Σ i λ i β i (2) /[2(1-ρ)]

18. 18 Work conservation: conservation law We may decompose the total amount of work in the system: E[V]=Σ i E[V i ] with V i representing amount of class i work in system Assume FCFS in each class: E[V i ]=E[L i ] β i +ρ i E[R i ] with L i number of waiting class i customers at arbitrary epoch in equilibrium, excluding possible class i customer R i remaining service time of class i customer, if any

19. 19 Work conservation: conservation law Little’s law E[L i ] =λ i E[W i ] with W i the waiting time in equil of arbitrary class i cust, excluding its own service time Further, via renewal argument: E[R i ]= β i (2) /[2β i ]

20. 20 Conservation law Inserting gives So that Which is called conservation law If we want to decrease E[W i ] for some class, then we must increase E[W j ] for at least one other class

21. 21 Polling model Polling as M/G/1: single server visits all the queues Polling as M/G/1 vacation queue includes switching times

22. 22 Work decomposition: pseudo conservation law Expectations in (*) EV I =0: conservation law, weighted sum of waiting times does not depend on scheduling discipline Remains to compute EV I

23. 23 Work decomposition (Borst sec 2.3)

24. 24 Exercises Consider the Bernoulli type service discipline of Resing, 1993, Lemma 1. Let the service discipline at Q i be Bernoulli type. Obtain an expression for the distribution of the amount of work left behind by the server at Q i at the completion of a visit. Consider cyclic polling. Obtain the expression for the amount of work at arbitrary time during switching as given on slide 22 and the expression for Z_ii on slide 23 for gated service.