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) Richard J. Boucherie department of Applied Mathematics University of Twente

2. 2 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 (.) This week: vacation queues Next week….. Multi-type branching processes (Resing) In two weeks: polling Then: 3 presentations by you

3. 3 Polling models Today: Vacation models S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper. 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 M/G/1 queue, PK formula M/G/1 queue with vacations M/G/1 queue with Generalized vacations

4. 4 M/G/1 queue Poisson arrival process rate λ, single server, General service times, mean 1/μ=E(B) Service time distribution F B (.), density f B (.), LST B state: pair (n,x), n=#customer, x= received service time Pollaczek-Khintchine formula for queue length

5. 5 Polling models Today: Vacation models S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper. 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 M/G/1 queue, PK formula M/G/1 queue with vacations M/G/1 queue with Generalized vacations

6. 6 Ancestral line I 0 group of customers I 1 set of customers who arrive while members of I 0 are being served: first generation offspring of I 0 I k, k>1 set of customers who arrive while members of I k-1 are being served: k-th generation offspring of I 0 ancestral line of I 0 For us: I 0 single customer, or set of customers arriving during some vacation Vacation customer: arrive while server is on vacation To each customer, C, corresponds a unique vacation customer, A, such that C is in ancestral line of A: A is ancestor of C.

7. 7 Vacation queue: exhaustive service Consider M/G/1 queue Server takes a vacation (general distribution) when the system becomes idle Upon return from vacation, if system idle new vacation otherwise serve until system idle: exhaustive service No preemption Theorem: Queue length decomposition N=N M/G/1 + N I where equality is in distribution, and N:= queue length at arbitrary epoch N M/G/1 := queue length at arbitrary epoch in corresponding M/G/1 N I := queue length at arbitrary epoch in vacation period N M/G/1 and N I are independent r.v.

8. 8 Vacation queue: Exhaustive service Alternative formulation: ψ(.) = p.g.f. stat distrib # cust random time π(.) = p.g.f. idem in corresp M/G/1 queue α(.) = p.g.f. # arrivals during vacation period = Vac(λ-λz) π(z)=

9. 9 Polling models Today: Vacation models S.C. Borst. Polling Systems, CWI: Tract, chapters 1 – 3. S.W. Fuhrmann, R.B. Cooper. 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 M/G/1 queue, PK formula M/G/1 queue with vacations M/G/1 queue with Generalized vacations

10. 10 Generalized vacations: assumptions Server can take vacation at any time: vacation = server unavailable Service time independent of sequence of vacation periods preceding that service time. Order of service independent of service times. Service non-preemptive Rules that govern when server begins and ends vacations do not anticipate future jumps of the arrival process.

11. 11 Generalized vacations: examples Standard vacation model: vacation upon idling N-policy: server waits until exactly N customers present before starting service, then work continues until system empty M/G/1 with gated vacations: when server returns from vacation, he accepts only those customers waiting upon return, service of other customers deferred to next visit Limited service: serve at most k customers Polling model Priority system

12. 12 Generalized vacations Decomposition property holds for any vacation system under assumptions stated. Theorem: Consider a random (tagged) customer C. Let A the ancestor, I 0 the set of vacation customers who arrived during the same vacation to which A arrived. Let I the ancestral line of I 0, and let X the number of members of I present in system when C departs. X has p.g.f. Proof: there may have been other customers in system at start of vacation. Due to LIFO these will remain in system, rest as proof last time

13. 13 Generalized vacations Assume that the number of customers that arrive during a vacation is independent of the number of customers present in the system when vacation began. Theorem: Under this assumption Proof: LIFO, p.g.f. of number of customers already present at beginning of vacation. Independence yields product.

14. 14 Generalised vacations Theorem: Queue length decomposition N=N M/G/1 + N I where equality is in distribution, and N:= queue length at arbitrary epoch N M/G/1 := queue length at arbitrary epoch in corresponding M/G/1 N I := queue length at arbitrary epoch in vacation period N M/G/1 and N I are independent r.v. Theorem: Work decomposition V=V M/G/1 + V I where equality is in distribution, and V:= work at arbitrary epoch V M/G/1 := work at arbitrary epoch in corresponding M/G/1 V I := work at arbitrary epoch in vacation period V M/G/1 and V I are independent r.v.

15. 15 Generalized vacations: Gated service Gated service: At time t n server finds X n customers. Gate closes. Server serves those X n then takes vacation of length V n and returns at time t n+1 Recall Recursion Markov chain; from recursion:

16. 16 Generalized vacations: Gated service Assume limiting distribution of X n n  ∞ exists, with pgf X(z), then Let then

17. 17 Generalized vacations: Gated service Let M  ∞ in resulting infinite product exists if ρ<1, and Hence, if ρ<1

18. 18 Convergence hence, indeed if ρ<1 Observe So that infinite product indeed converges

19. 19 Interpretation: Branching processes (Wolff, sec 3-9) Let Y r (i.i.d) be the first generation off-spring of individual r X n n-th generation off-spring of particular individual Pgf n-th generation off-spring individual

20. 20 Branching processes (Wolff, sec 3-9) X n+1 is sum of descendants of the j individuals of the first generation:

21. 21 Generalized vacations: Gated service Gated service: recall: Define is p.g.f. number of the k-th generation offspring first previous gated period, second previous gated period

22. 22 Generalized vacations Server can take vacation at any time Service time independent of sequence of vacation periods preceding that service time. Order of service independent of service times. Service non-preemptive Rules that govern when server begins and ends vacations do not anticipate future jumps of the arrival process. Number of customers that arrive during a vacation is independent of the number of customers present in the system when vacation began. Theorem:

23. 23 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 (.) Next week….. Multi-type branching processes and polling (Resing)