1. Richard J. Boucherie department of Applied Mathematics
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. Polling models N infinite buffer queues, Q1, …, QN
Service time distribution at queue j: Bj(.), mean βj, LST βj(.)
Poisson arrivals to queue j at rate λj
Single server in cyclic order
Switch over times: random variable Sj, mean σ j, LST σj(.)
This week: vacation queues
Next week….. Multi-type branching processes (Resing)
In two weeks: polling
Then: 3 presentations by you

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, p
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
Polling model

4. M/G/1 queue
Poisson arrival process rate λ, single server, General service times, mean 1/μ=E(B)
Service time distribution FB(.), density fB(.)
state: pair (n,x), n=#customer, x= received service time
# customers left behind by k-th departing cust.
# customers in the system at time t
# customers seen by k-th arriving customer
PASTA & frequency argument:

5. M/G/1 queue # arrivals during k-th service time
Embedded Markov chain at departure epochs pr n arrivals during service

6. M/G/1 queue Equilibrium equations Gen. functions, for z<1,
From equil. eq.
Hence

7. M/G/1 queue Furthermore Pollaczek-Khintchine formula for queue length
By analogy for sojourn time
And for Waiting time

8. 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, p
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
Polling model

9. M/G/1 queue with server vacations
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
Queue length? Sojourn time?
Mild conditions: in distribution N=NM/Q/1 + NI V=VM/Q/1 + VI
NM/Q/1 number at arbitrary moment in corresp M/G/1 NI number at arbitrary moment in non-serving interval Independent r.v.

10. Ancestral line I0 group of customers
I1 set of customers who arrive while members of I0 are being served: first generation offspring of I0
Ik, k>1 set of customers who arrive while members of Ik-1 are being served: k-th generation offspring of I0
ancestral line of I0
For us: I0 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.

11. Exhaustive service No preemption: # cust under FIFO = # under LIFO
Assume LIFO
Tag arbitrary customer C upon departure let A be its ancestor
present at departure of C are all vacation customers that arrived before A, and all remaining customers in ancestral line of A
A has started busy period (that ends with dept of A), C is arbitrary cust in this busy period, this busy period equivalent to standard busy period, thus number of customers in ancestral line of A left behind by C = number of customers left behind in standard M/G/1 queue.

12. Exhaustive service
present at departure of C are all vacation customers that arrived before A, and all remaining customers in ancestral line of A
C is arbitrary customer, all vacation customers are iid, therefore customer C corresponds to arbitrarily selected vacation customer. Thus : number of vacation customers left behind = number of customers at arbitrary epoch during vacation.
Theorem: Queue length decomposition N=NM/G/1 + NI where equality is in distribution, and N:= queue length at arbitrary epoch NM/G/1 := queue length at arbitrary epoch in corresponding M/G/1 NI := queue length at arbitrary epoch in vacation period NM/G/1 and NI are independent r.v.

13. Exhaustive service
NI distrib as # cust seen by arbitr arrival in vacation period (PASTA)
Nend,k # customers end k-th vac period, Nend generic rv.
Lemma:
Proof

14. Exhaustive service Alternative formulation:
ψ(.) = p.g.f. stat distrib # cust random dept cust leaves behind or finds or at arbitrary 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)=

15. 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, p
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
Polling model

16. 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.

17. 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

18. Generalized vacations
Decomposition property holds for any vacation system under assumptions stated.
Theorem: Consider a random (tagged) customer C. Let A the ancestor, I0 the set of vacation customers who arrived during the same vacation to which A arrived. Let I the ancestral line of I0, and let X the number of members of I present in system when C departs. X has p.g.f.
Proof:

19. 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.

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

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. 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, p
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
Polling model

23. Polling models N infinite buffer queues, Q1, …, QN
Service time distribution at queue j: Bj(.), mean βj, LST βj(.)
Poisson arrivals to queue j at rate λj
Single server in cyclic order
Switch over times: random variable Sj, mean σ j, LST σj(.)
Next week…..
Multi-type branching processes
and polling (Resing)