IBM System x Private Cloud Offering, Advanced - IBM Redbooks

1 Advanced Queueing Theory •  Networks of queues

(reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's theorem, sojourn times)

•  Analytical-numerical techniques (matrix-analytical methods, compensation method, error bound method, approximate decomposition method)

•  Polling systems (cycle times, queue lengths, waiting times, conservation laws, service policies, visit orders)

Richard J. Boucherie department of Applied Mathematics

University of Twente http://wwwhome.math.utwente.nl/~boucherierj/onderwijs/Advanced Queueing Theory/AQT.html

2 Advanced Queueing Theory Today (lecture 9): Polling models / branching

•  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

•  M/G/1 vacation model •  Branching processes •  Polling model •  Multi-type branching process •  Polling model as branching process •  Work decomposition

3

M/G/1 queue with server vacations

•  Consider M/G/1 queue •  Server takes a vacation (general distribution) at

arbitrary times: 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.

4

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

•  P.g.f. # customers present at start vacation •  P.g.f. # customers at arbitrary time in exhaustive

corresp, M/G/1 vacation model during vacation •  P.g.f. # customers at arbitrary time in corresp. M/G/1

5

Generalized vacations: examples

•  Exhaustive service: no customers at start vacation

•  M/G/1 with gated vacations: when server returns from vacation, a gate closes, and server serves only those customers present when gate closes, service of other customers deferred to next visit

6

Generalized vacations: Gated service

•  recall:

•  Off-spring: replace customer by arrivals during his service

•  Define

•  is p.g.f. number of the k-th generation offspring, and

7 Advanced Queueing Theory Today (lecture 9): Polling models / branching





8

Branching processes (Wolff, sec 3-9)

•  Let Yr (i.i.d) be the first generation off-spring of individual r

•  Xn n-th generation off-spring of particular individual

•  Pgf n-th generation off-spring individual

9

Branching processes (Wolff, sec 3-9)

•  Xn+1 is sum of descendants of the j individuals of the first generation:

10 Advanced Queueing Theory Today (lecture 8): Vacation models





11

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(.)

12

Polling models: 2 queues with gated service

•  Xij = # customers at Qj when server polls Qi

•  Yij = # customers at Qj when server leaves Qi

•  Generating functions

•  Clearly, with σi(.) the LST of switchover time after visit to Qi

in words: the number when server polls Q2 are the sums of the numbers when server left Q1 + the numbers of arrivals during the subsequent switch

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Fi(z1,z2) = E{z1X i1 z2

X i 2 }, i =1,2

Gi(z1,z2) = E{z1Yi1 z2

Yi 2 }, i =1,2

F2(z1,z2) =G1(z1,z2)σ1(λ1(1− z1) + λ2(1− z2)),F1(z1,z2) =G2(z1,z2)σ 2(λ1(1− z1) + λ2(1− z2))

13

Polling models: 2 queues Lemma

•  With the additional relation we have four foumulas

•  And we may express, via iteration

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G1(z1,z2) = F1(B1(λ1(1− z1) + λ2(1− z2)),z2)

F1(z1,z2) =G2(z1,z2)σ 2(λ1(1− z1) + λ2(1− z2))G2(z1,z2) = F2(z1,B2(λ1(1− z1) + λ2(1− z2)))F2(z1,z2) =G1(z1,z2)σ1(λ1(1− z1) + λ2(1− z2))G1(z1,z2) = F1(B1(λ1(1− z1) + λ2(1− z2)),z2)

Σ = λ1(1− z1) + λ2(1− z2)F1(k+1)(z1,z2) =σ 2(Σ)σ1(λ1(1− z1) + λ2(1− B2(Σ)))

× F1

(k )(B1(λ1(1− z1) + λ2(1− B2(Σ))),B2(Σ))

14

Polling : offspring

•  While served at queue i, customer is replaced by random population with p.g.f.

•  Exhaustive

•  gated

15

Polling models: 2 queues

•  Starting with the initial relations

•  Iterating, we get

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F1(z1,z2) =G2(z1,z2)σ 2(λ1(1− z1) + λ2(1− z2))G2(z1,z2) = F2(z1,h2(z1,z2))F2(z1,z2) =G1(z1,z2)σ1(λ1(1− z1) + λ2(1− z2))G1(z1,z2) = F1(h1(z1,z2)),z2)

Σ = λ1(1− z1) + λ2(1− z2)F1(k+1)(z1,z2) =σ 2(Σ)σ1(λ1(1− z1) + λ2(1− h2(z1,z2)))

F1

(k )(h1(z1,h2(z1,z2)),h2(z1,z2))

16

Polling : offspring

•  While served at queue i, customer is replaced by random population with p.g.f.

•  Exhaustive

•  gated

17

Polling : offspring

•  Assumption If the server arrives at Qi to find ki customers, then during the course of the servers visit, each of these ki 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 si

•  Bernoulli-type. When server arrives all customers present are handled as follows: customer is served, and all offspring served with probability pi.

20

Multitype branching processes

•  Polling model is MTBP with immigration

•  Switchover times: immigration in each state

•  No switchover times: immigration in state zero

21

MTBP with immigration in each state

22


•  Recall result for M/G/1 gated vacation!

23


•  Each particle replaced by its off-spring, and independently by immigration

24

MTBP with immigration in state zero

•  Immigration only in state zero, not in all states

25


26


28

Polling model with switchover times as MTBP

29

•  Proof: ancestral line Consider two successive times server at Q1: t(n) and t(n+1). Let CA 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 CB 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


30


31

Polling model with switchover timesas MTBP

32

Polling model without switchover times as MTBP

34

Work conservation: multi class M/G/1

•  Amount of work in system independent of service discipline

•  E[V]=E[VFCFS] for any work conserving discipline

•  Multi class queue, i=1,…,N arrival rates λi service times Bi, meanβi, second moment βi,

(2) finite •  Traffic intensity ρ=Σi λiβi

•  Stability ρ<1

35

Work conservation: multi class M/G/1

•  Observe system as single class M/G/1

•  Arrival rate λ=Σi λi •  Service times B distributed as Bi w.p. λi/λ •  In particular β=E[B]=Σi βiλi/λ=ρ/λ, β(2)=E[B2]=Σi βi

(2) λi/λ

•  Pollaczek-Khintchine:

E[VFCFS]=Σi λiβi(2) /[2(1-ρ)]

36

Work conservation: conservation law

•  We may decompose the total amount of work in the system:

E[V]=Σi E[Vi]

with Vi representing amount of class i work in system

•  Assume FCFS in each class:

E[Vi]=E[Li] βi+ρi E[Ri]

with Li number of waiting class i customers at arbitrary epoch in equilibrium, excluding possible class i customer Ri remaining service time of class i customer, if any

37

Work conservation: conservation law

•  Little’s law E[Li] =λiE[Wi]

with Wi the waiting time in equil of arbitrary class i cust, excluding its own service time

•  Further, via renewal argument: E[Ri]= βi(2) /[2βi]

€

E[V ] =i=1

N

∑ λiE[Wi]β i +i=1

N

∑ ρiβi(2)

2β i

=i=1

N

∑ ρiE[Wi]+12 i=1

N

∑ λiβ i(2)

38

Conservation law

€

i=1

N

∑ ρiE[Wi]+12 i=1

N

∑ λiβ i(2) = i=1

N

∑ λiβi(2)

2(1− ρ)

i=1

N

∑ ρiE[Wi] = ρ i=1

N

∑ λiβ i(2)

2(1− ρ)

• Inserting gives

• So that

• Which is called conservation law • If we want to decrease E[Wi] for some class, then we must increase E[Wj] for at least one other class

39

Polling model

• Polling as M/G/1: single server visits all the queues

•  Polling as M/G/1 vacation queue includes switching times

40

Work decomposition: pseudo conservation law

•  Expectations in (*)

•  EVI =0: conservation law, weighted sum of waiting times does not depend on scheduling discipline

•  Remains to compute EVI

41

Work decomposition (Borst sec 2.3)

42

Exercise

•  Consider the Bernoulli type service discipline of Resing, 1993, Lemma 1. Let the service discipline at Qi be Bernoulli type. Obtain an expression for the distribution of the amount of work left behind by the server at Qi at the completion of a visit.

•  Consider the vacation queue with Bernoulli service type service discipline. Obtain an expression for the LST of the amount of work in the queue, ς(z), when the server leaves on vacation.

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