1 CS SOR: Polling models Vacation models Multi type branching processes Polling systems (cycle...

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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. Boucheriedepartment of Applied Mathematics

University of Twente

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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

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Polling modelsToday: 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

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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(.), LST B

• state: pair (n,x), n=#customer, x= received service time

• Pollaczek-Khintchine formulafor queue length

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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.

kkI

0

7Vacation 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=NM/G/1 + NI

where equality is in distribution, and N:= queue length at arbitrary epochNM/G/1 := queue length at arbitrary epoch in corresponding M/G/1NI := queue length at arbitrary epoch in vacation periodNM/G/1 and NI are independent r.v.

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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)=

)()1(')1(

)(1)( z

z

zz

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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.

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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

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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: 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

)()1(')1(

)(1)( z

z

zz

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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.

)()1(')1(

)(1)()( z

z

zzz

14Generalised vacations

• Theorem: Queue length decomposition N=NM/G/1 + NI

where equality is in distribution, and N:= queue length at arbitrary epochNM/G/1 := queue length at arbitrary epoch in corresponding M/G/1NI := queue length at arbitrary epoch in vacation periodNM/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 epochVM/G/1 := work at arbitrary epoch in corresponding M/G/1VI := work at arbitrary epoch in vacation periodVM/G/1 and VI are independent r.v.

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Generalized vacations: Gated service

• Gated service: At time tn server finds Xn customers. Gate closes. Server serves those Xn then takes vacation of length Vn and returns at time tn+1

• Recall

Recursion

Markov chain; from recursion:

€

B(λ − λz) = PA (z)

Xn+1 = A(I=1

X n

∑ Bi) + A(Vn )

E[zX n+1 ] =V (λ − λz)E[(B(λ − λz))X n ]

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• Assume limiting distribution of Xn n ∞ exists, with pgf X(z), then

Let

then

€

X(z) =V (λ − λz)X(B(λ − λz))

h(z) := B(λ − λz)

X(z) =V (λ (1− z))X(h(z))

=V (λ (1− z))V (λ (1− h(1)(z))X(h(2)(z))

=j= 0

M

∏ V (λ (1− h( j )(z))X(h(M +1)(z))

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• Let M ∞ in

resulting infinite product exists if ρ<1, and

• Hence, if ρ<1

€

X(z) =j= 0

M

∏ V (λ (1− h( j )(z))X(h(M +1)(z))

h(M +1)(z) →1

X(z) =j= 0

∞

∏ V (λ (1− h( j )(z))

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Convergence

hence, indeed if ρ<1

• Observe

• So that infinite product indeed converges

€

|1− h( j )(z) |=|1− h(h( j−1)(z)) |≤ ρ |1− h( j−1)(z) |

h(M +1)(z) →1

|1−V (λ (1− h( j )(z)) |≤ λE[V ] |1− h( j )(z) |≤ λE[V ]ρ j |1− z |

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Interpretation: 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

jj

j

Xn

ijijnn

jjnn

jr

zzAzA

zA

XzEzA

jipiXjXp

pXjXp

jjYP

n

01

0

0

*1

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)()(

1)(

}1|{)(

,...1,0,,)|(

)1|(

,...,1,0,)(

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Branching processes (Wolff, sec 3-9)

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

))((}1|)]({[)(

)]([}|{}1,|{

01

10111

zAAXzAEzA

zAjXzEXjXzE

nj

nn

jn

XX nn

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• Gated service: recall:

Define

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

first previous gated period, second previous gated period

))(()(

2 )),(()(

)()(

)()(

)(

1

)1()1()(

)1(

zRz

kzRRzR

zBzR

zPzB

k

k

kk

A

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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:)(

)1(')1(

)(1)()( z

z

zzz

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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)

1 CS SOR: Polling models Vacation models Multi type branching processes Polling systems (cycle...

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