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
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1 CS SOR: Polling models Vacation models Multi type branching processes Polling systems (cycle times, queue lengths, waiting times, conservation laws,
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CS SOR: Polling models• Vacation models• Multi type branching processes• Polling systems
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
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.
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)
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
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
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.
)()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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Generalized vacations: Gated service
• Assume limiting distribution of Xn n ∞ exists, with pgf X(z), then
• 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
11
)()(
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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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
))(()(
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)