networks of queues
DESCRIPTION
Networks of queues. 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. Richard J. Boucherie Stochastic Operations Research - PowerPoint PPT PresentationTRANSCRIPT
1Networks of queues
• 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
Richard J. Boucherie
Stochastic Operations Researchdepartment of Applied Mathematics
University of Twente
2
Networks of Queues: lecture 4 MSc assignment: ambulance planning …
3
Networks of Queues: lecture 4 Nelson, sec 10.3.5-10.6.6• Last time on NoQ …
– Jackson network– Kelly-Whittle network– Partial balance– Time reversed process– Product form
• Quasi reversibility• Network of quasi reversible queues• Queue disciplines, Symmetric queues, BCMP networks• Summary / Exercises
4Last time on NoQ: Jackson network : Definition
• M/M/1 queues, exponential service queue j, j=1,…,J
• state
move
depart
arrive
• Transition rates
),...,1,...,(
),...,1,...,(
),...,1,...,1,...,(position at 1 )0...,0,1,0,...0(
),...,(
1
1
1
1
Jkk
Jjj
Jjiji
i
J
nnnen
nnnen
nnnneenie
nnn
kk
jjj
jkjkj
ennq
pennq
peennq
),(
),(
),(
0
5Last time on NoQ :closed network : equilibrium distribution
• Theorem: The equilibrium distribution for the closed Jackson network containing N jobs is
and satisfies partial balance
traffic equations
}:{/)(1
NnnSnBn jj
Njjjnj
J
jN
j
kk
jjj
jkjkj
ennq
pennq
peennq
),(
),(
),(
0
kjkk
jkjk
pp
),()(),()(11
neenqeeneennqn kj
J
kkjkj
J
k
6Last time on NoQ : Open network : equilibrium distribution
Theorem: The equilibrium distribution for the open Jackson network is
and satisfies partial balance
},...,1,0:{/)(1
JjnnnBn jjjjnj
J
j
j
),()(),()(00
neenqeeneennqn kj
J
kkjkj
J
k
7Last time on NoQ : Partial balance
• Detailed balance: Prob flow between each two states matches
• Partial balance: prob flow out of state n due to departure from queue j is balanced by prob flow into state n due to arrival to queue j, for each queue j, j=0,…,J
• Global balance: total prob flow out of state n equals total prob flow into state n
• Probability flow in/out queue in relation to network
),()(),()(
),()(),()(
),()(),()(
0000
00
neenqeeneennqn
neenqeeneennqn
neenqeeneennqn
kj
J
kkj
J
jkj
J
k
J
j
kj
J
kkjkj
J
k
kjkjkj
8Last time on NoQ :Kelly Whittle network
),()(),()(00
neenqeeneennqn kj
J
kkjkj
J
k
kk
jjj
j
jkjj
kj
nnennq
pnen
ennq
pnen
eennq
)()(),(
)()(
),(
)()(
),(
0
Theorem: The equilibrium distribution for the Kelly Whittle network is
where
and π satisfies partial balance
SnnBn jjjnj
J
j
j
/)()(1
kjkk
jj p
9
Insert equilibrium distribution and rates in partial balance
This is the beauty of partial balance!
kjkk
J
k
ns
J
sj
jkjj
J
k
ns
J
sj
kjkkj
j
j
kns
J
skj
J
k
jkjjn
s
J
s
J
k
kj
J
kkjkj
J
k
pen
pen
peenen
een
pnen
n
neenqeeneennqn
sjs
sjs
s
s
01
01
10
10
00
)(
)(
)()(
)(
)()(
)(
),()(),()(
• Last time on NoQ
10Last time on NoQ : Time reversed process• Theorem: If X(t) is a stationary Markov process with transition
rates q(j,k), and equilibrium distribution π(j), jεS, then the reversed process X(τ-t) is a stationary Markov process with transition rates
and the same equilibrium distribution.
• Theorem: Kelly’s lemmaLet X(t) be a stationary Markov process with transition rates q(j,k). If we can find a collection of numbers q’(j,k) such that q’(j)=q(j), jεS, and a collection of positive numbers π (j), jεS, summing to unity, such that
then q’(j,k) are the transition rates of the time-reversed process, and π (j), jεS, is the equilibrium distribution of both processes.
)(),()(),('
jjkqkkjq
Skj ,
),(')(),()( jkqkkjqj Skj ,
11
Alternative proof: use Kelly’s lemma
Forward rates
Guess backward rates
Check conditions
Jkjpnen
eennq jkjj
kj ,...,0, ,)(
)(),(
jj
jjk
jj
jjj
j
kj
kjjj
kkj
j
kjkj
r
kj
nen
pnen
pnen
eennq
)()(
)()(
)()(
),(
,
,,
jj
jjkj
j
kjkj
kj nen
pnen
eennq
)()(
)()(
),(,,
kjjj
kkrjk
rjkj
jkj
r
pp
Jkjpnen
eennq
,...,0, ,)(
)(),(
Last time on NoQ : Time reversed process
12
Alternative proof: use Kelly’s lemma
Guess for equilibrium distribution
insert
SnnBn jjjnj
J
j
j
/)()(1
kjkkj
jns
J
sj
kkj
rjkj
jns
J
s
kjkjkjr
peenen
een
pnen
n
neenqeeneennqn
s
s
)()(
)(
)()(
)(
),()(),()(
1
1
Last time on NoQ : Time reversed process
13
Networks of Queues: lecture 4 Nelson, sec 10.3.5-10.6.6• Last time on NoQ …• Quasi reversibility• Network of quasi reversible queues• Queue disciplines, Symmetric queues, BCMP networks• Summary / Exercises
Quasi-reversibility
• Multi class queueing network, class c ε C
• A queue is quasi-reversible if its state x(t) is a stationary Markov process with the property that the state of the queue at time t0, x(t0), is independent of(i) arrival times of class c customers subsequent to time t0 (ii) departure times of class c customers prior to time t0.
• TheoremIf a queue is QR then(i) arrival times of class c customers form independent Poisson processes(ii) departure times of class c customers form independent Poisson processes.
Quasi-reversibility
• S(c,x) set of states queue contains one more class c than in state x
• Arrival rate class c customerindependent of state x
• Thus arrival rate independent of prior events, and has constant rate Poisson process
)',()(),('
xxqcxcSx
Quasi-reversibility
• Multi class queueing network, class c ε C• S(c,x) set of states in which queue contains one more
class c than in state x• Arrival rate class c customer
independent of state x• Departure rate class c customer
independent of state x
• Characterise QR, combine
• A form of partial balance
)',()(),('
xxqcxcSx
)',()(
),'()'()',()(
),('
xxqc
xxqxxxqxr
xcSx
r
),'()'()',()()(),('),('
xxqxxxqxcxcSxxcSx
17
Networks of Queues: lecture 4 Nelson, sec 10.3.5-10.6.6• Last time on NoQ …• Quasi reversibility• Network of quasi reversible queues• Queue disciplines, Symmetric queues, BCMP networks• Summary / Exercises
18
Network of quasi reversible queues• Multiclass queueing network, type i=1,..,I• J queues• Customer type identifies route• Poisson arrival rate per type i=1,…,I• Route r(i,1),r(i,2),…,r(i,S(i))• Type i at stage s in queue r(i,s)• S(c,x) set of states in which queue contains one more
class c than in state x
• State X(t)=(x1(t),…,xJ(t))
• Fixed number of visits; cannot use Markov routing• 1, 2, or 3 visits to queue: use 3 types
19Network of quasi reversible queues• Construct network by multiplying rates for individual
queues• Transition rates• Arrival of type i causes queue k=r(i,1) to change at
• Departure type i from queue j = r(i,S(i))
• Routing
• Internal
),1,(')',( kkkkkk xiSxxxq
)'),(,()',( jjjjjj xiSiSxxxq
)',,(),1,('
)1,()',()',(
)',()',()',(
),1,('
jjjkkk
k
kkkjjj
kkxsiSx
kkkjjj
xsiSxxsiSx
sixxqxxq
xxqxxqxxq
kk
)',( jjj xxq
Network of Quasi-reversible queues• Rates
• Theorem : For an open network of QR queues(i) the states of individual queues are independent at fixed time
(ii) an arriving customer sees the equilibrium distribution(ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming a Poisson process.(iii) time-reversal: another open network of QR queues(iv) system is QR, so departures form Poisson process
)()...()( 11 JJ xxx
)',,(),1,(')1,()',()',(
)'),(,()',(),1,(')',(
)',(
jjjkkkk
kkkjjj
jjjjjj
kkkkkk
xsiSxxsiSxsixxqxxq
xiSiSxxxqxiSxxxq
xxq
Network of Quasi-reversible queues• Proof of part (i): Kelly’s lemma• Rates
• Transition rates reversed process (guess)
)',,()',1,(),(
)',(')',('
)',1,()',(')),(,(')',('
)',('
jjjkkkj
jjjkkk
jjjjjj
kkkkkk
xsiSxxsiSxsixxq
xxq
xiSxxxqxiSiSxxxq
xxq
)',,(),1,(')1,()',()',(
)'),(,()',(),1,(')',(
)',(
jjjkkkk
kkkjjj
jjjjjj
kkkkkk
xsiSxxsiSxsixxqxxq
xiSiSxxxqxiSxxxq
xxq
Network of Quasi-reversible queues• Proof of part (i): Kelly’s lemma• For
• We have
• Satisfied due to )(),()1,(
),(),'('
),'(')'()'()1,()',()',()()(
)',,(),',1,(
isisi
sixxq
xxqxxsixxqxxqxx
xsiSxxsiSx
jk
j
jjjkkkkkjj
k
kkkjjjkkjj
jjjkkk
23
Networks of Queues: lecture 4 Nelson, sec 10.3.5-10.6.6• Last time on NoQ …• Quasi reversibility• Network of quasi reversible queues• Queue disciplines, Symmetric queues, BCMP networks• Summary / Exercises
Queue disciplines
• Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion γ j(k,nj) of this effort directed to job in position k, k=1,…, nj ; when this job leaves, his service is completed, jobs in positions k+1,…, nj move to positions k,…, nj -1.(iv) When a job arrives at queue j he moves into position k with probability δj(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +1. 0 if 0)(1),(1),(
11
nnnknk jj
n
kj
n
k
• Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion γ j(k,nj) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δj(k,nj + 1)
• Examples: FCFSLCFSPSinfinite server queue
• BCMP network
Queue disciplines
Symmetric queues
• Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion γ j(k,nj) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δj(k,nj + 1)
• Examples: IS, LCFS, PS• Symmetric queue QR (for general service requirement) • Instantaneous attention• Note: FCFS with identical service rate for all types is QR
),(),( nknk jj
27
Networks of Queues: lecture 4 Nelson, sec 10.3.5-10.6.6• Last time on NoQ …• Quasi reversibility• Queue disciplines, Symmetric queues, BCMP networks• Network of quasi reversible queues• Summary / Exercises
Quasi-reversibility and partial balance• QR: fairly general queues, service disciplines,
Markov routing, product form equilibrium distribution factorizes over queues.
• PB: fairly general relation between service rate at queues, state-dependent routing (blocking), product form equilibrium distribution factorizes over service and routing parts.
• Identical for single type queueing network with Markov routing
• Note: in Nelson proof for Markov routing
• QR partial balance• NOT partial balance QR (exercise)• NOT QR Reversibility (see Nelson, ex 10.14)• NOT Reversibility QR (see Nelson, ex 10.12)
29
Summary / next / exercises:• Jackson network• Kelly Whittle network• Partial balance• Quasi reversibility• Customer types• Queue disciplines• BCMP networks
• Next:– Insensitivity– Aggregation / decomposition / Norton’s theorem
• Exercises: 20,22,24,25,26,27,29,30