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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 Researchdepartment of Applied Mathematics
University of Twente
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Networks of Queues: lecture 4
MSc assignment: ambulance planning …
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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
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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
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6Last time on NoQ : Open network : equilibrium distribution
Theorem: The equilibrium distribution for the open Jackson network is
and satisfies partial balance
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Last 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
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8Last time on NoQ :Kelly Whittle network
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Theorem: The equilibrium distribution for the Kelly Whittle network is
where
and π satisfies partial balance
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Insert equilibrium distribution and rates in partial balance
This is the beauty of partial balance!
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• 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.
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11
Alternative proof: use Kelly’s lemma
Forward rates
Guess backward rates
Check conditions
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Last time on NoQ : Time reversed process
12
Alternative proof: use Kelly’s lemma
Guess for equilibrium distribution
insert
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Last time on NoQ : Time reversed process
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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
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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
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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
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Network 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
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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
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Network of Quasi-reversible queues• Proof of part (i): Kelly’s lemma• Rates
• Transition rates reversed process (guess)
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Network of Quasi-reversible queues• Proof of part (i): Kelly’s lemma• For
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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
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• 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
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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)
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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
