1 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 department of Applied Mathematics University of Twente

2 Networks of Queues: lecture 4 MSc assignment: ambulance planning …

3 Networks of Queues: lecture 4 Nelson, sec 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

4 Last time on NoQ: Jackson network : Definition M/M/1 queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates

5 Last 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

6 Last time on NoQ : Open network : equilibrium distribution Theorem: The equilibrium distribution for the open Jackson network is and satisfies partial balance

7 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

8 Last time on NoQ :Kelly Whittle network Theorem: The equilibrium distribution for the Kelly Whittle network is where and π satisfies partial balance

9 Insert equilibrium distribution and rates in partial balance This is the beauty of partial balance! Last time on NoQ

10 Last 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 lemma Let 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.

11 Alternative proof: use Kelly’s lemma Forward rates Guess backward rates Check conditions Last time on NoQ : Time reversed process

12 Alternative proof: use Kelly’s lemma Guess for equilibrium distribution insert Last time on NoQ : Time reversed process

13 Networks of Queues: lecture 4 Nelson, sec 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 t 0, x ( t 0 ), is independent of (i) arrival times of class c customers subsequent to time t 0 (ii) departure times of class c customers prior to time t 0. Theorem If 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 customer independent of state x Thus arrival rate independent of prior events, and has constant rate  Poisson process

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

17 Networks of Queues: lecture 4 Nelson, sec 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 )=( x 1 ( t ),…, x J ( t )) Fixed number of visits; cannot use Markov routing 1, 2, or 3 visits to queue: use 3 types

19 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

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

Network of Quasi-reversible queues Proof of part (i): Kelly’s lemma Rates Transition rates reversed process (guess)

Network of Quasi-reversible queues Proof of part (i): Kelly’s lemma For We have Satisfied due to

23 Networks of Queues: lecture 4 Nelson, sec 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 φ j (n j ) (iii) Proportion γ j (k,n j ) of this effort directed to job in position k, k=1,…, n j ; when this job leaves, his service is completed, jobs in positions k+1,…, n j move to positions k,…, n j -1. (iv) When a job arrives at queue j he moves into position k with probability δ j (k,n j + 1), k=1,…, n j +1; jobs previously in positions k,…, n j move to positions k+1,…, n j +1.

Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate φ j (n j ) (iii) Proportion γ j (k,n j ) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δ j (k,n j + 1) Examples: FCFS LCFS PS infinite 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 φ j (n j ) (iii) Proportion γ j (k,n j ) of this effort directed to job in position k, (iv) job arriving at queue j moves into position k with prob. δ j (k,n j + 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

27 Networks of Queues: lecture 4 Nelson, sec 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