1. 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. 2 Networks of Queues: lecture 5 Nelson, sec 10.3.5-10.6.6 Last time on NoQ … –Kelly-Whittle network –Partial balance –Quasi reversibility PASTA / MUSTA Norton’s theorem

3. 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 Υ 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. FCFS, LCFS, PS, infinite server queue BCMP network

4. 4 Network of queues Multiclass queueing network, type or class i =1,.., I J queues Customer type identifies route Poisson arrival rate per type ν ( i ), i =1,…, I Route r ( i,1), r ( i,2),…, r ( i, S ( i )) Fixed number of visits; cannot use Markov routing 1, 2, or 3 visits to queue: use 3 types Type i at stage s in queue r ( i, s ) t j (l): type of customer in position l in queue j s j (l): stage of this customer along his route c j (l)=(t j (l),s j (l)): class of this customer c j =(c j (1),…, c j (n j )): state of queue j C=(c 1,…, c J ): Markov process representing states of the system

5. Arrival rate class i to network ν ( i ), so also for each stage, say Transition rates Equilibrium distribution queue j Theorem: equilibrium distribution for open network (closed): (and departure process from the network is Poisson) Product form: BCMP network

6. 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

7. 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

8. 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

9. 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

10. 10 Networks of Queues: lecture 6 Nelson, sec 10.3.5-10.6.6 Last time on NoQ … –Kelly-Whittle network –Partial balance –Quasi reversibility PASTA / MUSTA Norton’s theorem

11. 11 PASTA: Poisson Arrivals See Time Averages fraction of time system in state n probability outside observer sees n customers at time t probability that arriving customer sees n customers at time t (just before arrival at time t there are n customers in the system) in general

12. 12 PASTA: Poisson Arrivals See Time Averages Let C(t,t+h) event customer arrives in ( t,t+h) For Poisson arrivals q(n,n+1)=λ so that Alternative explanation; PASTA holds in general!

13. 13 PASTA: Poisson Arrivals See Time Averages Transient In equilibrium

14. 14 MUSTA: Moving Units See Time Averages Palm probabilities: Each type of transition n  n’ for Markov chain associated with subset H of SxS \diag(SxS) Example:transition in which customer queue i  queue j Transition in which customer leaves queue i Transition in which customer enters queue j

15. 15 MUSTA: Moving Units See Time Averages N H process counting the H-transitions Palm probability P H (C) of event C given that H occurs: Probability customer queue i  queue j sees state m Probability customer arriving to queue j sees state m

16. 16 Last time on NoQ :Kelly Whittle network Theorem: The equilibrium distribution for the Kelly Whittle network is where and π satisfies partial balance

17. 17 MUSTA :Kelly Whittle network Theorem: The distribution seen by a customer moving from queue i tot queue j is Entering queue j is where

18. 18 MUSTA

19. 19 Networks of Queues: lecture 5 Nelson, sec 10.3.5-10.6.6 Last time on NoQ … –Kelly-Whittle network –Partial balance –Quasi reversibility PASTA / MUSTA Norton’s theorem

20. 20 Theorem: Insensitivity Suppose subsets A i of S are such that there is no single transition of positive intensity in which more than one A i is vacated or more than one A i entered. Then the equilibrium distribution π ( x ) is insensitive to the nominal sojourn times in the A i if and only if the Markov process shows partial balance in all the A i. Norton’s theorem: state aggregation, flow equivalent servers, Nelson, sec 10.6.14-15 Consider network of subnetworks, each subnetwork represented by auxiliary process. Then we may lump subnet into single node if and only if partial balance over the subnets Insensitivity and partial balance, Norton’s theorem