1. Networks Plan for today (lecture 8): Last time / Questions? Quasi reversibility Network of quasi reversible queues Symmetric queues, insensitivity Partial balance vs quasi reversibility Proof of insensitivity? Summary Exercises Questions

2. Customer types : routes Customer type/class identified route Poisson arrival rate per type Type i: arrival rate  (i), i=1,…,I Route r(i,1), r(i,2),…,r(i,S(i)) Type i at stage s in queue r(i,s) Fixed number of visits; cannot use Markov routing 1, 2. or 3 visits to queue: use 3 types

3. Quasi-reversibility Multi class queueing network, class c  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 custmers prior to time t0. 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.

4. Quasi-reversibility Multi class queueing network, class c  C S(c,x) set of states queue contains one more class c than in state x Arrival rate class c customer Departure rate class c customer Characterise QR, combine QR vs R: stronger condition on arrival process, but weaker form of balance

5. Quasi-reversibility: network Multi class queueing network, type i=1,…,I J queues Customer type identifies route Poisson arrival rate per type  (i), i=1,…,I Route r(i,1), r(i,2),…,r(i,S(i)) Type i at stage s in queue r(i,s) State X(t)=(x 1 (t),…,x J (t)) Construct a network by multiplying the rates for the 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

6. Quasi-reversibility: network Transition rates Theorem : For an open network of QR queues (i) the states of individual queues are independent (ii) an arriving customer sees the equilibrium distri bution (ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming Poisson process. (iii) time-reversal: another open network of QR queues (iv) system is QR, so departures form Poisson process Proof of part (i)

7. Quasi-reversibility: network Proof of part (i) Transition rates Transition rates reversed process (guess)

8. Quasi-reversibility: state aggregation, Transition rates x may be complicated state, consider only total number in component: flow equivalent server

9. Symmetric queues; insensitivity Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate  j (n j )  (iii) Proportion  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  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. Examples: infinite server queue, lcfs, ps Symmetric queue QR for general service requirement Instanteneous attention Symmetric queue is insensitive

10. Quasi-reversiblity vs 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

11. Networks (3TU): summary stochastic networks Contents 1.Introduction; Markov chains 2.Birth-death processes; Poisson process, simple queue; reversibility; detailed balance 3.Output of simple queue; Tandem network; equilibrium distribution 4.Jackson networks; Partial balance 5.Sojourn time simple queue and tandem network 6.Performance measures for Jackson networks: throughput, mean sojourn time, blocking 7.Application: service rate allocation for throughput optimisation Application: optimal routing 8.Quasi reversibility, customer types, insensitivity further reading[R+SN] chapter 3: customer types; chapter 4: examples

12. Exercises Exercise Blocking Consider a tandem network of two simple queues. Let the arrival rate to queue 1 be Poisson , and let the service rate at each queue be exponential  i, i=1,2. Let queue 1 have capacity N1. For N1= , give the equilibrium distribution. For N1<  formulate three distinct protocols that preserve product form, indicate graphically what the implication of these protocols is on the transition diagram, and proof (by partial balance) that the equilbrium distribution is of product form. Can the product form results also be obtained via Quasi reversibility? If so, provide the proof of the product form result via quasi reversibility. Exercise: Proof Theorem 3.12 Provide the proof of Theorem 3.12 [R+SN] 3.1.2, 3.2.3, 3.1.4, 3.1.3, 3.1.6, 3.3.2