Networks Plan for today (lecture 6):

1. NetworksPlan for today (lecture 6): • Last time / Questions? • Customer types • Network queues • Results • Exercises • Questions

2. Customer types : routes • Customer type 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. Customer types : queue discipline • Customers ordered at queue • Consider queue j, containing nj jobs • Queue j contains jobs in positions 1,…, nj • Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate j(nj)(iii) Proportion  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 j(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +1.

4. queue discipline: examples • Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate j(nj)(iii) Proportion  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 j(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +1. • M/M/1 FIFO • M/M/1 LIFO • M/M/1 PS • M/M/s FIFO • M/M/∞ • No priority queue based on type of a job

5. Customer types : • Transition ratestype i job arrival (note that queue which job arrives is determined by type)type i job completion (job must be on last stage of route through the network)type i job towards next stage of its route • Notice that each route behaves as tandem network, where each stage is queue in tandemThus: arrival rate of type i to stage s : (i) class • type, stage along route of customer in position k in queue j

6. Customer types • Transition rates • Job in position b in queue j is at last stage of its route and leaves the system • Job in position b in queue j is not in last stage, leaves queue j to route to next stage and moves into position m in queue k • Job of type i enters the system and moves into position m in queue k, k=r(i,1)

7. Customer types • Notice that each route behaves as tandem network, where each stage is queue in tandemThus: arrival rate of type i to stage s is (i)Let • State of the network: • Equilibrium distribution

8. Proof: Kelly’s lemma Transition rates forward: • Job in position b in queue j is at last stage of its route and leaves the system • Job in position b in queue j is not in last stage, leaves queue j to route to next stage and moves into position m in queue • Job of type i enters the system and moves into position m in queue k, k=r(i,1) Transition rates reversed process: Balance per job!

9. Results Theorem 3.1: The equilibrium distribution of the network is Theorem 3.2: The reversed process is also a stationary open network of queues. Corollary 3.3: In equilibrium customers of type i leave the system in a Poisson stream at rate (i). These streams are independent, and C(t0) is independent of departures from the system prior to t0. Corollary 3.4: In equilibrium the state of queue j is independent of the state of the rest of the system. The probability that queue j contains n jobs is Corollary 3.5: MUSTA Note that the process recording n is not Markov!

10. Remarks: All jobs exponential (1) service requirement Different exponential service requirements? Different distributions? Symmetric queues: Quasi reversible queues

11. Exercises • [R+SN] 3.1.2, 3.1.4, 3.1.3, 3.1.6