Flows and Networks Plan for today (lecture 6):

1. Flows and NetworksPlan for today (lecture 6): • Last time / Questions? • Kelly / Whittle network • Optimal design of a Kelly / Whittle network:optimisation problem • Intermezzo: mathematical programming • Optimal design of a Kelly / Whittle network:Lagrangian and interpretation • Optimal design of a Kelly / Whittle network: Solution optimisation problem • Optimal design of a Kelly / Whittle network: network structure • Summary • 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. Customer types : equilibrium distribution • 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)Let • State of the network: • Equilibrium distribution

5. 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(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. • Symmetric queue is insensitive

6. Contents • Introduction; Markov chains • Birth-death processes; Poisson process, simple queue;reversibility; detailed balance • Output of simple queue; Tandem network; equilibrium distribution • Jackson networks;Partial balance • Sojourn time simple queue and tandem network • Performance measures for Jackson networks:throughput, mean sojourn time, blocking • Application: service rate allocation for throughput optimisationApplication: optimal routing • further reading[R+SN] chapter 3: customer types; chapter 4: examples Flows and network: summary stochastic networks

7. Exercises • [R+SN] 3.1.2, 3.2.3, 3.1.4.

8. Consider an open Jackson networkwith transition rates Assume that the service rates and arrival rates are given Let the costs per time unit for a job residing at queue j be Let the costs for routing a job from station i to station j be (i) Formulate the design problem (allocation of routing probabilities) as an optimisation problem. (ii) Provide the solution to this problem Exercise: Optimal design of Jackson network (1)

9. Consider an open Jackson networkwith transition rates Assume that the routing probabilities and arrival rates are given Let the costs per time unit for a job residing at queue j be Let the costs for routing a job from station i to station j be Let the total service rate that can be distributed over the queues be , i.e., (i) Formulate the design problem (allocation of service rates) as an optimisation problem. (ii) Provide the solution to this problem (iii) Now consider the case of a tandem network, and provide the solution to the optimisation problem for the casefor all j,k Exercise: Optimal design of Jackson network (2)