Flows and Networks

Plan 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

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

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.

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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 tandem

Thus: arrival rate of type i to stage s : (i) Let

• State of the network:

• Equilibrium distribution

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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

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Flows and network: 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 optimisationApplication: optimal routing

• further reading[R+SN] chapter 3: customer types; chapter 4: examples

Exercises

• [R+SN] 3.1.2, 3.2.3, 3.1.4.

Exercise: Optimal design of Jackson network (1)

• Consider an open Jackson network

with 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

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Exercise: Optimal design of Jackson network (2)

• Consider an open Jackson network

with 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 case

for all j,k

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