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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. Customer types : routes.
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NetworksPlan 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
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.
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
Quasi-reversibility: network • Multi class queueing network, class c C • 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)=(x1(t),…,xJ(t)) • Construct a network by multiplying the rates for the individual queues • Arrival of type i causes queue k=r(i,1) to change at • Departure type i from queue j = r(i,S(i)) • Routing
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. • TheoremIf 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.
Exercises • [R+SN] 3.1.2, 3.2.3, 3.1.4, 3.1.3, 3.1.6, 3.3.2
