Consider an M/G/1 queue with FCFS scheduling . Let the random variablesA, B, and D
Chapter 9, Problem 2(choose chapter or problem)
Consider an M/G/1 queue with FCFS scheduling . Let the random variablesA, B, and D, respectively, denote the interarrival time, the service time, and theinterdeparture time. By conditioning on the number of jobs in the system andthen using the theorem of total Laplace transforms, show that in the steady stateLD(s) = LB(s) + (1 )LA(s)LB(s),where is the traffic intensity so that = E[B]/E[A].Point out why the assumption of Poisson arrival stream is needed to derive thisresult. Then, specializing to the case of M/M/1 queue, show thatLD(s) = LA(s).This verifies Burkes result that the output process of an M/M/1 FCFS queue isPoissonian. Note that the independence of successive interdeparture times needsto be shown in order to complete the proof.
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