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The sojourn-time distribution in the M/G/1 queue with processor sharing. (English) Zbl 0544.60087

The author considers a queueing system in which customers arrive for service at event times of a Poisson process and demand a random service duration. Service time demands are independent and identically distributed. When n customers are present they are simultaneously served at an instantaneous rate of 1/n. This is the M/G/1 queue with processor sharing.
Analysis of this system proceeds by assuming that \(1\leq n<\infty\) customers are present at time zero and that they have fixed (residual) service requirements \(0<x_ 1\leq...\leq x_ n\leq \infty\). For \(x\leq x_ n\) define T(x) to be the time until customer n has received an additional amount of service x and let N(t) be the number of customers in the system at time t. Thus \(T(x_ k)\) is the departure time of customer k and \(N(T^{\pm}(x_ k))\) is the queue length just after or before \(T(x_ k)\). The author’s main result is an expression for the \(E[\exp(-sT(x_ k)z^{N(T^+(x_ n))}| x_ 1,...,x_ n,n].\)
This result is used to derive a number of others, including the equilibrium distributions of queue length and sojourn time of an arbitrary customer. The author gives a separate and simpler treatment of the special case where all customers have the same fixed service need. This paper unifies and generalises earlier discussions of this queueing system.
Reviewer: A.Pakes

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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