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We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then pn(t) = P(X(t) = n) converges to the stationary probability πn = P(X = n). The integrals ∫0∞[E(X)-E(X(t))]dt and ∫0∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

We consider the repairman problem which corresponds to the finite population M/M/1 queue. Asymptotic approximations for the transient probability distribution of the number of broken machines constructed when the number M of machines is large and the service (repair) rate is also large, specifically, O(M). The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions. Extensive numerical comparisons show the quality of our approximations.

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On the convergence to stationarity of the many-server Poisson queue

On the transient behavior of the repairman problem

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2 results

On the convergence to stationarity of the many-server Poisson queue

On the transient behavior of the repairman problem