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

Abstract

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.

On the convergence to stationarity of birth-death processes

“…Comparing our results with those of Stadje and Parthasarathy [10], we find agreement for c = 1, but a discrepancy for c = 2. As a check, we evaluated the integral I j of (11) directly by using the representation for p j (t) derived in Karlin and McGregor [6] for j = λ = µ = 1, and found that it equals 0, which is consistent with (12), but not with Theorem 3 of Stadje and Parthasarathy [10].…”

“…The expression will be evaluated for some specific birth-death processes in Section 4. In particular, we will compare our findings with those of Stadje and Parthasarathy [10] (and find a discrepancy). Finally, in Section 5, we consider birth-death processes in discrete time, and show that a similar result may be obtained in this setting by performing a suitable transformation, provided the birth and death probabilities satisfy certain requirements.…”

“…We will finally apply our results to the process of the number of customers in an M/M/c queueing system -the setting in which Stadje and Parthasarathy [10] proposed the integral (1) as a measure of the speed of convergence to stationarity -and compare our findings with those in [10]. The process at hand is a birth-death process X with rates…”

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“…Condition (22) holds, in particular, if the functions a(t) and b(t) are periodic (in general, with different periods) or asymptotically periodic.…”

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“…In [14], it is calculated in the ergodic case, ρ < 1, and when i = 0 the distance of the mean to its asymptote in the L 1 (R + ) norm…”