M/M/1 queue in two alternating environments and its heavy traffic approximation

Abstract

We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of the process in terms of a generalized mixture of two geometric distributions. In the special case when only one kind of switch is allowed, we analyze the transient distribution, and investigate the busy period problem. The analysis is also performed by means of a suitable heavy-traffic approximation which leads to a continuous random process. Its distribution satisfies a partial differential equation with randomly alternating infinitesimal moments. For the approximating process we determine the steady-state distribution, the transient distribution and a first-passage-time density.

The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion

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Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

“…In these cases, it is necessary take into account birth-death processes with a reflecting condition in the zero state (see, for instance, Di Crescenzo et al [8], Crawford and Suchard [9], Giorno and Nobile [10], Lenin et al [11] and Tavaré [12]). These processes provide interesting applications in queuing models in which a reflecting boundary must be imposed to describe the number of customers in the system (cf., for instance, Crawford et al [13], Di Crescenzo et al [14] and Giorno et al [15]).…”

Fractional Erlang queues

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