An infinite-server queue influenced by a semi-Markovian environment

Abstract

We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that is independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that is either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue-length in the case that the amount of work brought by a given arrival is of an arbitrary distribution.

Tail asymptotics of a Markov-modulated infinite-server queue

“…O'Cinneide and Purdue [10] provide explicit expressions for the moments of the stationary number of customers, and systems of partial differential equations for the corresponding transient moments, in the context of the model variant studied in the present paper (with state-dependent hazard rate, that is). Related results, for a considerably broader class of models, are given in [9]; we also mention [8] for extensions to a semi-Markovian background process. As mentioned above, [6] presents the useful observation that M (t) has a Poisson law with a random parameter (that depends on the path of the background process in [0, t]), as was highlighted in (1) and (2).…”

Large deviations of an infinite-server system with a linearly scaled background process

“…The main focus in the literature so far has been on characterizing (through the derivation of moments, or even the full probability generating function) the steady-state number of jobs in the system. The most striking feature is that the number of jobs in the system still has a Poisson distribution, but now with a random parameter; a few key references are [5,8,11,15]. Interestingly, under an appropriate time-scaling [2,9] in which the transitions of the background process occur at a faster rate than the Poisson arrivals, we retrieve the Poisson distribution (with a deterministic parameter, that is) for the steady-state number of jobs in the system.…”

Rare Event Analysis of Markov-Modulated Infinite-Server Queues: A Poisson Limit

“…The Markov-modulated infinite-server queue has attracted some (but relatively limited) attention in recent years. The main focus in the literature so far has been on characterizing (through the derivation of moments, or even the full probability generating function) the steady-state number of jobs in the system [5,7,9,10]. Interestingly, under an appropriate time scaling [3,8] in which the transitions of the background process occur at a faster rate than the Poisson arrivals, we retrieve the Poisson distribution for the steady-state number of jobs in the system.…”