On the number of departures from the $M/M/\infty$ queue in a finite time interval

TL;DR

This paper uses spectral theory to analyze the distribution of departures in an $M/M/\infty$ queue over a finite interval, extending methods to finite capacity systems.

Contribution

It introduces a spectral approach to compute transient characteristics of the $M/M/\infty$ queue and extends the analysis to finite capacity $MM/c/c$ systems.

Findings

Derived Laplace transforms for the number of departures and transitions

Extended analysis to finite capacity queue systems

Provided explicit spectral representations for transient metrics

Abstract

In this paper, we analyze the number of departures from an initially empty $M/M/\infty$ system in a finite time interval. We observe the system during an exponentially distributed period of time starting from the time origin. We then consider the absorbed Markov chain describing the number of arrivals and departures in the system until the observer leaves the system, triggering the absorption of the Markov chain. The generator of the absorbed Markov chain induces a selfadjoint operator in some Hilbert space. The use of spectral theory then allows us to compute the Laplace transform of several transient characteristics of the $M/M/\infty$ system (namely, the number of transitions of the Markov chain until absorption, the number of departures from the system, etc.). The analysis is extended to the finite capacity $MM/c/c$ system for some finite integer $c$}.