Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach

TL;DR

This paper analyzes how small, environment-dependent perturbations in the server rate of an M/M/1 queue affect the busy period, evaluating the accuracy of the Reduced Service Rate approximation through a probabilistic approach.

Contribution

It provides a probabilistic analysis of the impact of small, environment-dependent perturbations on the busy period of an M/M/1 queue, including the first two terms of the perturbation expansion.

Findings

The first-order approximation confirms the RSR equivalence.

Second-order analysis reveals the importance of environment process correlations.

The approach quantifies the limitations of the RSR approximation in perturbed queues.

Abstract

Motivated by the problem of the coexistence on transmission links of telecommunication networks of elastic and unresponsive traffic, we study in this paper the impact on the busy period of an M/M/1 queue of a small perturbation in the server rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter \eps \ll 1. We specifically compute the two first terms of the power series expansion in \eps of the mean value of the busy period duration. This allows us to study the validity of the Reduced Service Rate (RSR) approximation, which consists in comparing the perturbed M/M/1 queue with the M/M/1 queue where the service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of the environment process (X(t)) play a key role.