Perturbation analysis of a variable M/M/1 queue: a probabilistic approach

Abstract

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 ε ≪ 1. We specifically compute the two first terms of the power series expansion in ε 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.

Integration of streaming services and TCP data transmission in the Internet

“…The average of the duration B ε of the corresponding busy period has been studied by Antunes et al (4) and can be expanded in power series of ε as follows.…”

“…In this context, a usual approximation (referred to as Reduced Service Rate, RSR) consists of assuming that everything happens as if the service rate for data were reduced by the mean bit rate offered by streaming flows. This approximation has been investigated for the number of active data flows by Antunes et al (4), when the load offered by streaming flows is very small. We note that the same kind of problem has been addressed in the technical literature by Núñez-Queija and Boxma (5) in the context of ABR service in ATM networks and more recently by Núñez-Queija (6; 7) via matrix analysis for systems described by means of quasi birth and death processes.…”

“…Along the same line of investigations as Antunes et al (4), because of the real difficulty of the problems, the mean load offered by streaming flows is supposed to be very small (controlled by a parameter ε ≪ 1) and a perturbation analysis for the analysis of the mean bit rate is done. It is assumed that elastic flows arrive according to a Poisson process and share the available bandwidth according to the processor sharing discipline.…”

Perturbation analysis of Markov modulated fluid models

“…For a long time there has been a recurrent interest in perturbation analysis, see for instance Cao and Chen [5], Heidergott, et al [7], Antunes et al [1].…”

Sojourn time asymptotics in Processor Sharing queues with varying service rate

“…Other versions of the reduced-load approximation for queues with varying service rate are given in e.g. [1,3,5]. Somewhat related results for workloads in a GPS queue can be found in [25].…”