Queueing Systems with Some Versions of Limited Processor Sharing Discipline

TL;DR

This paper analyzes a queueing system with limited processor sharing, deriving explicit formulas for state and loss probabilities, applicable to wireless and computer network models, and explores the impact of different job loss rules.

Contribution

It introduces relations between limited and unlimited processor sharing systems and provides explicit formulas for stationary and loss probabilities under Poisson arrivals.

Findings

Explicit formulas for state probabilities derived

Loss probability equals server capacity exhaustion probability

Probabilities invariant under job length distribution with fixed mean

Abstract

The paper considers a queueing system with limited processor sharing. No more than n jobs may be served simultaneously. This system may be used for modeling bandwidth sharing in wireless communication systems and processes of service in computer networks. If there are n jobs in the considered queueing system and a new job arrives, then the arriving job is lost or the service of a job is interrupted and this job is lost. We study two rules to choose the job to be lost. In accordance with one of these rules, the job with the shortest remaining length is lost. Relations are obtained between the state probabilities of considered system and the state probabilities of the corresponding unlimited processor sharing system. These relations allow to compute the state probabilities for considered system if the state probabilities for the unlimited processor sharing system are known. In the case of Poisson arrival process, the probability that the server capacity is exhausted is equal to the probability that a job is lost. We have obtained an explicit formulas for the stationary state probabilities and the loss probability for this case. These probabilities are invariant under the job length distribution under the condition that the average value of the length is fixed.