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Processor-sharing and random-service queues with semi-Markovian arrivals

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

De-An Wu  and
Hideaki Takagi
De-An Wu*
Affiliation:
University of Tsukuba
Hideaki Takagi*
Affiliation:
University of Tsukuba
*
∗Postal address: Doctoral Program in Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan.
∗∗Postal address: Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan. Email address: [email protected]
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Abstract

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We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

Keywords

Queuesemi-Markov arrival processprocessor sharingrandom servicesojourn timewaiting time

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 478 - 490
Copyright
© Applied Probability Trust 2005 

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