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On the stationary workload distribution of work-conserving single-server queues: a general formula via stochastic intensity

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Naoto Miyoshi
Naoto Miyoshi*
Affiliation:
Tokyo Institute of Technology
*
∗ Postal address: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan. Email address: [email protected]
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Abstract

It is well known that a simple closed-form formula exists for the stationary distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.

Keywords

Work-conserving single-server queuesstationary workload distributionPalm-martingale calculusstochastic intensity kernelLCFS-PR discipline

MSC classification

Primary: 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 793 - 798
Copyright
Copyright © by the Applied Probability Trust 2001 

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