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On the subexponential properties in stationary single-server queues: a Palm-martingale approach

Part of: Mathematical economics Stochastic processes Special processes

Published online by Cambridge University Press:  01 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: miyoshi@is.titech.ac.jp
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Abstract

This paper studies the subexponential properties of the stationary workload, actual waiting time and sojourn time distributions in work-conserving single-server queues when the equilibrium residual service time distribution is subexponential. This kind of problem has been previously investigated in various queueing and insurance risk settings. For example, it has been shown that, when the queue has a Markovian arrival stream (MAS) input governed by a finite-state Markov chain, it has such subexponential properties. However, though MASs can approximate any stationary marked point process, it is known that the corresponding subexponential results fail in the general stationary framework. In this paper, we consider the model with a general stationary input and show the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS governed by a finite-state Markov chain inherently possesses them. The approach used here is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The proof is essentially an extension of the M/GI/1 case to cover ‘Poisson-like’ arrival processes such as Markovian ones, where the stochastic intensity is admitted.

Keywords

Stationary work-conserving single-server queuesubexponential distributionPalm-martingale calculusstochastic intensity kernel

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 872 - 892
Copyright
Copyright © Applied Probability Trust 2004 

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