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M/GI/1 queues with services of both positive and negative customers

Part of: Special processes Computer system organization

Published online by Cambridge University Press:  14 July 2016

Yijun Zhu  and
Zhe George Zhang
Yijun Zhu*
Affiliation:
Jiangsu University
Zhe George Zhang*
Affiliation:
Western Washington University and Simon Fraser University
*
∗ Postal address: Faculty of Science, Jiangsu University, Zhenjiang 212013, P. R. China
∗∗ Postal address: Department of Decision Sciences, Western Washington University, Bellingham, WA 98225-9077, USA. Email address: george.zhang@wwu.edu
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Abstract

We consider an M/GI/1 queue with two types of customers, positive and negative, which cancel each other out. The server provides service to either a positive customer or a negative customer. In such a system, the queue length can be either positive or negative and an arrival either joins the queue, if it is of the same sign, or instantaneously removes a customer of the opposite sign at the end of the queue or in service. This study is a generalization of Gelenbe's original concept of a queue with negative customers, where only positive customers need services and negative customers arriving at an empty system are lost or need no service. In this paper, we derive the transient and the stationary probability distributions for the major performance measures in terms of generating functions and Laplace transforms. It has been shown that the previous results for the system with negative arrivals of zero service time are special cases of our model. In addition, we obtain the stationary waiting time distribution of this system in terms of a Laplace transform.

Keywords

Queue with negative customersupplementary variableLaplace transformLaurent series

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1157 - 1170
Copyright
Copyright © Applied Probability Trust 2004 

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