Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-20T07:44:30.815Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artalejo, J. R., and Gomez-Corral, A. (1999). On a single-server queue with negative arrival and request repeated. J. Appl. Prob. 36, 907–918.Google Scholar
Boucherie, R. J., and van Dijk, N. M. (1994). Local balance in queueing networks with positive and negative customers. Ann. Operat. Res. 48, 463–492.Google Scholar
Cox, D. R. (1955). The analysis of non-Markovian processes by the inclusion of supplementary variables. Proc. Camb. Phil. Soc. 51, 433–441.Google Scholar
Feller, W. (1957). An Introduction to Probability Theory and its Applications, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Foster, F. G. (1953). On the stochastic matrices associated with certain queuing processes. Ann. Math. Statist. 24, 355–360.Google Scholar
Gaver, D. P. (1963). Time to failure and availability of paralleled redundant systems with repair. IEEE Trans. Reliab. 12, 30–38.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product form networks with negative and positive customers. J. Appl. Prob. 28, 656–663.Google Scholar
Harrison, P. G., and Pitel, E. (1993). Sojourn times in single-server queues with negative customers. J. Appl. Prob. 30, 943–963.CrossRefGoogle Scholar
Harrison, P. G., and Pitel, E. (1995). Response time distributions in tandem G-networks. J. Appl. Prob. 32, 224–246.Google Scholar
Harrison, P. G., and Pitel, E. (1996). The M/G/1 queue with negative customers. Adv. Appl. Prob. 28, 540–566.Google Scholar
Henderson, W. (1993). Queueing networks with negative customers and negative queue length. J. Appl. Prob. 30, 931–942.Google Scholar
Henderson, W., Northcote, B. S., and Taylor, P. G. (1994). Geometric equilibrium distributions for queues with interactive batch departures. Ann. Operat. Res. 48, 493–511.Google Scholar
Kosten, L. (1973). Stochastic Theory of Service Systems. Pergamon, Oxford.Google Scholar
Ohashi, M., and Nishida, T. (1980). A two-unit paralleled system with general distribution. J. Operat. Res. Soc. Japan 23, 313–325.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Takács, L. (1962). Introduction to the Theory of Queues. Oxford University Press.Google Scholar
Takagi, H. (1992). Analysis of an M/G/1/N queue with multiple server vacations, and its application to a polling model. J. Operat. Res. Japan 35, 300–315.Google Scholar
Widder, D. V. (1941). The Laplace Transform. Princeton University Press.Google Scholar