Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-15T05:22:17.015Z Has data issue: false hasContentIssue false
  • English
  • Français

On a single server queue with negative arrivals and request repeated

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

J. R. Artalejo  and
A. Gomez-Corral
J. R. Artalejo*
Affiliation:
Universidad Complutense de Madrid
A. Gomez-Corral*
Affiliation:
Universidad Complutense de Madrid
*
∗Postal address: Departamento de Estadística e I.O., Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain.
∗Postal address: Departamento de Estadística e I.O., Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.

Keywords

Queueing theorynegative arrivalslinear request repeatedFredholm integral equationstable recursion schemes

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 907 - 918
Copyright
Copyright © Applied Probability Trust 1999 

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. (1994). New results in retrial queueing systems with breakdown of the servers. Statist. Neerlandica 48, 23–36.CrossRefGoogle Scholar
Artalejo, J. R., and Gomez-Corral, A. (1997). Steady state solution of a single-server queue with linear request repeated. J. Appl. Prob. 34, 223–233.Google Scholar
Artalejo, J. R., and Gomez-Corral, A. (1998). Generalized birth and death processes with applications to queues with repeated attempts and negative arrivals. OR Spektrum 20, 5–14.CrossRefGoogle Scholar
Bayer, N., and Boxma, O. J. (1996). Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks. Queueing Systems 23, 301–316.Google Scholar
Boucherie, R. J., and Boxma, O. J. (1995). The workload in the M/G/1 queue with work removal. Prob. Eng. Inf. Sci. 10, 261–277.Google Scholar
Chao, X. (1995). A queueing network model with catastrophes and product form solution. Operat. Res. Lett. 18, 75–79.Google Scholar
Falin, G. I. (1990). A survey of retrial queues. Queueing Systems 7, 127–168.Google Scholar
Falin, G. I., and Templeton, J. G. C. (1997). Retrial Queues. Chapman and Hall, London.CrossRefGoogle Scholar
Farahmand, K. (1990). Single line queue with repeated demands. Queueing Systems 6, 223–228.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28, 656–663.Google Scholar
Gelenbe, E. (1994). G-networks: a unifying model for neural and queueing networks. Ann. Operat. Res. 48, 433–461.CrossRefGoogle Scholar
Gomez-Corral, A. (1996). Queueing systems with repeated attempts and negative arrivals. , Department of Statistics and O.R., University Complutense of Madrid, Madrid. (In Spanish).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
Jain, G., and Sigman, K. (1996). A Pollaczek–Khintchine formula for M/G/1 queues with disasters. J. Appl. Prob. 33, 1191–1200.CrossRefGoogle Scholar
Kok, A. G. De (1984). Algorithmic methods for single server systems with repeated attempts. Statist. Neerlandica 38, 23–32.CrossRefGoogle Scholar
Martin, M., and Artalejo, J. R. (1995). Analysis of an M/G/1 queue with two types of impatient units. Adv. Appl. Prob. 27, 840–861.Google Scholar
Neuts, M. F., and Rao, B. M. (1990). Numerical investigation of a multi-server retrial model. Queueing Systems 7, 169–190.CrossRefGoogle Scholar
Schellhaas, H. (1986). Computation of the state probabilities in a class of semi-regenerative queueing models. In Semi-Markov Models: Theory and Applications, ed. Janssen, J. Plenum Press, New York, pp. 111–130.CrossRefGoogle Scholar
Takács, L. (1962). Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Tijms, H. C. (1994). Stochastic Models: An Algorithmic Approach. Wiley, Chichester, UK.Google Scholar
Yang, T., and Templeton, J. G. C. (1987). A survey on retrial queues. Queueing Systems 2, 203–233.Google Scholar
Zabreyko, P. P., Koshelev, A. I., Krasnosel'skii, M. A., Mikhlin, S. G., Rakovshchik, L. S. and Stet'senko, V. Ya. (1986). Integral Equations – A Reference Text. Noordhoff, Leyden, The Netherlands.Google Scholar