Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-07T22:23:06.875Z Has data issue: false hasContentIssue false

An analysis of a modified M/G/1 queue using a martingale technique

Published online by Cambridge University Press:  14 July 2016

Matthew Roughan*
Affiliation:
CSSIP, Adelaide
*
Postal address: Cooperative Research Centre for Sensor Signal and Information Processing, SPRI Building, Technology Park, The Levels, Adelaide, South Australia 5095, Australia.

Abstract

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1996 

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

Baccelli, F. and Makowski, A. M. (1985) Direct martingale arguments for stability: the M/GI/1 case. Syst. Cont. Lett. 6, 181186.Google Scholar
Baccelli, F. and Makowski, A. M. (1989) Dynamic, transient and stationary behaviour of the M/GI/1 queue via martingales. Ann. Prob. 17, 16911699.Google Scholar
Barnett, S. and Storey, S. (1970) Matrix Methods in Stability. Nelson, London.Google Scholar
Cooper, R. B. (1972) Introduction to Queueing Theory. Macmillian, London.Google Scholar
Golub, G. H. and Van Loan, C. F. (1983) Matrix Computations. North Oxford Academic, Oxford.Google Scholar
Householder, A. S. (1964) The Theory of Matrices in Numerial Analysis. Blaisdell, New York.Google Scholar
Nakagawa, T. and Osaki, S. (1976) Markov renewal processes with some non-regeneration points and their applications to reliability theory. Microelect. Reliability 15, 633636.CrossRefGoogle Scholar
Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and their Applications. Marcel Dekker, New York.Google Scholar
Neveu, J. (1975) Discrete-Parameter Martingales. North-Holland, Amsterdam.Google Scholar
Roughan, M. (1994) An Application of Martingales to Queueing Theory. , Department of Applied Mathematics, University of Adelaide.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, Oxford.Google Scholar
Williams, D. (1991) Probability with Martingales. Cambridge University Press, Cambridge.Google Scholar
Wolff, R. W. (1989) Stochastic Modelling and the Theory of Queues. Prentice Hall, New York.Google Scholar
Yeo, G. F. (1962) Single server queue with modified service mechanisms. J. Operat. Res. 1, 499507.Google Scholar