Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T14:42:07.651Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating last-exit probabilities of a random walk, by application to conditional queue length moments within busy periods of M/GI/1 queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

D. J. Daley  and
L. D. Servi
Get access Rights & Permissions [Opens in a new window]

Abstract

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion.

The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

Keywords

M/D/1 QUEUEM/GI/1 QUEUETABOO PROBABILITIESCONJUGATE DISTRIBUTIONS

MSC classification

Secondary: 60K25: Queueing theory
Type
Part 4 Random Walks
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 251 - 267
Copyright
Copyright © Applied Probability Trust 1994 

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

Blumenthal, R. M. (1992) Excursions of Markov Processes. Birkhäuser, Boston.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities , 2nd edn. Springer-Verlag, Heidelberg.Google Scholar
Daley, D. J. and Servi, L. D. (1992) Exploiting Markov chains to infer queue-length from transactional data. J. Appl. Prob. 29, 713–732.Google Scholar
Daley, D. J. and Servi, L. D. (1993) A two-point Markov chain boundary value problem. Adv. Appl. Prob. 25, 607–630.Google Scholar
Feller, W. (1959) On combinatorial methods in fluctuation theory. In Probability and Statistics (The Harald Cramér Volume) , ed. Grenander, Ulf, pp. 74–91. Almqvist & Wiksell, Stockholm, and Wiley, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Hooghiemstra, G. (1983) Conditioned limit theorems for waiting-time processes of the M/G/1 queue. J. Appl. Prob. 20, 675–688.Google Scholar
Iglehart, D. L. (1975) Conditioned limit theorems for random walks. In Stochastic Processes and Related Topics , ed. Puri, M. L., Vol. 1. Academic Press, New York.Google Scholar
Ito, K. and Mckean, H. P. Jr (1965) Diffusion Processes and their Sample Paths . Springer-Verlag, Berlin.Google Scholar
Keilson, J. (1965) Green's Function Methods in Probability Theory. Griffin, London.Google Scholar
Larson, R. C. (1990) The queue inference engine: deducing queue statistics from transactional data. Management Sci. 36, 586–601.CrossRefGoogle Scholar
Moran, P. A. P. (1958) Approximate relations between series and integrals. Math. Tables and other Aids to Computation 12, 34–37.Google Scholar
Moran, P. A. P. (1968) An Introduction to Probability Theory. Clarendon Press, Oxford.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton. 2nd edn 1976, Springer-Verlag, New York.CrossRefGoogle Scholar