Hostname: page-component-cb9f654ff-plnhv Total loading time: 0.001 Render date: 2025-08-24T02:21:00.044Z Has data issue: false hasContentIssue false
  • English
  • Français

On the transient behavior of the repairman problem

Published online by Cambridge University Press:  01 July 2016

Charles Knessl
Charles Knessl*
Affiliation:
University of Illinois at Chicago
*
∗ Postal address: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago (MC/249), Chicago, IL 60680, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the repairman problem which corresponds to the finite population M/M/1 queue. Asymptotic approximations for the transient probability distribution of the number of broken machines constructed when the number M of machines is large and the service (repair) rate is also large, specifically, O(M). The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions. Extensive numerical comparisons show the quality of our approximations.

Keywords

TRANSIENT PROBABILITIESSINGULAR PERTURBATIONSRAY METHOD

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 327 - 354
Copyright
Copyright © Applied Probability Trust 1991 

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.)

Article purchase

Temporarily unavailable

References

1. Abate, J. and Whitt, W. (1987) Transient behaviour of the M/M/1 queue: starting at the origin. QUESTA 2, 41–65.Google Scholar
2. Abate, J. and Whitt, W. (1988) Approximations for the M/M/1 busy-period distribution. Liber Amicorum for Professor J. W. Cohen. Google Scholar
3. Barlow, R. E. (1962) Repairman problems. In Studies in Applied Probability and Management Science , ed Arrow, K. J., Karlin, S. and Scarf, H., Stanford University Press, pp. 18–33.Google Scholar
4. Bender, C. M. and Orszag, S. A. (1978) Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.Google Scholar
5. Feller, W. (1950) An Introduction to Probability Theory and its Applications , Vol. 1. Wiley, New York.Google Scholar
6. Gaver, D. P. and Lehoczky, J. P. (1977) A diffusion approximation solution for a repairman problem with two types of failures. Management Sci. 24, 71–81.CrossRefGoogle Scholar
7. Iglehart, D. L. (1965) Limit diffusion approximations for the many server queue and the repairman problem. J. Appl. Prob. 2, 429–441.CrossRefGoogle Scholar
8. Karmeshu, and Jaiswal, N. K. (1981) A machine interference model with threshold effect. J. Appl. Prob. 18, 491–498.CrossRefGoogle Scholar
9. Keller, J. B. (1978) Rays, waves and asymptotics. Bull. Amer. Math. Soc. 84, 727–750.CrossRefGoogle Scholar
10. Kevorkian, J. and Cole, J. D. (1981) Perturbation Methods in Applied Mathematics . Springer-Verlag, New York.CrossRefGoogle Scholar
11. Kleinrock, L. (1975) Queueing Systems , Vol. I. Wiley, New York.Google Scholar
12. Kleinrock, L. (1976) Queueing Systems , Vol. II. Wiley, New York.Google Scholar
13. Mitra, D. and Weiss, A. (1988) The transient behavior in Erlang's model for large trunk groups and various traffic conditions. Proc. 12th Internat. Teletraffic Congr., Torino , pp. 5. 1B4.1–5.1B4.8.Google Scholar