Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T08:27:58.752Z Has data issue: false hasContentIssue false
  • English
  • Français

A service model in which the server is required to search for customers

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts  and
M. F. Ramalhoto
Marcel F. Neuts*
Affiliation:
University of Delaware
M. F. Ramalhoto*
Affiliation:
Instituto Superior Tecnico, Lisbon
*
∗ Postal address: Applied Mathematics Institute, University of Delaware, Newark, DE 19711, U.S.A.
∗∗ Postal address: Department of Mathematics, Instituto Superior Tecnico, Av. Rovisco Pais, 1500 Lisbon, Portugal.
Get access Rights & Permissions [Opens in a new window]

Abstract

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.

Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

Keywords

THEORY OF QUEUESCOMPUTATIONAL PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 1 , March 1984 , pp. 157 - 166
Copyright
Copyright © Applied Probability Trust 1984 

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.)

Footnotes

Research supported by the National Science Foundation under Grant No. ENG-7908351 and the Air Force Office of Scientific Research under Grant No. AFOSR-77-3236.

Support from the Fulbright travelling scholarship is acknowledged.

References

[1] Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. U.S. Dept, of Commerce, National Bureau of Standards, Appl. Math. Series No. 55.Google Scholar
[2] Aleksandrov, A. M. (1974) A queueing system with repeated orders. Engineering Cybernetics 12, 1–4.Google Scholar
[3] Falin, G. I. (1979) A single-line system with secondary orders. Engineering Cybernetics 17, 76–83.Google Scholar
[4] Kamke, E. (1971) Differentialgleichungen-Lösungsmethoden und Lösungen I. Chelsea, New York.Google Scholar
[5] Keilson, J., Cozzolino, J. and Young, M. (1968) A service system with unfilled requests repeated. Operat. Res. 16, 1126–1137.CrossRefGoogle Scholar
[6] Kulkarni, V. G. (1982) Letter to the editor. J. Appl. Prob. 19, 901–904.CrossRefGoogle Scholar
[7] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore, Md.Google Scholar
[8] Schassberger, R. (1973) Warteschlangen. Springer-Verlag, New York.CrossRefGoogle Scholar