Hostname: page-component-84b7d79bbc-l82ql Total loading time: 0 Render date: 2024-07-30T07:25:19.272Z Has data issue: false hasContentIssue false
  • English
  • Français

General customer impatience in the queue GI/G/1

Published online by Cambridge University Press:  14 July 2016

D. J. Daley
D. J. Daley*
Affiliation:
University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

For the queueing system GI/G/1 with both waiting-line and service-line customer impatience an integral equation for the limiting waiting-time distribution function W(x) is derived and the existence of W(x) proved when the system satisfies certain conditions. Two counter examples are constructed showing that W(x) does not necessarily exist. A sufficient condition for its existence in GI/G/1 with deterministic waiting-line impatience is given. Formal solutions for M/G/l are derived for deterministic impatience and negative exponential impatience.

Type
Research Papers
Information
Journal of Applied Probability , Volume 2 , Issue 1 , June 1965 , pp. 186 - 205
Copyright
Copyright © Sheffield: Applied Probability Trust 

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

[1] Ancker, C. J. and Gafarian, A. V. (1962) Queueing with impatient customers who leave at random. J. Industrial Eng. 13, 86–87.Google Scholar
[2] Ancker, C. J. and Gafarian, A. V. (1963) Some queueing problems with balking and reneging, I. Operat. Res. 11, 88–100.Google Scholar
[3] Barrer, D. Y. (1957) Queueing with impatient customers and ordered service. Operat. Res. 5, 650–656.Google Scholar
[4] Brody, S. M. (1959) On an integro-differential equation for systems with t-waiting. (In Ukrainian). Dopov. Akad. Nauk Ukrain. SSR. 571–573. (Transl. in Selected Translations in Mathematical Statistics and Probability 4, 13–16.).Google Scholar
[5] Daley, D. J. Single-server queueing systems with uniformly limited queueing time. (Submitted to J. Aust. Math. Soc.) .Google Scholar
[6] Feller, W. (1941) On the integral equation of renewal theory. Ann. Math. Statist. 12, 243–267.Google Scholar
[7] Feller, W. (1957) Introduction to Probability Theory and its Applications (Second Edition). John Wiley, New York.Google Scholar
[8] Finch, P. D. (1960) Deterministic customer impatience in the queueing system GI/M/1 Biometrika 47, 45–52.Google Scholar
[9] Finch, P. D. (1961) Deterministic customer impatience in the queueing system GI/M/1: a correction. Biometrika 48, 472–473.CrossRefGoogle Scholar
[10] Gurk, H. M. (1957) Single-server, time-limited queues (Abstract). Bull. Amer. Math. Soc. 63, 400.Google Scholar
[11] Hardy, G. H. and Wright, E. M. (1960) An Introduction to the Theory of Numbers (Fourth Edition). Oxford U. P., London.Google Scholar
[12] Kovalenko, I. N. (1961) Some queueing problems with restrictions. (In Russian). Teor. Veroyat. Primen 6, 222–228. [Transl. in Theory of Probability and its Applications (1961) 6, 204–208].Google Scholar
[13] Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
[14] Loève, M. (1963) Probability Theory (Third Edition). D. van Nostrand, New York.Google Scholar
[15] Pollaczek, F. (1962) Sur une théorie unifiée des problèmes stochastiques soulevés par l'encombrement d'un faisceau parfait de lignes téléphoniques. C. R. Acad. Sci. Paris 254, 3965–3967.Google Scholar
[16] Prabhu, N. U. (1958) On the integral equation for the finite dam. Quart. J. Math. (2)9, 183–188.Google Scholar
[17] Saaty, T. L. (1961) Elements of Queueing Theory with Applications. McGraw-Hill, New York.Google Scholar
[18] Störmer, H. (1956) Wartezeitlenkung in Handbedienten Vermittlungsanlagen. Arch. Elektrotech. 10, 58–64.Google Scholar
[19] Takács, L. (1955) Investigation of waiting-time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6, 101–129.Google Scholar