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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
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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 

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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