Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T00:17:46.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Single-server queues with impatient customers

Published online by Cambridge University Press:  01 July 2016

F. Baccelli ,
P. Boyer  and
G. Hebuterne
F. Baccelli*
Affiliation:
INRIA
P. Boyer*
Affiliation:
CNET
G. Hebuterne*
Affiliation:
CNET
*
∗Postal address: INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France.
∗∗ Postal address: CNET, LAA-SLC-EVP, Route de Tregastel, 22301 Lannion A, France.
∗∗ Postal address: CNET, LAA-SLC-EVP, Route de Tregastel, 22301 Lannion A, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

Keywords

QUEUEING THEORYLIMITED WAITING TIMESERGODIC MARKOV CHAINACTUAL WAITING TIMESVIRTUAL WAITING TIMESREGENERATIVE PROCESSESINVARIANT MEASUREFUNCTIONAL EQUATION
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 4 , December 1984 , pp. 887 - 905
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.)

References

[1] Baccelli, F. and Hebuterne, G. (1981) On queues with impatient customers. In Performance 81, ed. Kylstra, F. J. Elsevier, Amsterdam.Google Scholar
[2] Barrer, D. Y. (1957) Queueing with impatient customers and ordered service. Operat. Res. 5, 650–656.Google Scholar
[3] Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, Berlin.Google Scholar
[4] Boyer, P. and Hebuterne, G. (1983) Relations de conservation pour une file d'attente avec clients impatients. Ann. Telecomm. 38 (5-6), 226–230.Google Scholar
[5] Callahan, J. R. (1973) A queue with waiting time dependent service time. Naval Res. Logist. Quart. 20, 321–324.Google Scholar
[6] Charlot, F. and Pujolle, G. (1978) Recurrence in single server queues with impatient customers. Ann. Inst. H. Poincaré, B14, 399–410.Google Scholar
[7] Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[8] Cohen, J. W. (1969) Single server queues with restricted accessibility. J. Engineering Math. 3, 265–284.Google Scholar
[9] Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Springer-Verlag, Berlin.Google Scholar
[10] Daley, D. J. (1964) Single server queueing systems with uniformly limited queueing time. J. Austral. Math. Soc. 4, 489–505.Google Scholar
[11] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[12] Gavish, B. and Schweitzer, P. J. (1977) The Markovian queue with bounded waiting time. Management Sci. 23, 1349–1357.Google Scholar
[13] Gnedenko, B. V. and Kovalenko, I. N. (1968) Introduction to Queueing Theory. Israel Program for Scientific Translations, Jerusalem.Google Scholar
[14] Haugen, R. and Skogan, E. (1980) Queueing systems with stochastic time out. IEEE Trans. Comm. 28, 1984–1989.Google Scholar
[15] Hokstad, P. A. (1979) Single server queue with constant service time and restricted accessibility. Management Sci. 25, 205–208.Google Scholar
[16] Laslett, G. M., Pollard, D. B. and Tweedie, R. L. (1978) Techniques for establishing ergodic and recurrence properties of continuous valued Markov chains. Naval Res. Logist. Quart. 25, 455–472.CrossRefGoogle Scholar
[17] Loève, M. (1977) Probability Theory, 4th edn. Springer-Verlag, Berlin.Google Scholar
[18] Mikhlin, S. G. (1957) Integral Equations. Pergamon Press, Oxford.Google Scholar
[19] 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 A 254, 3965–3967.Google Scholar
[20] Stidham, S. (1972) Regenerative processes in the theory of queues. Adv. Appl. Prob. 4, 532–577.Google Scholar
[21] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[22] Takács, L. (1974) A single server queue with limited virtual waiting time. J. Appl. Prob. 11, 612–617.Google Scholar
[23] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
[24] Tweedie, R. L. (1977) Hitting times of Markov chains with application to state-dependent queues. Bull. Austral. Math. Soc. 17, 97–107.Google Scholar