Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-22T00:39:22.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for generalized single server queues

Published online by Cambridge University Press:  01 July 2016

Austin J. Lemoine
Austin J. Lemoine*
Affiliation:
University of Texas at Austin
Get access Rights & Permissions [Opens in a new window]

Abstract

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.

Keywords

GENERALIZED SINGLE SERVER QUEUESWAITING TIMEWORKLOADQUEUE-LENGTHWEAK CONVERGENCEDELAYED RANDOM WALKSOPTIONAL RANDOM VARIABLESCUMULATIVE PROCESSESFUNCTIONAL LIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 1 , March 1974 , pp. 159 - 174
Copyright
Copyright © Applied Probability Trust 1974 

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] Chung, K. (1967) Markov Chains with Stationary Transition Probabilities. 2nd Ed. Springer-Verlag, Berlin.Google Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley, New York.Google Scholar
[3] Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. I, 3rd Ed. John Wiley, New York.Google Scholar
[4] Finch, P. D. (1959) A probability limit theorem with application to a generalization of queueing theory. Acta Math. Acad. Aci. Hung. 10, 317–325.Google Scholar
[5] Harris, C. M. (1966) A queueing system with multiple service time distributions. Naval Res. Log. Quart. 14, 231–239.Google Scholar
[6] Iglehart, D. L. (1971) Functional limit theorems for the queue GI/G/1 in light traffic. Adv. Appl. Prob. 3, 269–281.Google Scholar
[7] Lemoine, A. J. (1973) Delayed random walks. Stochastic Processes and their Applications. 1, 251–268.Google Scholar
[8] Lemoine, A. J. (1972) Limit theorems for generalized single server queues, I: Basic processes. Center for Cybernetic Studies Report No. 98, The University of Texas at Austin.Google Scholar
[9] Lemoine, A. J. (1973) Limit theorems for generalized single server queues, II: Cumulative processes and extreme values. Center for Cybernetic Studies Report No. 99, The University of Texas at Austin.Google Scholar
[10] Lemoine, A. J. (1973) Limit theorems for generalized single server queues: The exceptional system. To appear.Google Scholar
[11] Loris-Teghem, J. (1971) On the waiting-time distribution in a generalized GI/G/1 queueing system. J. Appl. Prob. 8, 241–251.CrossRefGoogle Scholar
[13] Neuts, M. F. (1971) A queue subject to extraneous phase changes. Adv. Appl. Prob. 3, 79–119.Google Scholar
[13] Pollaczek, F. (1957) Problèmes stochastiques posés par le phénomène de formation d'une queue d'attente à un guichet et par des phénomènes apparentés. Memor. Sci. Math. 136. Gauthiers-Villars, Paris.Google Scholar
[14] Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A 232, 6–31.Google Scholar
[15] Takács, L. (1961) The transient behavior of a single server queueing process with a Poisson input. Proc. Fourth Berkeley Symp. on Math. Statist. and Prob. II, 535–567.Google Scholar
[16] Takács, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service times. Sankhyā A 25, 91–100.Google Scholar
[17] Welch, P. D. (1964). On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736–752.Google Scholar