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Time and customer processes in queues with stationary inputs

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Tokyo Institute of Technology, Japan
*
*Now at Science University of Tokyo, Noda City, Chiba, Japan.
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Abstract

Two types of processes occurring in queues with stationary inputs are considered. They are called ‘time processes’ and ‘customer processes’. Sufficient conditions for the convergence of sample averages and the existence of limiting distributions for each type of processes are given. The results generalize those of Loynes (1962). The results are applied to three queueing processes and the Little's formula L = λW is obtained under rather general conditions.

Keywords

QUEUESSTATIONARY INPUTSPOINT PROCESSERGODIC PROPERTYCONVERGENCE OF SAMPLE AVERAGESEXISTENCE OF LIMITING DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 2 , June 1977 , pp. 349 - 357
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Benés, V. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading, Mass.Google Scholar
[2] Billingsley, P. (1965) Ergodic Theory and Information. Wiley, New York.Google Scholar
[3] Brumelle, S. L. (1971) On the relation between customer and time averages in queues. J. Appl. Prob. 8, 508–520.Google Scholar
[4] Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes, ed. Lewis, P. A. W., 299–383.Google Scholar
[5] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[6] Little, J. D. C. (1961) A proof for the queueing formula: L = ?W. Opns Res. 9, 383–387.Google Scholar
[7] Loynes, R. M. (1962) The stability of a queue with nonindependent interarrival and service time. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[8] Mecke, J. (1967) Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.Google Scholar
[9] Miller, D. R. (1974) Limit theorems for path functionals of regenerative processes. Stoch. Proc. Appl. 2, 141–161.Google Scholar
[10] Miller, D. R. and Sentilles, F. D. (1975) Translated renewal processes and the existence of a limiting distribution for the queue length of the GI/G/s queue. Ann. Prob. 3, 424–439.Google Scholar
[11] Miyazawa, M. (1975a) Time and customer quantities in queues with stationary loads. Research Report on Information Sciences B-19, Tokyo Institute of Technology.Google Scholar
[12] Miyazawa, M. (1975b) On the existence of some limit distributions in queues with stationary loads, Research Report on Information Sciences B-24, Tokyo Institute of Technology.Google Scholar
[13] Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 455–465.Google Scholar