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On the waiting-time and busy period distributions for a general birth-and-death queueing model

Published online by Cambridge University Press:  14 July 2016

Bent Natvig
Bent Natvig*
Affiliation:
University of Sheffield
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Abstract

A general birth-and-death queueing model is considered with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and a first-come-first-served queueing discipline. The first and second order moments of the steady-state waiting-time (excluding service) for a non-lost arriving customer are given. By setting the busy period equal to the time where at least one service is in progress, we obtain the first and second order moments of the length of a busy period and also the distribution of the number served during it, given an arbitrary number of customers present originally. Using a direct approach all expressions are given in explicit forms which, although being far from elegant, are suitable for evaluation on a computer.

Keywords

GENERAL BIRTH-AND-DEATH QUEUEING MODELSTATE-DEPENDENTWAITING TIMEBUSY PERIODDIRECT METHODEXPLICIT RESULTS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 3 , September 1975 , pp. 524 - 532
Copyright
Copyright © Applied Probability Trust 1975 

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References

Conolly, B. W. (1974) The generalised state dependent Erlangian queue: The busy period. J. Appl. Prob. 11, 618–623.Google Scholar
Conolly, B. W. (1975) The generalised state dependent Erlangian queue: Speculations on calculation of measures of effectiveness. J. Appl. Prob. 12, 358–363.CrossRefGoogle Scholar
Erdélyi, A. et al. (1954) Tables of Integral Transforms 1. McGraw-Hill, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. I. (Third edition) Wiley, New York.Google Scholar
Hadidi, N. (1969) On the service time distribution and waiting time process of a potentially infinite capacity queueing system. J. Appl. Prob. 6, 594–603.Google Scholar
Hadidi, N. and Conolly, B. W. (1969) On the reduction of congestion. Research Report, Inst. of Math., University of Oslo.Google Scholar
Karlin, S. and Mcgregor, J. (1957) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Knopp, K. (1945) Theory of Functions, Part 1 . Dover, New York.Google Scholar
Natvig, B. (1974) The probability law of the busy period for D/M/1 obtained by an immediate method. Research Report, Department of Probability and Statistics, University of Sheffield.Google Scholar
Natvig, B. (1975a) On a queueing model where potential customers are discouraged by queue length. Scand. J. Statist , 2, 34–42.Google Scholar
Natvig, B. (1975b) On the input and output processes for a general birth-and-death queueing model. Adv. Appl. Prob. 7, 576–592.Google Scholar
Wolff, R. W. (1965) Problems of statistical inference for birth and death queueing models. Operat. Res. 13, 343–357.Google Scholar