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On the transient state probabilities for a queueing model where potential customers are discouraged by queue length

Published online by Cambridge University Press:  14 July 2016

Bent Natvig
Bent Natvig*
Affiliation:
University of Trondheim — NTH, Norway
*
*Now at the University of Sheffield.
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Abstract

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p∗i0, p∗in (n ≥ 1) being determined recursively. An exact expression for p10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.

Keywords

BIRTH AND DEATH QUEUEING PROCESSSTATE DEPENDENT QUEUEING MODELDISCOURAGED CUSTOMERSQUEUE LENGTHOPERATIVE ADVANTAGESTRANSIENT BEHAVIOURNUMERICAL EVALUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 2 , June 1974 , pp. 345 - 354
Copyright
Copyright © Applied Probability Trust 1974 

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References

Erdélyi, A. et al. (1954) Tables of Integral Transforms, 1. McGraw-Hill, New York.Google Scholar
Hadidi, N. (1969) On the service time distribution and the waiting time process of a potentially infinite capacity queueing system. J. Appl. Prob. 6, 594–603.Google Scholar
Hadidi, N. and Conolly, B. W. (1969a) On the reduction of congestion. Statistical Research Report No. 6, Inst. of Mathematics, University of Oslo.Google Scholar
Hadidi, N. and Conolly, B. W. (1969b) On the improvement of the operational characteristics of single server queues by the use of a queue length dependent service mechanism. Appl. Statist. 18, 229–240.Google Scholar
Karlin, S. and Mcgregor, J. (1957a) The differential equations of birth and death processes and Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Natvig, B. (1975) On a queueing model where potential customers are discouraged by queue length. Scand. J. Statist. To appear.Google Scholar