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On a new formula for the transient state probabilities for M/M/1 queues and computational implications

Part of: Approximations and expansions Special processes

Published online by Cambridge University Press:  14 July 2016

B. W. Conolly  and
Christos Langaris
B. W. Conolly*
Affiliation:
Queen Mary and Westfield College
Christos Langaris*
Affiliation:
University of Ioannina
*
∗ Postal address: Queen Mary and Westfield College, University of London, Mile End Road, London E1 4NS, UK.
∗∗ Postal address: University of Ioannina, Department of Mathematics, 45110 Ioannina, Greece.
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Abstract

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.

Keywords

EXPONENTIAL QUEUESCOMPUTATIONAL FEATURES AND PERFORMANCE

MSC classification

Primary: 60K25: Queueing theory
Secondary: 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 237 - 246
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Abate, J. and Whitt, W. (1988) Transient behavior of the M/M/1 queue via Laplace transforms. Adv. Appl. Prob. 20, 145–178.CrossRefGoogle Scholar
[2] Abate, J. and Whitt, W. (1989) Calculating time-dependent performance measures for the M/M/1 queue. IEEE Trans. Comm. 37, 1102–1104.Google Scholar
[3] Abramowitz, M. and Stegun, I. (1970) Handbook of Mathematical Functions. Dover.Google Scholar
[4] Baccelli, F. and Massey, W. A. (1989) A sample path analysis of the M/M/1 queue. J. Appl. Prob. 26, 418–422.Google Scholar
[5] Böhm, W. and Mohanty, S. G. (1990) The transient solution of M/M/1 queues under (M, N) policy. A combinatorial approach. J. Statist. Planning Inf.Google Scholar
[6] Boxma, O. J. (1984) The joint arrival and departure process for the M/M/1 queue. Statist. Neerlandica 38, 199–208.Google Scholar
[7] Champernowne, D. G. (1956) An elementary method of solution of the queueing problem with a single server and constant parameters. J. R. Statist. Soc. B18, 125–128.Google Scholar
[8] Cohen, J. W. (1982) The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
[9] Conolly, B. W. (1975) Lecture Notes on Queueing Systems. Ellis Horwood. Chichester.Google Scholar
[10] Parthasarathy, P. R. (1987) A transient solution to an M/M/1 queue: a simple approach. Adv. Appl. Prob. 19, 997–998.Google Scholar
[11] Parthasarathy, P. R. and Sharafali, M. (1989) Transient solution to the many-server Poisson queue. J. Appl. Prob. 26, 584–594.Google Scholar
[12] Pegden, C. D. and Rosenshine, M. (1982) Some new results for the M/M/l queue. Management Sci, 28, 821–828.Google Scholar
[13] Sharma, O. P. (1990) Markovian Queues. Ellis Horwood. Chichester.Google Scholar
[14] Syski, R. (1988) Further comments on the solution of the M/M/1 queue. Adv. Appl. Prob. 20, 693.Google Scholar
[15] Towsley, D. (1987) An application of the reflection principle to the transient analysis of the M/M/l queue. Naval. Res. Logist. 34, 451–456.3.0.CO;2-8>CrossRefGoogle Scholar
[16] Vetterling, W. T., Teukolsky, S. A., Flannery, B. P. and Press, W. H. (1986) Numerical Recipes. Cambridge University Press.Google Scholar