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The departure process for the M/M/1 queue

Published online by Cambridge University Press:  14 July 2016

John R. Hubbard ,
Claude Dennis Pegden  and
Matthew Rosenshine
John R. Hubbard*
Affiliation:
The University of Richmond
Claude Dennis Pegden
Affiliation:
The Pennsylvania State University
Matthew Rosenshine*
Affiliation:
The Pennsylvania State University
*
∗Postal address: Mathematics and Computer Science, The University of Richmond, VA 23173, USA.
∗∗Postal address: to Dept, of Industrial and Management Systems Engineering, The Pennsylvania State University, 207 Hammond Building, University Park, PA 16802, USA.
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Abstract

The problem of determining the probability of j departures during the time interval (0, t) from an M/M/l queue empty at t = 0 is considered. A closed-form solution is obtained. It is shown that this solution is unique and invariant under interchanging the arrival rate and service rate. Finally, sample computational representations of the solution are developed and results of a simple computation are provided.

Keywords

TRANSIENT M/M/1 DEPARTURESARRIVAL RATE–SERVICE RATE INTERCHANGEABILITYDEPARTURE PROCESS COMPUTATIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 249 - 255
Copyright
Copyright © Applied Probability Trust 1986 

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References

[1] Boxma, O. J. (1983) The joint arrival and departure process for the M/M/l queue. Reprint No. 307, Department of Mathematics, University of Utrecht.Google Scholar
[2] Daniels, S. C. (1982) A Computer Implementation of the Transient M/M/1 Queue Departure Process. M.Sc. Paper in Computer Science and Operations Research, The Pennsylvania State University.Google Scholar
[3] Hirsch, M. W. and Smale, S. (1974) Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York.Google Scholar
[4] Hubbard, J. R. (1983) Some New Results for the M/M/1 Queue and its Departure Process. M.Sc. Paper in Computer Science, The Pennsylvania State University.Google Scholar
[5] Pegden, C. D. and Rosenshine, M. (1982) Some new results for the M/M/1 queue. Management Sci. 28, 821–828.Google Scholar
[6] Rosenshine, M. and Pegden, C. D. (1982) The departure process for the M/M/1 queue. ISME Working Paper 82-122, The Pennsylvania State University.Google Scholar
[7] Towsley, D. (1983) An application of the reflection principle to the transient analysis of the M/M/1 queue. IBM Report RC 9940.Google Scholar