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On the length and number of served customers of the busy period of a generalised M/G/1 queue with finite waiting room

Published online by Cambridge University Press:  01 July 2016

Stig I. Rosenlund
Stig I. Rosenlund*
Affiliation:
University of Stockholm
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Abstract

Customers arrive in groups to a single server queue with finite waiting room. Two-dimensional distributions for times and numbers of served customers between occurrences of states in the embedded Markov chain are obtained by linear algebra giving systems of equations for joint Laplace-Stieltjes transforms. For M/M/1 a simple recursion relation for the joint transform of the two variables in the title is derived and used to obtained the first and second moments.

Keywords

QUEUEFINITE WAITING ROOMGENERALISED POISSON PROCESSEMBEDDED MARKOV CHAINBUSY PERIOD LENGTHJOINT TRANSFORMSLINEAR EQUATIONS FOR TRANSFORMSLIMIT THEOREM FOR BUSY PERIOD LENGTH
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 2 , August 1973 , pp. 379 - 389
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Cohen, J. W. (1971) On the busy periods for the M/G/1 queue with finite and with infinite waiting room. J. Appl. Prob. 8, 821–827.Google Scholar
[2] Rosenlund, S. I. (1973) An M/G/1 model with finite waiting room in which a customer remains during part of service. J. Appl. Prob. 10, No. 4. To appear.Google Scholar
[3] Råde, L. (1972) A model for interaction of a Poisson and a renewal process and its relation with queueing theory. J. Appl. Prob. 9, 451–456.Google Scholar
[4] Råde, L. (1972) Limit theorems for thinning of renewal point processes. J. Appl. Prob. 9, 847–851.Google Scholar
[5] Tomko, J. (1967) A limit theorem for a queue when the input rate increases indefinitely. (In Russian) Studia Sci. Math. Hung. 2, 447–454.Google Scholar