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Conditioned limit theorems for waiting-time processes of the M/G/1 queue

Published online by Cambridge University Press:  14 July 2016

G. Hooghiemstra
G. Hooghiemstra*
Affiliation:
Delft University of Technology
*
∗ Postal address: Technische Hogeschool Delft, Department of Mathematics and Informatics, P.O. Box 356, 2600 AJ Delft, The Netherlands.
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Abstract

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.

Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.

The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

Keywords

BROWNIAN EXCURSIONBROWNIAN MEANDERCONDITIONED LIMIT THEOREMSFUNCTIONAL LIMIT THEOREMSM/G/1 QUEUEWEAK CONVERGENCE

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983, pp. 675 - 688
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

The paper is based on a part of the author's doctoral dissertation, which was written under the supervision of Professor J. W. Cohen.

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