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Invariance principles in queueing theory

Published online by Cambridge University Press:  14 July 2016

Michael Alex and
Josef Steinebach
Michael Alex*
Affiliation:
University of Marburg
Josef Steinebach*
Affiliation:
University of Hannover
*
∗ Postal address: Fachbereich Mathematik, Universität Marburg, Hans-Meerwein Strasse, D-3550 Marburg, W. Germany.
∗∗ Postal address: Institute für Math. Stochastik, Universität Hannover, Welfengarten 1, D-3000 Hannover, W. Germany.
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Abstract

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk }and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Keywords

STRONG APPROXIMATIONSGI/G/I QUEUEDARLING—ERDÖS TYPE THEOREMSEXTREME VALUE LIMITING DISTRIBUTIONS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989, pp. 845 - 857
Copyright
Copyright © Applied Probability Trust 1989 

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