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Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Władysław Szczotka  and
Krzysztof Topolski
Władysław Szczotka
Affiliation:
Wrocław University
Krzysztof Topolski*
Affiliation:
Wrocław University
*
* Postal address for both authors: Mathematical Institute of Wrocław University, Pl. Grunwałdzki 2/4, 50–384 Wrocław, Poland.
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Abstract

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

Keywords

SINGLE-SERVER QUEUEING SYSTEMBUSY PERIODBROWNIAN MEANDERWIENER PROCESSCONDITIONED LIMIT THEOREM

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 1 , March 1994 , pp. 242 - 257
Copyright
Copyright © Applied Probability Trust 1994 

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