Queues with Workload-Dependent Arrival and Service Rates

Authors:
R. Bekker;S. C. Borst;O. J. Boxma;O. Kella
Affiliations:
CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands, and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ...;CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands, and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ...;CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands, and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ...;Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel offer.kella@huji.ac.il
Venue:
Queueing Systems: Theory and Applications
Year:
2004

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Abstract

We consider two types of queues with workload-dependent arrival rate and service speed. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet. First, in the M/G/1 case, we compare the steady-state distribution of the workload (both at arbitrary epochs and at arrival instants) in two models, in which the ratio of arrival rate and service speed is equal. Applying level crossing arguments, we show that the steady-state distributions are proportional. Second, we consider a G/G/1-type queue with workload-dependent interarrival times and service speed. Using a stochastic mean-value approach, several well-known relations for the workload at various epochs in the ordinary G/G/1 queue are generalized.