The busy period in the fluid queue

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The busy period in the fluid queue

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ACM SIGMETRICS Performance Evaluation Review archive
Volume 26 , Issue 1 (June 1998) table of contents

Pages: 100 - 110

Year of Publication: 1998

ISSN:0163-5999

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Authors

O. J. Boxma	CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
V. Dumas	MAB - Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France

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ACM New York, NY, USA

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Downloads (6 Weeks): 2, Downloads (12 Months): 31, Citation Count: 10

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ABSTRACT

Consider a fluid queue fed by N on/off sources. It is assumed that the silence periods of the sources are exponentially distributed, whereas the activity periods are generally distributed. The inflow rate of each source, when active, is at least as large as the outflow rate of the buffer.We make two contributions to the performance analysis of this model. Firstly, we determine the Laplace-Stieltjes transforms of the distributions of the busy periods that start with an active period of source i, i = 1,…,N, as the unique solution in [0, 1]^N of a set of N equations. Thus we also find the Laplace-Stieltjes transform of the distribution of an arbitrary busy period.Secondly, we relate the tail behaviour of the busy period distributions to the tail behaviour of the activity period distributions. We show that the tails of all busy period distributions are regularly varying of index - ν iff the heaviest of the tails of the activity period distributions are regularly varying of index - ν We provide explicit equivalents of the former in terms of the latter, which show that the contribution of the sources with lighter associated tails is equivalent to a simple reduction of the outflow rate. These results have implications for the performance analysis of networks of fluid queues.

REFERENCES

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CITED BY 10

	Yong Liu , Weibo Gong, Perturbation Analysis for Stochastic Fluid Queueing Systems, Discrete Event Dynamic Systems, v.12 n.4, p.391-416, October 2002
	Weixin Shang , Liming Liu , Quan-Lin Li, Tail asymptotics for the queue length in an M/G/1 retrial queue, Queueing Systems: Theory and Applications, v.52 n.3, p.193-198, March 2006
	Miranda Van Uitert , Sem Borst, A Reduced-Load Equivalence for Generalised Processor Sharing Networks with Long-Tailed Input Flows, Queueing Systems: Theory and Applications, v.41 n.1-2, p.123-163, June 2002
	O. J. Boxma , D. Perry , F. A. Van Der DuynSchouten, FLUID QUEUES AND MOUNTAIN PROCESSES, Probability in the Engineering and Informational Sciences, v.13 n.4, p.407-427, October 1999
	Krzysztof Debicki , Michel Mandjes , Miranda Van Uitert, A Tandem Queue With LÉVY Input: A New Representation Of The Downstream Queue Length, Probability in the Engineering and Informational Sciences, v.21 n.1, p.83-107, January 2007
	O. J. Boxma , S. G. Foss , J. -M. Lasgouttes , R. Queija, Waiting Time Asymptotics in the Single Server Queue with Service in Random Order, Queueing Systems: Theory and Applications, v.46 n.1-2, p.35-73, January-February 2004
	Sem Borst , Onno Boxma , Miranda Van Uitert, The Asymptotic Workload Behavior of Two Coupled Queues, Queueing Systems: Theory and Applications, v.43 n.1-2, p.81-102, January-February 2003
	A. P. Zwart , O. J. Boxma, Sojourn time asymptotics in the M $/$ G $/$ 1 processor sharing queue, Queueing Systems: Theory and Applications, v.35 n.1-4, p.141-166, 2000
	Sem Borst , Rudesindo Núñez-Queija , Bert Zwart, Sojourn time asymptotics in processor-sharing queues, Queueing Systems: Theory and Applications, v.53 n.1-2, p.31-51, June 2006
	S. C. Borst , O. J. Boxma , R. Núñez-Queija , A. P. Zwart, The impact of the service discipline on delay asymptotics, Performance Evaluation, v.54 n.2, p.175-206, 1 October 2003