On the asymptotic behaviour of the distributions of the busy period and service time in <i>M/G/</i>1

Meyer, Anne-Marie De; Teugels, Jozef L.

doi:10.2307/3212973

Cited by 79 publications

(40 citation statements)

References 10 publications

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“…Indeed, it is the distribution of an uninterrupted period in which at least one source is active, active sources "emerging" according to a Poisson process with rate A. As a by-product of our investigations we have hence found the interesting result that the busy period of an M/G/oo queue has a regularly varying tail of index -v iff the tail of the service time distribution is regularly varying of index -v. For the M/G/1 queue a similar statement is known [12].…”

Section: The Cumulative Activity Period Distributionmentioning

confidence: 62%

Fluid queues and regular variation

Boxma¹

1996

Performance Evaluation

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This paper considers a fluid queueing system, fed by N independent sources that alternate between silence and activity periods. We assume that the distribution of the activity periods of one or more sources is a regularly varying function of index l;. We show that its fat tail gives rise to an even fatter tail of the buffer content distribution, viz., one that is regularly varying of index l; + I. In the special case that l.; E ( -2, -1 ), which implies long-range dependence of the input process, the buffer content does not even have a finite first moment.As a queueing-theoretic by-product of the analysis of the case of N identical sources, with N ~ oo, we show that the busy period of an M/G/oo queue is regularly varying of index l; iff the service time distribution is regularly varying of index l;.

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Section: The Cumulative Activity Period Distributionmentioning

confidence: 62%

Fluid queues and regular variation

Boxma¹

1996

Performance Evaluation

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“…The busy-period obviously has the same distribution for LCFS-PR as for FCFS. The tail behavior of the busy-period distribution in the M/G/1 queue has been studied by De Meyer and Teugels [43] for the case of a regularly varying service requirement distribution. This yields the next theorem (S LPR denoting the steadystate sojourn-time).…”

Section: Theorem 22mentioning

confidence: 99%

“…As observed in Section 2, the sojourn-time in the M/G/1 LCFS-PR queue has the same distribution as the busy-period in the M/G/1 queue. De Meyer and Teugels [43] have studied the tail of the latter distribution in the case of a regularly varying service requirement distribution. Their starting-point is the fact that the LST µ{s} of the steady-state busy-period length P is the unique solution of the equation…”

Section: The Single-class Casementioning

confidence: 99%

The impact of the service discipline on delay asymptotics

Borst

Boxma

Núñez-Queija

et al. 2003

Performance Evaluation

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This paper surveys the M/G/1 queue with regularly varying service requirement distribution. It studies the effect of the service discipline on the tail behavior of the waiting-time and/or sojourn-time distribution, demonstrating that different disciplines lead to quite different tail behavior. The orientation of the paper is methodological: We outline four different methods for determining tail behavior, illustrating them for service disciplines like FCFS, Processor Sharing and LCFS.

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“…For example, results exist for the GI/GI/1 queue under both heavy-and light-tailed job sizes for first come first served (FCFS) (Asmussen 2003, Borovkov 1976, Cohen 1973, Pakes 1975, preemptive last come first served (LCFS) (Meyer andTeugels 1980, Zwart 2001), processor sharing (PS) (Borst et al 2006), shortest remaining processing time (SRPT) Zwart 2006, Nuyens et al 2008), and other disciplines. Complete surveys can be found in Borst et al (2003) and Boxma and Zwart (2007).…”

Section: Introductionmentioning

confidence: 99%

Is Tail-Optimal Scheduling Possible?

Wierman

Zwart

2012

Operations Research

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This paper focuses on the competitive analysis of scheduling disciplines in a large deviations setting. Although there are policies that are known to optimize the sojourn time tail under a large class of heavy-tailed job sizes (e.g., processor sharing and shortest remaining processing time) and there are policies known to optimize the sojourn time tail in the case of light-tailed job sizes (e.g., first come first served), no policies are known that can optimize the sojourn time tail across both light-and heavy-tailed job size distributions. We prove that no such work-conserving, nonanticipatory, nonlearning policy exists, and thus that a policy must learn (or know) the job size distribution in order to optimize the sojourn time tail.

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On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1

Abstract: For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

Cited by 79 publications

References 10 publications

Fluid queues and regular variation

Fluid queues and regular variation

The impact of the service discipline on delay asymptotics

Is Tail-Optimal Scheduling Possible?

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