Superimposed renewal processes and storage with gradual input

Cohen, J. W.

doi:10.1016/0304-4149(74)90011-8

Cited by 58 publications

(49 citation statements)

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“…1 and [9]) that Determination of BO is a far from trivial problem if N > 1, because the number of active sources may fluctuate during a cumulative activity period. Cohen tackles this problem in [7] for N identical sources with r 1 = · · · = rN = l, and in [9] for N different sources. He observes that the activity and silence periods of the N sources give rise to N alternating renewal processes, and uses this to obtain the LST of the joint distribution of B and C (B and C denote generic random variables with joint distribution the joint limiting distribution of the random variables Bn and Cn).…”

Section: The Fluid Queuementioning

confidence: 99%

“…Activity periods hence begin according to a Poisson process of rate A. Cohen ([7], Formula (2.2.7)) derives the following expression for the joint LST of B and C (he takes r = 1): for with (cf. [7], Formula (2.2.9)): fort ~ 0, Rew~ 0, and some y > 0,…”

Section: N=oomentioning

confidence: 99%

“…Taking w = 0 in (7 Now exploit the fact that -after some scaling -an LST of a proper probability distribution appears in the RHS of (7.3), so that we can apply Lemma 4.1. If Ac(x) = x-vl(x) for x -+ oo, with v E (1, 2), so that (see also (4.5)) fx':,r Ac(x)dx = v~1 t 1 -vl(t) fort-+ oo, then…”

Section: The Cumulative Activity Period Distributionmentioning

confidence: 99%

“…Formula (2.3.2) of [7]). Yet another application of Lemma4.1 leads to the conclusion that, fort-+ oo and v E (1, 2), (7.6) One can easily check that the reverse statement also holds.…”

Section: The Cumulative Activity Period Distributionmentioning

confidence: 99%

“…The fluid queueing system under consideration is described in Section 2. Section 3 displays for this model some key results of [7,9) that form the starting-point of our approach. Section 4 summarizes the main ingredients of the theory of regular variation.…”

Section: Introductionmentioning

confidence: 99%

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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 Fluid Queuementioning

confidence: 99%

Section: N=oomentioning

confidence: 99%

Section: The Cumulative Activity Period Distributionmentioning

confidence: 99%

“…Formula (2.3.2) of [7]). Yet another application of Lemma4.1 leads to the conclusion that, fort-+ oo and v E (1, 2), (7.6) One can easily check that the reverse statement also holds.…”

Section: The Cumulative Activity Period Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fluid queues and regular variation

Boxma¹

1996

Performance Evaluation

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Asymptotic Analysis of Queues with Subexponential Arrival Processes

Jelenković

2000

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One of the major challenges in designing modern communication networks is providing quality of service to the individual users. An important part of this design process is understanding statistical characteristics of network traf®c streams and their impact on network performance. Unlike the conventional voice traf®c, modern data traf®c exhibits an increased level of``burstiness'' that spans over multiple time scales. It was observed that sample paths of these data sequences show evidence of self-similarity. Their autocorrelation structure is characterized by long-range dependency and the empirical distributions are easily matched with subexponential and long-tailed distributions. Early discovery of the self-similar nature of Ethernet traf®c was reported in Leland et al. [42] (see also Leland et al. [43]). More recently, Crovella [22] attributed the long-range dependency of Ethernet traf®c to the longtailed ®le sizes that are transferred over the network. Long-range dependency of the variable bit rate video traf®c was demonstrated by Beran et al. [9]. Long-tailed characteristics of the scene length distribution of MPEG video streams were explored in Heyman and Lakshman [30] and Jelenkovic Â et al. [37]. Practical importance, novelty, and the intriguing nature of these phenomena have attracted a great number of scientists to develop new traf®c models and to understand the impact of these models on network performance. In this development there

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