Slow, fast and arbitrary growth conditions for renewal-reward
          processes when both the renewals and the rewards are heavy-tailed

Pipiras, Vladas; Taqqu, Murad S.; Levy, Joshua B.

doi:10.3150/bj/1077544606

Cited by 49 publications

(43 citation statements)

References 16 publications

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“…In the literature it is almost invariably a stable Lévy motion, i.e., a process with independent and stationary infinite-variance increments. Results of this type were obtained in Leland et al [24], Taqqu et al [37], Willinger et al [38], and Konstantopoulos [23] for the ON/OFF model and further extended (also to superpositions of iid copies of the input A) in Mikosch et al [28] and Pipiras et al [32] for the ON/OFF and the infinite-source Poisson models. We will see below that the limit may be much more general than stable Lévy motion.…”

Section: Now G 3r1 T ≤ Cp Z > T/2 Tp K > T = O Tp K > Tmentioning

confidence: 84%

“…This is perhaps the most popular model for teletraffic. Consider a single ON/OFF source such as a workstation as described in Heath et al [15], Leland et al [24], Taqqu et al [37], Pipiras and Taqqu [31], Levy and Taqqu [25], Mikosch et al [28], Stegeman [35], Gaigalas and Kaj [12], and Pipiras et al [32]. During an ON period, the source generates traffic at a constant rate one, for example, one byte per time unit.…”

Section: Proposition 21 the Variance Of The Input A T Is Given Bymentioning

confidence: 99%

“…During an ON period, the source generates traffic at a constant rate one, for example, one byte per time unit. (However, Pipiras and Taqqu [31] and Pipiras et al [32] also allow for random rates (rewards).) During an OFF period, the source remains silent and the input rate is zero.…”

Section: Proposition 21 the Variance Of The Input A T Is Given Bymentioning

confidence: 99%

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Scaling Limits for Cumulative Input Processes

Mikosch

Samorodnitsky

2007

Mathematics of OR

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We study different scaling behavior of very general telecommunications cumulative input processes. The activities of a telecommunication system are described by a marked-point process T n Z n n∈ , where T n is the arrival time of a packet brought to the system or the starting time of the activity of an individual source, and the mark Z n is the amount of work brought to the system at time T n . This model includes the popular ON/OFF process and the infinite-source Poisson model. In addition to the latter models, one can flexibly model dependence of the interarrival times T n − T n−1 , clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process T n and the marks Z n . Similarly to the ON/OFF and infinite-source Poisson model, we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources. The memory in the input process depends on a variety of factors, such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival T n , or the number of activities starting at T n . It turns out that, as in standard results on the scaling behavior of cumulative input processes in telecommunications, fractional Brownian motion or infinite-variance Lévy stable motion can occur in the scaling limit. However, the fractional Brownian motion is a much more robust limit than the stable motion, and many other limits may occur as well.

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Section: Now G 3r1 T ≤ Cp Z > T/2 Tp K > T = O Tp K > Tmentioning

confidence: 84%

Section: Proposition 21 the Variance Of The Input A T Is Given Bymentioning

confidence: 99%

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Scaling Limits for Cumulative Input Processes

Mikosch

Samorodnitsky

2007

Mathematics of OR

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“…In the case 1 < γ < δ < 2 of infinite variance rewards, we have F R (dr) ∼ r −δ−1 dr as r → ∞. Lemma 3 and the scaling representation (27) yield…”

Section: 32mentioning

confidence: 96%

“…Observe that there are several ways to change variables in the integrals. We will use (27) and other variations. We used here the scaling property…”

Section: Scaling Behavior Of the Workload Processmentioning

confidence: 99%

In and Out of Equilibrium 2

Sidoravičius¹,

Vares²

2008

Progress in Probability

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This is an author-produced version of a chapter published in In and Out of Equilibrium 2. It does not include the final publisher proof-corrections or pagination. Abstract. It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise.

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Limiting Fractal Random Processes in Heavy-Tailed Systems

Kaj

Fractals in Engineering

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Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed

Cited by 49 publications

References 16 publications

Scaling Limits for Cumulative Input Processes

Scaling Limits for Cumulative Input Processes

In and Out of Equilibrium 2

Limiting Fractal Random Processes in Heavy-Tailed Systems

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