We study the asymptotics for the maximum on a random time interval of a random walk with a long-tailed distribution of its increments and negative drift. We extend to a general stopping time a result by Asmussen (1998), simplify its proof, and give some converses.

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We propose a new methodology for modeling and analyzing heavy-tailed distributions, such as the Pareto distribution, in communication networks. The basis of our approach is a fitting algorithm which approximates a heavy-tailed distribution by a hyperexponential distribution. This algorithm possesses several key properties. First, the approximation can be achieved within any desired degree of accuracy. Second, the fitted hyperexponential distribution depends only on a few parameters. Third, only a small number of exponentials are required in order to obtain an accurate approximation over many time scales. Once equipped with a fitted hyperexponential distribution, we have an integrated framework for analyzing queueing systems with heavy-tailed distributions. We consider the GI=G=1 queue with Pareto distributed service time and show how our approach allows to derive both quantitative numerical results and asymptotic closed-form results. This derivation shows that classical teletraffic me...

The superposition process of independent counting renewal processes associated with a heavy-tailed interarrival time distribution is shown to converge weakly after rescaling in time and space to fractional Brownian motion, as the number of renewal processes tends to infinity. Corresponding results for continuous arrival fluid processes are discussed. Keywords: Renewal process, heavy-tails, fractional Brownian motion, arrival process modeling 1 Introduction It is well known that various classes of arrival processes in telecommunications traffic modeling based on heavy-tailed interarrival time distributions exhibit long-range dependence. This includes arrival rate processes of Anick-MitraSondhi (AMS) type where the rate process is an on/off-process with heavytailed on-period distribution and/or off-period distribution, as well as generalized Kosten type models with rate process the M/G/1 queuing model with heavy-tailed service time distribution. It is shown in Taqqu, Willinger and Sher...

We consider a fluid queue in discrete time with random service rate. Such a queue has been used in several recent studies on wireless networks where the packets can be arbitrarily fragmented. We provide conditions on finiteness of moments of stationary delay, its Laplace-Stieltjes transform, various approximations under heavy traffic and asymptotics of its tail distribution. Results are extended to the case where the wireless link can transmit in only a few slots during a frame.

the single server queue