Cumulative broadband network traffic is often thought to be well modelled by fractional Brownian motion. However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection length distribution tails, then stable L'evy motion is a sensible approximation to cumulative traffic over a time period. If connection rates are large relative to heavy tailed connection length distribution tails, then FBM is the appropriate approximation. The results are framed as limit theorems for a sequence of cumulative input processes whose connection rates are varying in such a way as to remove or induce long range dependence.

. A fluid queue with ON periods arriving according to a Poisson process and having a long--tailed distribution has long range dependence. As a result, its performance deteriorates. The extent of this performance deterioration depends on a quantity determined by the average values of the system parameters. In the case when the the performance deterioration is the most extreme, we quantify it by studying the time until the amount of work in the system causes an overflow of a large buffer. This turns out to be strongly related to the tail behavior of the increase in the buffer content during a busy period of the M=G=1 queue feeding the buffer. A large deviation approach provides a powerful method of studying such tail behavior. 1. Introduction We consider the following fluid queuing model. Sessions arrive to a network server (multiplexer) according to a Poisson process with rate ? 0. Each session remains active for a random length of time with distribution F and a finite mean ¯. We assum...

We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog W A1+A2 ;c in a buffer fed by a combined fluid process A 1 + A 2 and drained at a constant rate c. The fluid process A 1 is an (independent) on-off source with average and peak rates ae 1 and r 1 , respectively, and with distribution G for the activity periods. The fluid process A 2 of average rate ae 2 is arbitrary but independent of A 1 . These bounds are used to identify subexponential distributions G and fairly general fluid processes A 2 such that the asymptotic equivalence P \Theta W A1+A2 ;c ? x P \Theta W A1 ;c\Gammaae 2 ? x (x ! 1) holds under the stability condition ae 1 + ae 2 ! c and under the non-triviality condition c \Gamma ae 2 ! r 1 . The stationary backlog W A1 ;c\Gammaae 2 in these asymptotics results from feeding source A 1 into a buffer drained at reduced rate c \Gamma ae 2 . This reduced load asymptotic equivalence extends to a larger class o...

. We consider a fluid queue with sessions arriving according to a Poisson process. A long--tailed distribution of session lengths induces long range dependence in the system and causes its performance to deteriorate. The deterioration is due to occurrence of load regimes far from average ones. Nonetheless, the extent of this performance deterioration is shown to depend crucially on the average values of the system parameters. 1. Introduction We consider the following fluid queuing model. Sessions (ON periods) are initiated at a network server or multiplexer according to a Poisson process with rate ? 0. Each session is active for a random length of time with distribution F and a finite mean ; during this time it generates network traffic at unit rate. We assume that the lengths of different sessions are independent, and they are also independent of the Poisson arrival process. The service rate is r ? 0 units of traffic per unit time. If X(t) denotes the amount of work (measured in unit...

Consider a single server queue with i.i.d. arrival and service processes, fA; A n ; n 1g and fC; C n ; n 1g, respectively, and finite buffer B. The queue content process fQ B n ; n 0g is recursively defined as Q B n+1 = min((Q B n + A n+1 \Gamma C n+1 ) + ; B), q + = max(0; q). When E(A \Gamma C) ! 0, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue E(Q B n + A n+1 \Gamma C n+1 \Gamma B) + has the following explicit asymptotic characterization E(Q B n +A n+1 \Gamma C n+1 \Gamma B) + ¸ E(A \Gamma B) + as B !1; independently of the server process C n . For a fluid queue with capacity c, M/G/1 arrival process a t , characterized by intermediately regularly varying On periods ø on , that arrive with Poisson rate , the average loss rate B loss satisfies B loss ¸ E(ø on j \Gamma B) + as B !1; where j = r + ae \Gamma c, ae = Ea t ! c, and r; r c, is the rate at which the fluid is arriving during an On per...

Extensive studies indicate that traffic in high-speed communication networks exhibits long-range dependence (LRD) and impulsiveness, thus posing new challenges in network engineering. While many models have recently appeared for capturing the traffic LRD, far less models exist that account for impulsiveness as well as LRD. One of the few existing constructive models for network traffic is the celebrated On/Off model, or Alternating Fractal Renewal Process (AFRP). However, while the AFRP results in aggregated traffic with LRD, it fails to capture impulsiveness, yielding traffic with Gaussian marginal distribution. A new constructive model, namely the Extended AFRP (EAFRP), is proposed here, which overcomes the limitations of the AFRP model. We show that, for both single-user and aggregated traffic, it results in impulsiveness and long-range dependence, the LRD being defined here in a generalized sense. We provide queueing analysis of the proposed model, which clearly demonstrates the implications of the impulsiveness in traffic engineering, and validate all theoretical findings based on real traffic data.

This paper studies rare events simulation for the heavy--tailed case, where some of the underlying distributions fail to have the exponential moments required for the standard algorithms for the light--tailed case. Several counterexamples are given to indicate that in the heavy--tailed case, there are severe problems with the approach of developing limit results for the conditional distribution given the rare event and use this as basis for importance sampling. On the positive side, two algorithms having a relative error which is almost bounded are presented, one based upon order statistics and the other upon a different importance sampling idea. Keywords and phrases conditional Monte Carlo, importance sampling, large deviations, logarithmic efficiency, M/G/1 queue, order statistics, random walk, regular variation, subexponential distribution 1 Introduction Estimation of small probabilities by simulation is one of the key issues of todays simulation literature. The reason is two--fol...

This article reviews various models within the queueing framework which have been suggested for teletraffic data. Such models aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on- and off-periods and long-range dependence in teletraffic transmission. Subexponential distributions constitute a large class of heavy-tailed distributions, and we investigate their (sometimes disastrous) influence within teletraffic models. We demonstrate some of the above effects in an explorative data analysis of Munich Universities' intranet data.

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.

We study the asymptotic probability that a random walk with heavy-tailed increments crosses a high boundary on a random time interval. We use new techniques to extend results of Asmussen [Ann. Appl. Probab. 8 (1998) 354–374] to completely general stopping times, uniformity of convergence over all stopping times and a wide class of nonlinear boundaries. We also give some examples and counterexamples.