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.

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.

. 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...

We investigate the distribution of the waiting time V in a stable M/G/1 processor sharing queue with trac intensity < 1. When the distribution of a customer service request B belongs to a large class of subexponential distributions with tails heavier than e , it is shown that P[V > x] = P[B > (1 )x](1 + o(1)) as x !1: Furthermore, we demonstrate that the preceding relationship does not hold if the service distribution has a lighter tail than e .

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...

Recently, the class of SMART scheduling policies has been introduced in order to formalize the common heuristic of “biasing toward small jobs. ” We study the tail of the sojourn-time (response-time) distribution under both SMART policies and the Foreground-Background policy (FB) in the GI/GI/1 queue. We prove that these policies behave very well under heavy-tailed service times. Specifically, we show that the sojourn-time tail under all SMART policies and FB is similar to that of the service-time tail, up to a constant, which makes the SMART class superior to First-Come-First-Served (FCFS). In contrast, for light-tailed service times, we prove that the sojourn-time tail under FB and SMART is larger than that under FCFS. However, we show that the sojourn-time tail for a job of size y under FB and all SMART policies still outperforms FCFS as long as y is not too large.

. We consider a fluid queue with ON periods arriving according to a Poisson process and having a long--tailed distribution. This queue has long range dependence, and we compute the asymptotic behavior of the steady state distribution of the buffer content. The tail of this distribution is much heavier than the tail of the buffer content distribution of a queue which does not possess long range dependence and which has light tailed ON periods and the same traffic intensity. 1. Introduction and preliminaries We consider a model of a network server (multiplexer) defined as follows. Sessions arrive to the server according to a Poisson process with rate ? 0. Each session lasts a random length of time with distribution F that has a finite mean . The lengths of different sessions are independent of each other and of the Poisson arrival process. A session generates work or traffic or fluid at unit rate, commonly measured in some units of network traffic, e.g. packets; the work that cannot be ...

Consider a fluid queue with a finite buffer B and capacity c fed by a superposition of N independent On-Off processes. An On-Off process consists of a sequence of alternating independent activity and silence periods. Successive activity, as well as silence, periods are identically distributed. The process is active with probability p on and during its activity period produces fluid with constant rate r. For this queueing system, under the assumption that the residual activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary loss probability and loss rate. In the case of homogeneous sources with residual activity periods equal in distribution to on r , the queue overflow probability is asymptotically, as B !1, equal to P[Q B = B] = ` N k 0 ' p k 0 on P on r ? B k 0 (r \Gamma ae) +N ae \Gamma c k 0 (1 + o(1)); where ae = rp on , N ae ! c and k 0 is the smallest integer greater than (c...

Abstract. This paper considers 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- or sojourn time distribution, demonstrating that different disciplines may lead to quite different tail behavior. The orientation of the paper is methodological: We outline three different methods of determining tail behavior, illustrating them for service disciplines like FCFS, Processor Sharing and LCFS. This paper is dedicated to the memory of Vincent Dumas, a dear friend and gifted young mathematician. 1

Long Range Dependent time series are endemic in the statistical analysis of Internet traffic. The Hurst Parameter provides good summary of important self-similar scaling properties. We compare a number of different Hurst parameter estimation methods and some important variations. This is done in the context of a wide range of simulated, laboratory generated and real data sets. Important differences between the methods are highlighted. Deep insights are revealed on how well the laboratory data mimic the real data. Non-stationarities, that are local in time, are seen to be central issues, and lead to both conceptual and practical recommendations. 1