Consider an aggregate arrival process AN obtained by multiplexing N On-Off processes with exponential Off periods of rate λ and subexponential On periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/ ∞ arrival process A ∞ t and capacity c. When On periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable QP t observed at the beginning of the arrival process activity periods P[Q P t +ρ−c> x] ∼ Λr P[τ c−ρ x/(r+ρ−c) on> u]du x → ∞, where ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. The asymptotic (time average) queuedistributionlower boundis obtained undermoregeneral assumptions on On periods than regular variation. In addition, we analyze a queueing system in which one On-Off process, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate Eet. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value Eet.

The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer Predrag R. Jelenkovi'c The main theme of this dissertation is the evaluation of the capacity of broadband multimedia network multiplexers. This problem calls for the modeling of network traffic streams and the analysis of a network multiplexer that is loaded with the corresponding models. For modeling we focus on MPEG video traffic streams that are expected to be predominant in the traffic mixture of future multimedia networks. We experimentally demonstrate that real-time MPEG video traffic exhibits multiple time scale characteristics, as well as subexponential first and second order statistics. Then we construct a model of MPEG video that captures both of these characteristics and accurately predicts queueing behavior for a broad range of buffer and capacity sizes. Depending on whether a network multiplexer (loaded with MPEG) is strictly or weakly stable the dominant effect o...

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 is a survey paper on fluid queues, with a strong emphasis on recent attempts to represent phenomena like long-range dependence. The central model of the paper is a fluid queueing system fed by N independent sources that alternate between silence and activity periods. The distribution of the activity periods of at least one source is assumed to be long-tailed, which may give rise to long-range dependence. We consider the effect of this tail behaviour on the steady-state distributions of the buffer content at embedded points in time and at arbitrary time, and on the busy period distribution. Both exact results and bounds are discussed.

. 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 survey the properties and uses of the class of subexponential probability distributions, paying particular attention to their use in modelling heavy-tailed data such as occurs in insurance and queueing applications. We give a detailed summary of the core theory and discuss subexponentiality in various contexts including extremes, random walks and L'evy processes with negative drift, and sums of random variables, the latter extended to cover random sums, weighted sums and moving averages. 1. Definition and first properties Subexponential distributions are a special class of heavy--tailed distributions. The name arises from one of their properties, that their tails decrease more slowly than any exponential tail; see (1.4). This implies that large values can occur in a sample with non--negligible probability, and makes the subexponential distributions candidates for modelling situations where some extremely large values occur in a sample compared to the mean size of the data. Such a p...

We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t \Gamma with 1 ! 2, so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a ! 1, then W, multiplied by an appropriate `coefficient of contraction' that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a ! 1, then W, multiplied by another appropriate `coefficient of contraction' that is a function of a, converges in distribution to the negative exponential distribution.

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Consider two M=G=1 queues that are coupled in the following way. Whenever both queues are non-empty, each server serves its own queue at unit speed. However, if server 2 has no work in its own queue, then it assists server 1, resulting in an increased service speed r 1 1 in the first queue. This kind of coupling is related to generalized processor sharing. We assume that the service request distributions at both queues are regularly varying at infinity of index 1 and 2 , viz., they are heavy-tailed. Under this assumption, we present a detailed analysis of the tail behaviour of the workload distribution at each queue. If the guaranteed unit speed of server 1 is already sufficient to handle its offered traffic, then the workload distribution at the first queue is shown to be regularly varying at infinity of index 1 1 . But if it is not sufficient, then the workload distribution at the first queue is shown to be regularly varying at infinity of index 1 min(1 ; 2 ). In partic...

We analyze the asymptotic behavior of long-tailed traffic sources under the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, have emerged as an important mechanism for achieving differentiated quality-of-service in integrated-services networks. Under certain conditions, we prove that in an asymptotic sense an individual source with longtailed traffic characteristics is effectively served at a constant rate, which may be interpreted as the maximum feasible average rate for that source to be stable. Thus, asymptotically, the source is only affected by the traffic characteristics of the other sources through their average rate. In particular, the source is essentially immune from excessive activity of sources with `heavier'-tailed traffic characteristics. This suggests that GPS-based scheduling algorithms provide an effective mechanism for extracting high multiplexing gains, while protecting individual connections.