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114
Fitting Mixtures Of Exponentials To LongTail Distributions To Analyze Network Performance Models
, 1997
"... Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and interva ..."
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Cited by 144 (13 self)
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Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have longtail distributions, being well described by distributions such as the Pareto and Weibull. It is known that longtail distributions can have a dramatic effect upon performance, e.g., longtail servicetime distributions cause longtail waitingtime distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component longtail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a longtail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...
Inverting Sampled Traffic
 In Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement
, 2003
"... Routers have the ability to output statistics about packets and flows of packets that traverse them. Since however the generation of detailed tra#c statistics does not scale well with link speed, increasingly routers and measurement boxes implement sampling strategies at the packet level. In this pa ..."
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Cited by 89 (0 self)
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Routers have the ability to output statistics about packets and flows of packets that traverse them. Since however the generation of detailed tra#c statistics does not scale well with link speed, increasingly routers and measurement boxes implement sampling strategies at the packet level. In this paper we study both theoretically and practically what information about the original tra#c can be inferred when sampling, or `thinning', is performed at the packet level. While basic packet level characteristics such as first order statistics can be fairly directly recovered, other aspects require more attention. We focus mainly on the spectral density, a second order statistic, and the distribution of the number of packets per flow, showing how both can be exactly recovered, in theory. We then show in detail why in practice this cannot be done using the traditional packet based sampling, even for high sampling rate. We introduce an alternative flow based thinning, where practical inversion is possible even at arbitrarily low sampling rate. We also investigate the theory and practice of fitting the parameters of a Poisson cluster process, modelling the full packet tra#c, from sampled data.
Squeezing The Most Out Of ATM
, 1996
"... Even though ATM seems to be clearly the wave of the future, one performance analysis indicates that the combination of stringent performance requirements (e.g., 10  9 cell blocking probabilities), moderatesize buffers and highly bursty traffic will require that the utilization of the network be ..."
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Cited by 72 (10 self)
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Even though ATM seems to be clearly the wave of the future, one performance analysis indicates that the combination of stringent performance requirements (e.g., 10  9 cell blocking probabilities), moderatesize buffers and highly bursty traffic will require that the utilization of the network be quite low. That performance analysis is based on asymptotic decay rates of steadystate distributions used to develop a concept of effective bandwidths for connection admission control. However, we have developed an exact numerical algorithm that shows that the effectivebandwidth approximation can overestimate the target small blocking probabilities by several orders of magnitude when there are many sources that are more bursty than Poisson. The bad news is that the appealing simple connectionadmissioncontrol algorithm using effective bandwidths based solely on tailprobability asymptotic decay rates may actually not be as effective as many have hoped. The good news is that the statistical multiplexing gain on ATM networks may actually be higher than some have feared. For one example, thought to be realistic, our analysis indicates that the network actually can support twice as many sources as predicted by the effectivebandwidth approximation. That discrepancy occurs because for a large number of bursty sources the asymptotic constant in the tail probability exponential asymptote is extremely small. That in turn can be explained by the observation that the asymptotic constant decays exponentially in the number of sources when the sources are scaled to keep the total arrival rate fixed. We also show that the effectivebandwidth approximation is not always conservative. Specifically, for sources less bursty than Poisson, the asymptotic constant grows exponentially in the numbe...
Waitingtime tail probabilities in queues with longtail servicetime distributions
 QUEUEING SYSTEMS
, 1994
"... We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to ..."
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Cited by 55 (21 self)
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We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to have a finite moment generating function. We have developed algorithms for computing the waitingtime distribution by Laplace transform inversion when the Laplace transforms of the interarrivaltime and servicetime distributions are known. One algorithm, exploiting Pollaczek’s classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of longtail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Paretolike tails, i.e., are of order x − r for r> 1. We use this family of longtail distributions to investigate the quality of approximations based on asymptotics for P(W> x) as x → ∞. We show that the asymptotic approximations with these longtail servicetime distributions can be remarkably inaccurate for typical x values of interest. We also derive multiterm asymptotic expansions for the waitingtime tail probabilities in the M/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typical x values of interest. Thus, we evidently must rely on numerical algorithms for determining the waitingtime tail probabilities in this case. When working with servicetime data, we suggest using empirical Laplace transforms.
Numerical inversion of probability generating functions
 Oper. Res. Letters
, 1992
"... Random quantities of interest in operations research models can often be determined conveniently in the form of transforms. Hence, numerical transform inversion can be an effective way to obtain desired numerical values of cumulative distribution functions, probability density functions and probabil ..."
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Cited by 40 (17 self)
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Random quantities of interest in operations research models can often be determined conveniently in the form of transforms. Hence, numerical transform inversion can be an effective way to obtain desired numerical values of cumulative distribution functions, probability density functions and probability mass functions. However, numerical transform inversion has not been widely used. This lack of use seems to be due, at least in part, to good simple numerical inversion algorithms not being well known. To help remedy this situation, in this paper we present a version of the Fourierseries method for numerically inverting probability generating functions. We obtain a simple algorithm with a convenient error bound from the discrete Poisson summation formula. The same general approach applies to other transforms. Key Words: numerical inversion of transforms, computational probability, generating functions, Fourierseries method, Poisson summation formula, discrete Fourier transform.
Asymptotics for M/G/1 lowpriority waitingtime tail probabilities
, 1997
"... We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptiveresume disciplines. We show that the lowpriority steadystate waitingtime can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waitingtime distribution. ..."
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Cited by 39 (6 self)
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We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptiveresume disciplines. We show that the lowpriority steadystate waitingtime can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waitingtime distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have nonexponential asymptotics. This phenomenon even occurs when both servicetime distributions are exponential. When nonexponential asymptotics holds, the asymptotic form tends to be determined by the nonexponential asymptotics for the highpriority busyperiod distribution. We obtain asymptotic expansions for the lowpriority waitingtime distribution by obtaining an asymptotic expansion for the busyperiod transform from Kendall’s functional equation. We identify the boundary between the exponential and nonexponential asymptotic regions. For the special cases of an exponential highpriority servicetime distribution and of common general servicetime distributions, we obtain convenient explicit forms for the lowpriority waitingtime transform. We also establish asymptotic results for cases with longtail servicetime distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the nonexponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waitingtime transform.
Exponential approximations for tail probabilities in queues, I: waiting times
 Oper. Res
, 1995
"... In this paper, we focus on simple exponential approximations for steadystate tail probabilities in G/GI/1 queues based on largetime asymptotics. We relate the largetime asymptotics for the steadystate waiting time, sojourn time and workload. We evaluate the exponential approximations based on th ..."
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Cited by 39 (20 self)
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In this paper, we focus on simple exponential approximations for steadystate tail probabilities in G/GI/1 queues based on largetime asymptotics. We relate the largetime asymptotics for the steadystate waiting time, sojourn time and workload. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/GI/1 queues. Numerical examples show that the exponential approximations are remarkably accurate at the 90 th percentile and beyond. Key words: queues; approximations; asymptotics; tail probabilities; sojourn time and workload.
Asymptotics for steadystate tail probabilities in structured Markov queueing models
 Commun. Statist.Stoch. Mod
, 1994
"... In this paper we establish asymptotics for the basic steadystate distributions in a large class of singleserver queues. We consider the waiting time, the workload (virtual waiting time) and the steadystate queue lengths at an arbitrary time, just before an arrival and just after a departure. We s ..."
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Cited by 38 (10 self)
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In this paper we establish asymptotics for the basic steadystate distributions in a large class of singleserver queues. We consider the waiting time, the workload (virtual waiting time) and the steadystate queue lengths at an arbitrary time, just before an arrival and just after a departure. We start by establishing asymptotics for steadystate distributions of Markov chains of M/GI/1 type. Then we treat steadystate distributions in the BMAP/GI/1 queue, which has a batch Markovian arrival process (BMAP). The BMAP is equivalent to the versatile Markovian point process or Neuts (N) process; it generalizes the Markovian arrival process (MAP) by allowing batch arrivals. The MAP includes the Markovmodulated Poisson process (MMPP), the phasetype renewal process (PH) and independent superpositions of these as special cases. We also establish asymptotics for steadystate distributions in the MAP/MSP/1 queue, which has a Markovian service process (MSP). The MSP is a MAP independent of the arrival process generating service completions during the time the server is busy. In great generality (but not always), the basic steadystate distributions have asymptotically exponential tails in all these models. When they do, the asymptotic parameters of the different distributions are closely related. 1.
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
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On the Laguerre method for numerically inverting Laplace transforms
 INFORMS Journal on Computing
, 1996
"... The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 TricomiWidder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coeff ..."
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Cited by 34 (7 self)
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The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 TricomiWidder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (1) using our previously developed variant of the Fourierseries method to calculate the coefficients of the Laguerre generating function, (2) developing systematic methods for scaling, and (3) using Wynn’s ɛalgorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux’s theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples. Subject classifications: Mathematics, functions: Laplace transforms. Probability, distributions: calculation by transform inversion. Queues, algorithms: Laplace transform inversion.