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The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remar ..."
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Cited by 197 (52 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
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 100 (3 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.
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 59 (19 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.
Computing QueueLength Distributions for PowerLaw Queues
 in Proc. INFOCOM
, 1998
"... : The interest sparked by observations of longrange dependent traffic in real networks has lead to a revival of interest in nonstandard queueing systems. One such queueing system is the M/G/1 queue where the servicetime distribution has infinite variance. The known results for such systems are as ..."
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Cited by 9 (2 self)
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: The interest sparked by observations of longrange dependent traffic in real networks has lead to a revival of interest in nonstandard queueing systems. One such queueing system is the M/G/1 queue where the servicetime distribution has infinite variance. The known results for such systems are asymptotic in nature, typically providing the asymptotic form for the tail of the workload distribution, simulation being required to learn about the rest of the distribution. Simulation however performs very poorly for such systems due to the large impact of rare events. In this paper we provide a method for numerically evaluating the entire distribution for the number of customers in the M/G/1 queue with powerlaw tail servicetime. The method is computationally efficient and shown to be accurate through careful simulations. It can be directly extended to other queueing systems and more generally to many problems where the inversion of probability generating functions complicatedby powerlaws...
Approximating heavy tailed behaviour with phase type distributions
"... In this paper two main problems are investigated. The first one is the effect of the goal function of the applied fitting method on the goodness of Phase type fitting. We discuss a numerical method based on a simple numerical optimization procedure that allows us to fit any nonnegative distribution ..."
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Cited by 8 (1 self)
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In this paper two main problems are investigated. The first one is the effect of the goal function of the applied fitting method on the goodness of Phase type fitting. We discuss a numerical method based on a simple numerical optimization procedure that allows us to fit any nonnegative distribution with a Phase type (PH) distribution according to any arbitrary distance measure. By comparing the fitting results obtained by minimizing different distance measures, conclusions are drawn regarding the role of the optimization criteria. The second considered problem is the tail behaviour of Phase type distributions obtained via different fitting methods. To limit the numerical complexity of fitting methods (basically the evaluation of distance measures) the computations (numerical integration) are truncated at some point. Hence the information on the tail behaviour of the distribution is not considered beyond this point. To approximate distributions with heavy tail we propose a complex method that uses different techniques to fit the main part and the tail of the distribution. The proposed method combines the advantages of fitting techniques and this way it overcomes some of their limitations. The goodness of the discussed fitting methods are compared in queuing behaviour as well. The behaviour of the M/G/1 queue is compared with the one of the approximating M/PH/1 queue.
QueueLength Distributions for MultiPriority Queueing Systems
 in IEEE INFOCOM
, 1999
"... The bottleneck in many telecommunication systems has often been modeled by an M/G/1 queueing system with priorities. While the probability generating function for the occupancy distribution of each traffic classes can be readily obtained, the occupancy distributions have been obtainable only rarely. ..."
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Cited by 2 (0 self)
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The bottleneck in many telecommunication systems has often been modeled by an M/G/1 queueing system with priorities. While the probability generating function for the occupancy distribution of each traffic classes can be readily obtained, the occupancy distributions have been obtainable only rarely. However, the occupancy distribution is of great importance, particularly in those cases where the moments are not all finite. We present a method of obtaining the occupancy distribution from the PGF and demonstrate its validity by obtaining the occupancy distributions for a number of cases, including those with regularly varying service time distributions. I. INTRODUCTION The bottleneck in telecommunication systems has often been modeled by an M/G/1 queueing system having nonpreemptive priority service, where the probability generating function (PGF) for the occupancies of the various traffic classes can be obtained using either classical approaches [17] or FuhrmannCooper decomposition [1...
Sojourntime Analysis of Nodal Congestion in Broadband Networks and its Impact on QoS Specifications
 University of Texas at Austin
, 1996
"... ..."
generating functions
, 1991
"... 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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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. numerical inversion of transforms; computational probability; generating functions; Fourierseries method; Poisson summation fi)rmula; discrete Fourier transform 1. Introduction and
Approximating nonMarkovian Behaviour by Markovian Models
"... vi The material presented in the dissertation is about approximating nonMarkovian behavior by Markovian models. Introduction to various aspects of the field is provided in Part I consisting of three chapters. Chapter 1 provides the background for discretetime Phasetype (DPH) distributions. A sho ..."
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vi The material presented in the dissertation is about approximating nonMarkovian behavior by Markovian models. Introduction to various aspects of the field is provided in Part I consisting of three chapters. Chapter 1 provides the background for discretetime Phasetype (DPH) distributions. A short introduction to Markovian modeling for traffic engineering is given in Chapter 2. Characteristics of random quantities and random processes of telecommunication networks with related statistical tests are introduced in