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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 remarkably easy ..."
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Cited by 149 (51 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...
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
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Transient Behaviour of Queueing Systems with Correlated Traffic
 Journal of Stochastic Models
, 1996
"... In this paper, we present the timedependent solutions of various stochastic processes associated with a finite QuasiBirthDeath queueing system. These include the transient queueing solutions, the transient departure and loss intensity processes and certain transient cummulative measures associate ..."
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Cited by 3 (0 self)
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In this paper, we present the timedependent solutions of various stochastic processes associated with a finite QuasiBirthDeath queueing system. These include the transient queueing solutions, the transient departure and loss intensity processes and certain transient cummulative measures associated with the queueing system. The focus of our study is the effect of the arrival process correlation on the queueing system before it reaches steadystate. With the aid of numerous examples, we investigate the strong relationship between the time scales of variation of the arrival process and those of the transient queueing, loss and departure processes. These timedependent solutions require the computation of the exponential of the stochastic generator matrix G which may be of very large order. This precludes the use of known techniques to solve the matrix exponential such as the eigenvalue decomposition of G. We present a numerical technique based on the computation of the Laplace Transfor...
Performance Analysis of Rate Based Feedback Control for ATM Networks
 IEEE/ACM Trans. Networking
, 1997
"... Closedloop input rate regulation schemes have come to play an important role in the transport of the Available Bit Rate (ABR) traffic service category for ATM. In this paper, we present a numerical approach to the performance study of a delayed feedback system with one congested node and multiple c ..."
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Cited by 2 (0 self)
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Closedloop input rate regulation schemes have come to play an important role in the transport of the Available Bit Rate (ABR) traffic service category for ATM. In this paper, we present a numerical approach to the performance study of a delayed feedback system with one congested node and multiple connections. This approach consists in modeling the feedback system as a finite QuasiBirthDeath (QBD) process. Due to the peculiar block tridiagonol nature of its generator, efficient techniques exist for its steadystate and transient solutions. Using these techniques, we examine a simple parsimonious feedback system for issues such as throughput/loss performance, fairness and stability. Our approach has the flexibility to study the effect of several additional factors such as asynchronous feedback, twolevel control and explicit rate notification in the presence of underlying highpriority traffic. This study brings to light the tradeoffs between system performance and the complexity of...
A Flexible Inverse Laplace Transform Algorithm and its Application
"... A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotientdifference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the qu ..."
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Cited by 1 (0 self)
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A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotientdifference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotientdifference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm. Key words. Inverse Laplace transform, timeintegration, transport equation, porous media. AMS subject classfications. 65M60, 65Y20. 1
On NonMonotone Solutions Of An Integrodifferential Equation In Linear Viscoelasticity
, 1996
"... . We consider the integrodifferential equation u(t; x) = R t 0 a(t \Gamma s)uxx (s; x)ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) = a 0 + a1 t + R t 0 a 1 (s)ds, where a 0 0, a1 0, and a 1 2 L 1 loc (R+ ) is of positive type ..."
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. We consider the integrodifferential equation u(t; x) = R t 0 a(t \Gamma s)uxx (s; x)ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) = a 0 + a1 t + R t 0 a 1 (s)ds, where a 0 0, a1 0, and a 1 2 L 1 loc (R+ ) is of positive type and satisfies the condition R 1 0 e \Gammafflt ja 1 (t)jdt ! 1 for every ffl ? 0. Solving the equation numerically and performing a careful error analysis we show that the solution u(t; x) need not be nondecreasing in t 0 for fixed x ? 0, if a 1 is nonnegative, nonincreasing, and convex. The same result is shown to hold under the assumption that a 1 is completely positive. This answers a question that remained unsolved in [J. Pruß, Math. Ann., 279 (1987), p. 330]. In the case where a1 is convex, piecewise linear, the solution is shown to be almost everywhere equal to a function which is discontinuous across infinitely many parallel lines. Key words. viscoelasticity, integrodifferentia...
A Hybrid Laplace Transform Mixed Multiscale FiniteElement Method for Flow Model in Porous Media ⋆
"... This paper presents a hybrid Laplace transform Mixed Multiscale Finiteelement Method (MMsFEM) to solve partial differential equations of flow in porous media. First, the time term of parabolic equation with unknown pressure term is removed by the Laplace transform. Then, to obtain the numerical app ..."
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This paper presents a hybrid Laplace transform Mixed Multiscale Finiteelement Method (MMsFEM) to solve partial differential equations of flow in porous media. First, the time term of parabolic equation with unknown pressure term is removed by the Laplace transform. Then, to obtain the numerical approximation of pressure and velocity directly, the transformed equations on coarse mesh are solved by mixed multiscale FEM, which utilizes the effects of finescale heterogeneities through basis function formulations computed from local flow problems. Finally, the associated pressure and velocity transform are inverted by the method of numerical inversion of the Laplace transform to obtain the numerical solution.