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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...
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.
Approximating Response Time Distributions
 Performance Evaluation Review
, 1989
"... : The response time is the most visible performance index to users of computer systems. Endusers see individual response times, not the average. Therefore the distribution of response times is important in performance evaluation and capacity planning studies. However, the analytic results cannot be ..."
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Cited by 2 (1 self)
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: The response time is the most visible performance index to users of computer systems. Endusers see individual response times, not the average. Therefore the distribution of response times is important in performance evaluation and capacity planning studies. However, the analytic results cannot be obtained in practical cases. A new method is proposed to approximate the responsetime distribution. Unlike the previous methods the proposed one takes into account the servicetime distributions and routing behaviour. The reported results indicate that the method provides reasonable approximations in many cases. 1 Introduction Queueing network modelling is a popular tool in the performance evaluation of computer systems. It has been successfully used in various applications of modelling computer systems. In most applications only the mean values of performance indices, such as mean device queuelengths and the mean system responsetime, have been considered. The increasing usage of comput...
2009), ‘Laplace transformation method for the Black–Scholes equations
 Int. J. Numer. Anal. Model
"... Abstract. In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the BlackScholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very ..."
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Cited by 2 (1 self)
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Abstract. In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the BlackScholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various option prices. Existence and uniqueness properties of the Laplace transformed BlackScholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.
Performance analysis of reassembly and multiplexing queueing with longrangedependent input traffic
 TELECOMMUNICATION SYSTEMS
, 2002
"... This paper studies the impact of longrangedependent (LRD) traffic on the performance of reassembly and multiplexing queueing. A queueing model characterizing the general reassembly and multiplexing operations performed in packet networks is developed and analyzed. The buffer overflow probabiliti ..."
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Cited by 1 (1 self)
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This paper studies the impact of longrangedependent (LRD) traffic on the performance of reassembly and multiplexing queueing. A queueing model characterizing the general reassembly and multiplexing operations performed in packet networks is developed and analyzed. The buffer overflow probabilities for both reassembly and multiplexing queues are derived by extending renewal analysis and Beneˇs fluid queue analysis, respectively. Tight upper and lower bounds of the frame loss probabilities are also analyzed and obtained. Our analysis is not based on existing asymptotic methods, and it provides new insights regarding the practical impact of LRD traffic. For the reassembly queue, the results show that LRD traffic and conventional Markov traffic yield similar queueing behavior. For the multiplexing queue, the results show that the LRD traffic has a significant impact on the buffer requirement when the target loss probability is small, including for practical ranges of buffer size or maximum delay.
Inversion of noisefree Laplace transforms: Towards a standardized set of test problems
 Inverse Problems in Engineering, 2001
"... The numerical inversion of Laplace transform arises in many applications of science and engineering whenever ordinary and partial differential equations or integral equations are solved. The increasing number of available numerical methods and computer codes has generated a need for welldocumented ..."
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Cited by 1 (0 self)
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The numerical inversion of Laplace transform arises in many applications of science and engineering whenever ordinary and partial differential equations or integral equations are solved. The increasing number of available numerical methods and computer codes has generated a need for welldocumented sets of test problems. Using such sets, algorithm developers can evaluate the relative merits and drawbacks of their suggested new methods, and endusers can make judgments on the applicability of an individual method for a specific problem. Many areas in science and engineering, lead to problems that share three important properties: i) the image function can be evaluated for real arguments, but not necessarily for complex ones; ii) the original is known to be infinitely differentiable for times t> 0, iii) the values of the image function can be obtained with any prescribed accuracy. The published test sets do not properly cover these applications, as many included problems are beyond of the specific class, while the remaining ones fail to address some of the potential difficulties arising in practice. The goal of this paper is to establish a common ground for problem classification, to list the requirements for the above class of problems, and to provide a carefully selected test set by addressing the
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...