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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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The Completion Time of Programs on Processors Subject to Failure and Repair
 IEEE Transactions on Computers
, 1993
"... AbstractThe objective of this paper is to describe a technique for computing the distribution of the completion time of a program on a server subject to failure and repair. Several realistic aspects of the system are included in the model. The server behavior is modeled by a semiMarkov process in ..."
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Cited by 14 (3 self)
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AbstractThe objective of this paper is to describe a technique for computing the distribution of the completion time of a program on a server subject to failure and repair. Several realistic aspects of the system are included in the model. The server behavior is modeled by a semiMarkov process in order to accommodate nonexponential repairtime distributions. More importantly, the effect on the job completion time of the work lost due to the occurrence of a server failure is modeled. We derive a closedform expression for the LaplaceStieltjes transform (LST) of the time to completion distribution of programs on such systems. We then describe an effective numerical procedure for computing the completion time distribution. We show how these results apply to the analysis of different computer system structures and organizations of faulttolerant systems. Finally, we use numerical solution methods to find the distribution of time to completion on several systems. Index Terms Computer performance, failurerepair models, Laplace transform inversion, multistate computer systems, preemptions, semiMarkov processes. 1.
CALCULATING TRANSIENT CHARACTERISTICS OF THE ERLANG LOSS MODEL BY NUMERICAL TRANSFORM INVERSION
 Stochastic Models
"... In this paper we consider the classical Erlang loss model, i.e., the M/M/c/0 system with Poisson arrival process, exponential service times, c servers and no extra waiting space, where blocked calls are lost. We let the individual service rate be 1 and the arrival rate (which coincides with the offe ..."
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Cited by 10 (6 self)
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In this paper we consider the classical Erlang loss model, i.e., the M/M/c/0 system with Poisson arrival process, exponential service times, c servers and no extra waiting space, where blocked calls are lost. We let the individual service rate be 1 and the arrival rate (which coincides with the offered load) be a. We show how to compute several transient characteristics by numerical transform inversion. Transience arises by considering arbitrary fixed initial states.
Numerical Transform Inversion to Analyze Teletraffic Models
 IN THE EVOLUTION OF TELECOMMUNICATIONS NETWORKS, PROCEEDINGS OF THE 14 TH INTERNATIONAL TELETRAFFIC CONGRESS
, 1994
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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...