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73
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 192 (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...
Radiation Boundary Condition for the Numerical Simulation of Waves
 Acta Numerica
, 1999
"... We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of ..."
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Cited by 76 (3 self)
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We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem. In this paper the theory of exact boundary conditions for constant coefficient timedependent problems is developed in detail, with many examples from physical applications. The theory is used to motivate various approximations and to establish error estimates. Complexity estimates are also derived to
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 ..."
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Cited by 66 (22 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.
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 44 (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.
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
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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 39 (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.
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 36 (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.
An Approximation for Performance Evaluation of Stationary Single Server Queues
 IEEE Transactions on Communications
, 1994
"... This paper provides a method for approzimating the probability distributions of stationary statistics in FIFO single server queues. The method is based on the Wiener Hopf factorization technique, and is applied to semiMarkov queues where the underlying state space is of unlimited size. A particular ..."
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Cited by 32 (6 self)
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This paper provides a method for approzimating the probability distributions of stationary statistics in FIFO single server queues. The method is based on the Wiener Hopf factorization technique, and is applied to semiMarkov queues where the underlying state space is of unlimited size. A particularly important case is a model for a statistical multiplezer where the net input process forms a stationary ergodic Gaussian discretetime stochastic process. 1
Closedloop parametric identification for continuoustime linear systems via new algebraic techniques
, 2008
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Pricing options on scalar diffusions: an eigenfunction expansion approach
 Management Science
"... This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All derivative securities are unbundled into portfolios of primitive securities termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator (present value operator). Pricing ..."
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Cited by 24 (9 self)
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This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All derivative securities are unbundled into portfolios of primitive securities termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator (present value operator). Pricing is then immediate by the linearity property of the pricing operator and the eigenvector property of eigensecurities. To illustrate the computational power of the method, we develop two applications: pricing vanilla, single and doublebarrier options under the constant elasticity of variance (CEV) process and interest rate knockout options in the CoxIngersollRoss (CIR) termstructure model.