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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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A Fast and Accurate FFTBased Method for Pricing EarlyExercise Options under Lvy Processes
 SIAM Journal of Scientific Computing
, 2008
"... Abstract. A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the wellknown riskneutral valuation formula ..."
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Cited by 12 (6 self)
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Abstract. A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the wellknown riskneutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within many regular affine models, among which the class of exponential Lévy models. For an Mtimes exercisable Bermudan option, the overall complexity is O(MN log 2 (N)) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.
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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Introduction to Engineering Design: http://www.ecsel.psu.edu/edg100
, 2006
"... Abstract. We evaluate the performance of a class of twohop relay protocols for mobile ad hoc networks. The interest is on the multicopy twohop relay (MTR) protocol, where the source may generate multiple copies of a packet and use relay nodes to deliver the packet (or a copy) to its destination, a ..."
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Cited by 5 (1 self)
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Abstract. We evaluate the performance of a class of twohop relay protocols for mobile ad hoc networks. The interest is on the multicopy twohop relay (MTR) protocol, where the source may generate multiple copies of a packet and use relay nodes to deliver the packet (or a copy) to its destination, and on the twohop relay protocol with erasure coding. Performance metrics of interest are the time to deliver a single packet to its destination, the number of copies of the packet at delivery instant, and the total number of copies that the source generates. The packet copies at relay nodes have limited lifetime (timetolive TTL). Via a Markovian analysis, the three performance metrics of the MTR protocol are obtained in closedfrom in the case where the number of the copies in the network is limited. Also, we develop an approximation analysis in the case where the intermeeting times between nodes are arbitrarily distributed and the TTLs of the copies are constant and all equal. In particular, we show that exponential intermeeting times yield stochastically smaller delivery delays than hyperexponential intermeeting times, and that exponential TTLs yield stochastically larger delivery delays than constant TTLs. Finally, we characterize the delivery delay and the number of transmissions in the twohop relay protocol with erasure coding and compare this scheme with the multicopy scheme. Keywords: Mobile ad hoc network, Twohop relay protocol, Erasure coding, Mobility
Capacity planning in clientserver systems
 Journal of Distributed Systems Engineering
, 1996
"... Abstract. The proliferation of client–server systems in business continues unabated, as applications are split into local tasks run on ‘client ’ workstations and resourceintensive computations run on a ‘server ’ mainframe. The complexity of such systems requires quantitative modelling for their eff ..."
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Cited by 4 (0 self)
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Abstract. The proliferation of client–server systems in business continues unabated, as applications are split into local tasks run on ‘client ’ workstations and resourceintensive computations run on a ‘server ’ mainframe. The complexity of such systems requires quantitative modelling for their efficient design and reconfiguration throughout their lifetime. The tools and techniques that are needed for the effective performance management of distributed client–server systems are discussed and illustrated by a case study taken from the financial sector. 1.
Simple Models for Performance Evaluation of a Class of TwoHop Relay Protocols
"... We evaluate the performance of a class of twohop relay protocols for mobile ad hoc networks via simple models. The focus is on the multicopy twohop relay protocol, where the source may generate multiple copies of a packet and use relay nodes to transmit the packet (or a copy) to its destination, a ..."
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Cited by 1 (0 self)
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We evaluate the performance of a class of twohop relay protocols for mobile ad hoc networks via simple models. The focus is on the multicopy twohop relay protocol, where the source may generate multiple copies of a packet and use relay nodes to transmit the packet (or a copy) to its destination, and on the twohop relay protocol with erasure coding, where a piece of information is fragmented into n blocks in such a way that the destination may decode the data if it receives at least k blocks. Performance metrics of interest are the time to deliver a single packet to its destination, the number of copies of the packet at delivery instant, and the total number of copies that the source generates; the latter number will be larger when TTLs are associated with the copies of a packet, a situation that we address. We also investigate the case where the number of copies of a packet currently in the network is limited so as to limit the energy consumption. Performance metrics are obtained in closedfrom for the multicopy twohop relay protocol in the case of exponential intermeeting times, exponential TTLs and when the number of copies of the packet in the network is limited. We evaluate the impact of constant TTLs as opposed to exponential TTLs, and we develop an approximation analysis in the case where the intermeeting times are arbitrarily distributed. In particular, we show that exponential intermeeting times yield stochastically smaller delivery delays than hyperexponential intermeeting times; we also show that exponential TTLs yield larger expected delivery delays than constant TTLs. Finally, we characterize the delivery delay in the twohop relay protocol with erasure coding and compare this scheme with the multicopy routing scheme.
materials by the Laplace transform Galerkin boundary element method
"... Transient heat conduction in homogeneous and nonhomogeneous ..."