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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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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.
Passage Time Distributions in Large Markov Chains
 In Proc. ACM SIGMETRICS 2002, Marina Del Rey
, 2002
"... Probability distributions of response times are important in the design and analysis of transaction processing systems and computercommunication systems. We present a general technique for deriving such distributions from highlevel modelling formalisms whose state spaces can be mapped onto finite M ..."
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Cited by 21 (10 self)
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Probability distributions of response times are important in the design and analysis of transaction processing systems and computercommunication systems. We present a general technique for deriving such distributions from highlevel modelling formalisms whose state spaces can be mapped onto finite Markov chains. We use a loadbalanced, distributed implementation to find the Laplace transform of the first passage time density and its derivatives at arbitrary values of the transform parameter s. Setting s=0 yields moments while the full passage time distribution is obtained using a novel distributed Laplace transform inverter based on the Laguerre method. We validate our method against a variety of simple densities, cycle time densities in certain overtakefree (treelike) queueing networks and a simulated Petri net model. Our implementation is thereby rigorously validated and has already been applied to substantial Markov chains with over 1 million states. Corresponding theoretical results for semiMarkov chains are also presented.
Response time densities in Generalised Stochastic Petri Net models
 In Proc. 3rd Int. Workshop on Software and Performance (WOSP 2002
, 2002
"... Generalised Stochastic Petri nets (GSPNs) have been widely used to analyse the performance of hardware and software systems. This paper presents a novel technique for the numerical determination of response time densities in GSPN models. The technique places no structural restrictions on the models ..."
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Cited by 18 (10 self)
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Generalised Stochastic Petri nets (GSPNs) have been widely used to analyse the performance of hardware and software systems. This paper presents a novel technique for the numerical determination of response time densities in GSPN models. The technique places no structural restrictions on the models that can be analysed, and allows for the highlevel specification of multiple source and destination markings, including any combination of tangible and vanishing markings. The technique is implemented using a scalable parallel Laplace transform inverter that employs a modified Laguerre inversion technique. We present numerical results, including a study of the full distribution of endtoend response time in a GSPN model of the Courier communication protocol software. The numerical results are validated against simulation. 1.
Parallel computation of response time densities and quantiles in large Markov and semiMarkov models
, 2004
"... Response time quantiles reflect userperceived quality of service more accurately than mean or average response time measures. Consequently, online transaction processing benchmarks, telecommunications Service Level Agreements and emergency services legislation all feature stringent 90th percenti ..."
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Cited by 14 (8 self)
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Response time quantiles reflect userperceived quality of service more accurately than mean or average response time measures. Consequently, online transaction processing benchmarks, telecommunications Service Level Agreements and emergency services legislation all feature stringent 90th percentile response time targets. This thesis presents techniques and tools for extracting response time densities, quantiles and moments from largescale models of reallife systems. This work expands the applicability, capacity and specification power of prior work, which was hitherto focused on the analysis of Markov models which only support exponential delays. Response time densities or cumulative distribution functions of interest are computed by calculating and subsequently numerically inverting their Laplace transforms. We develop techniques for the extraction of response time measures from Generalised Stochastic Petri Nets (GSPNs) and SemiMarkov Stochastic Petri Nets (SMSPNs). The latter is our proposed modelling formalism for the highlevel specification of semiMarkov models which support generallydistributed delays.
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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Numerical inversion of multidimensional Laplace transforms by the Laguerre method
 Eval
, 1998
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General methods for analysis of sequential “nstep” kinetic mechanisms: application to single turnover kinetics of helicasecatalyzed DNA unwinding
 Biophys. J
, 2003
"... ABSTRACT Helicasecatalyzed DNA unwinding is often studied using ‘‘all or none’ ’ assays that detect only the final product of fully unwound DNA. Even using these assays, quantitative analysis of DNA unwinding time courses for DNA duplexes of different lengths, L, using ‘‘nstep’ ’ sequential mechan ..."
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Cited by 8 (6 self)
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ABSTRACT Helicasecatalyzed DNA unwinding is often studied using ‘‘all or none’ ’ assays that detect only the final product of fully unwound DNA. Even using these assays, quantitative analysis of DNA unwinding time courses for DNA duplexes of different lengths, L, using ‘‘nstep’ ’ sequential mechanisms, can reveal information about the number of intermediates in the unwinding reaction and the ‘‘kinetic step size’’, m, defined as the average number of basepairs unwound between two successive rate limiting steps in the unwinding cycle. Simultaneous nonlinear leastsquares analysis using ‘‘nstep’ ’ sequential mechanisms has previously been limited by an inability to float the number of ‘‘unwinding steps’’, n, and m, in the fitting algorithm. Here we discuss the behavior of single turnover DNA unwinding time courses and describe novel methods for nonlinear leastsquares analysis that overcome these problems. Analytic expressions for the time courses, fss(t), when obtainable, can be written using gamma and incomplete gamma functions. When analytic expressions are not obtainable, the numerical solution of the inverse Laplace transform can be used to obtain fss(t). Both methods allow n and m to be continuous fitting parameters. These approaches are generally applicable to enzymes that translocate along a lattice or require repetition of a series of steps before product formation.
Performance Trees: Implementation and Distributed Evaluation
"... In this paper, we describe the first realisation of an evaluation environment for Performance Trees, a recently proposed formalism for the specification of performance properties and measures. In particular, we present details of the architecture and implementation of this environment that comprises ..."
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Cited by 5 (4 self)
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In this paper, we describe the first realisation of an evaluation environment for Performance Trees, a recently proposed formalism for the specification of performance properties and measures. In particular, we present details of the architecture and implementation of this environment that comprises a clientside model and performance query specification tool, and a serverside distributed evaluation engine, supported by a dedicated computing cluster. The evaluation engine combines the analytic capabilities of a number of distributed tools for steadystate, passage time and transient analysis, and also incorporates a caching mechanism to avoid redundant calculations. We demonstrate in the context of a case study how this analysis pipeline allows remote users to design their models and performance queries in a sophisticated yet easytouse framework, and subsequently evaluate them by harnessing the computing power of a Grid cluster backend.