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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...
Towards Reliable Modelling with Stochastic Process Algebras
 Department of Computer Science, University of Bristol, Bristol
, 1999
"... Abstract In this thesis, we investigate reliable modelling within a stochastic process algebra framework. Primarily, we consider issues of variance in stochastic process algebras as a measure of model reliability. This is in contrast to previous research in the field which has tended to centre aroun ..."
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Cited by 10 (6 self)
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Abstract In this thesis, we investigate reliable modelling within a stochastic process algebra framework. Primarily, we consider issues of variance in stochastic process algebras as a measure of model reliability. This is in contrast to previous research in the field which has tended to centre around mean behaviour and steadystate solutions. We present a method of stochastic aggregation for analysing generallydistributed processes. This allows us more descriptive power in representing stochastic systems and thus gives us the ability to create more accurate models. We improve upon two welldeveloped Markovian process algebras and show how their simpler paradigm can be brought to bear on more realistic synchronisation models. Now, reliable performance figures can be obtained for systems, where previously only approximations of unknown accuracy were possible. Finally, we describe reliability definitions and variance metrics in stochastic models and demonstrate how systems can be made more reliable through careful combination under stochastic process algebra operators. ii Acknowledgements My three years in the department in Bristol have been a lot of fun and the person I have most to thank for this is my friend and mentor, Neil Davies. I should also acknowledge the funding from NATS for my project and especially the help of Suresh Tewari (NATS) and Gordon Hughes (SSRC).
A matrixbased method for analysing stochastic process algebras
 University of Waterloo
, 2000
"... This paper demonstrates how three stochastic process algebras can be mapped on to a generallydistributed stochastic transition system. We demonstrate an aggregation technique on these stochastic transition systems and show how this can be implemented as a matrixanalysis method for finding steadys ..."
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Cited by 9 (5 self)
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This paper demonstrates how three stochastic process algebras can be mapped on to a generallydistributed stochastic transition system. We demonstrate an aggregation technique on these stochastic transition systems and show how this can be implemented as a matrixanalysis method for finding steadystate distributions. We verify that the time complexity of the algorithm is a considerable improvement upon a previous method and discuss how the technique can be used to generate partial steadystate distributions for SPA systems.
RESPONSE TIME DENSITIES AND QUANTILES IN LARGE MARKOV AND SEMIMARKOV MODELS
"... Abstract. 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 ..."
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Abstract. 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 chapter describes a range of techniques for extracting response time densities and quantiles from largescale Markov and semiMarkov models of reallife systems. We describe a method for the computation of response time densities or cumulative distribution functions which centres on the calculation and subsequent numerical inversion of their Laplace transforms. This can be applied to both Markov and semiMarkov models. We also review the use of uniformization to calculate such measures more efficiently in purely Markovian models. We demonstrate these techniques by using them to generate response time quantiles in a semiMarkov model of a highavailability webserver. We show how these techniques can be used to analyse models with state spaces of O 10 7 states and above. 1. Introduction. A
France,
"... Abstract: We show that the density of Z = argmax{W (t)−t 2}, sometimes known as Chernoff’s density, is logconcave. We conjecture that Chernoff’s density is strongly logconcave or “superGaussian”, and provide evidence in support of the conjecture. We also show that the standard normal density can ..."
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Abstract: We show that the density of Z = argmax{W (t)−t 2}, sometimes known as Chernoff’s density, is logconcave. We conjecture that Chernoff’s density is strongly logconcave or “superGaussian”, and provide evidence in support of the conjecture. We also show that the standard normal density can be written in the same structural form as Chernoff’s density, make connections with L. Bondesson’s class of hyperbolically completely monotone densities, and identify a large subclass thereof having logtransforms to R which are strongly logconcave.