Journal ArticleDOI
Distribution Estimation Using Laplace Transforms
TLDR
Two related methods for deriving probability distribution estimates using approximate rational Laplace transform representations are proposed, addressing the question of the number of terms, or the order, involved in a generalized hyperexponential, phase-type, or Coxian distribution, a problem not adequately treated by existing methods.Abstract:
We propose two related methods for deriving probability distribution estimates using approximate rational Laplace transform representations Whatever method is used, the result is a Coxian estimate for an arbitrary distribution form or plain sample data, with the algebra of the Coxian often simplifying to a generalized hyperexponentia l or phase-type The transform (or, alternatively, the moment-generating function) is used to facilitate the computations and leads to an attractive algorithm For method one, the first 2N - 1 derivatives of the transform are matched with those of an approximate rational function; for the second method, a like number of values of the transform are matched with those of the approximation The numerical process in both cases begins with an empirical Laplace transform or truncation of the actual transform, and then requires only the solution of a relatively small system of linear equations, followed by root finding for a low-degree polynomial Besides the computationally attractive features of the overall procedure, it addresses the question of the number of terms, or the order, involved in a generalized hyperexponential, phase-type, or Coxian distribution, a problem not adequately treated by existing methods Coxian distributions are commonly used in the modeling of single-stage and network queueing problems, inventory theory, and reliability analyses They are particularly handy in the development of large-scale model approximationsread more
Citations
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Journal ArticleDOI
Characterization of Matrix-Exponential Distributions
TL;DR: For a rational Laplace-Stieltjes transform that has a pole of maximal real part that is real and negative, this paper gave a geometric description of all admissible numerator polynomials that give rise to matrix-exponential distributions.
Journal ArticleDOI
Partitioning Customers Into Service Groups
TL;DR: Methodology for quantifying the tradeoff between economies of scale associated with larger systems and the benefit of having customers with shorter service times separated from other customers with longer service times, as is done in service systems with express lines is provided.
Journal ArticleDOI
Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis
TL;DR: This work exploits the exponentiality of the steady-state delay distributions for the G/M/1 and G/ M/ c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times.
Journal ArticleDOI
An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue
TL;DR: This paper develops a method for approximating Laplace transforms, and gives algorithms to compute the steady state probability distribution of the waiting time of an M/G/1 queue to a desired accuracy.
Journal ArticleDOI
Analysis of the M/G/1 processor-sharing queue with bulk arrivals
TL;DR: This work obtains an expression for the expected response time of a job as a function of its size, when the service times of jobs have a generalized hyperexponential distribution and more generally for distributions with rational Laplace transforms.
References
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Journal ArticleDOI
Maximum likelihood from incomplete data via the EM algorithm
Journal ArticleDOI
The Fourier-series method for inverting transforms of probability distributions
Joseph Abate,Ward Whitt +1 more
TL;DR: This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions and describes two methods for inverting Laplace transform based on the Post-Widder inversion formula.
Journal Article
Fitting Phase-type Distributions via the EM Algorithm
TL;DR: An extended EM algorithm is used to minimize the information divergence (maximize the relative entropy) in the density approximation case and fits to Weibull, log normal, and Erlang distributions are used as illustrations of the latter.
Journal ArticleDOI
The least variable phase type distribution is Erlang
Aldous David,Shepp Larry +1 more
TL;DR: In this article, it was shown that To is most nearly constant in the sense of minimizing the coefficient of variation var(T0)/(ET0)2 over all transition matrices Pij and exponential delay parameters λi- in each state when Pii = 1, i = n, n- l, l, 1 and λ i, ≡ constant.
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