Transform inversion and its application to stochastic models

Wu-Lin Chen, Purdue University

Abstract

This research deals with computational aspects of stochastic modeling. Contributions are made to inverting transforms and analyzing polling systems by transform inversion. Abate and Whitt proposed an algorithm, known as “Algorithm Euler”, to invert Laplace transforms of probability density functions efficiently. This algorithm is a version of the Fourier series method, in which the inverse transform is approximated by the Fourier series of a periodic function. At each point where the inverse transform is to be computed, the parameters in Algorithm Euler are chosen so that an almost alternating series is obtained. Then a convergence-acceleration technique for summing an alternating series, known as Euler summation, is utilized to improve the convergence rate. Algorithm Euler computes a new set of Fourier coefficients, which can be recognized as values of the transform, for each function evaluation. In our study, a sequence of non-overlapping intervals is constructed such that a single Fourier series is effective for each interval. Thus our refined algorithm is more efficient. We test this revised algorithm on some classical queueing models. A single-station polling system is a station where a single server serves multiple queues cyclically. The key to much of the analysis of these systems is to obtain the first two moments of queue lengths at “polling instants”, which are defined as the instants when the server starts to serve each queue. We propose an approach in which we compute the generating functions of queue lengths at polling instants recursively. Then these generating functions are inverted to get the first two moments. Several numerical experiments are provided to compare the performance of our algorithm to that of the existing approach. We also develop a queueing network analyzer for networks of polling stations. The decomposition approach, as used in the Queueing Networks Analyzer (QNA), is adopted to treat the interactions between stations. Then, each station is analyzed individually by transform inversion. We provide several examples to test this analyzer.

Degree

Ph.D.

Advisors

O'Cinneide, Purdue University.

Subject Area

Industrial engineering|Mechanics

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