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August, 1994 Multidimensional Transform Inversion with Applications to the Transient M/G/1 Queue
Gagan L. Choudhury, David M. Lucantoni, Ward Whitt
Ann. Appl. Probab. 4(3): 719-740 (August, 1994). DOI: 10.1214/aoap/1177004968

Abstract

We develop an algorithm for numerically inverting multidimensional transforms. Our algorithm applies to any number of continuous variables (Laplace transforms) and discrete variables (generating functions). We use the Fourier-series method; that is, the inversion formula is the Fourier series of a periodic function constructed by aliasing. This amounts to an application of the Poisson summation formula. By appropriately exponentially damping the given function, we control the aliasing error. We choose the periods of the multidimensional periodic function so that each infinite series is a finite sum of nearly alternating infinite series. Then we apply the Euler transformation to compute the infinite series from finitely many terms. The multidimensional inversion algorithm enables us, evidently for the first time, to calculate probability distributions quickly and accurately from several classical transforms in queueing theory. For example, we apply our algorithm to invert the two-dimensional transforms of the joint distribution of the duration of a busy period and the number served in that busy period, and the time-dependent transient queue-length and workload distributions in the M/G/1 queue. In other related work, we have applied the inversion algorithms here to calculate time-dependent distributions in the transient BMAP/G/1 queue (with a batch Markovian arrival process) and the piecewise-stationary $M_t/G_t/1$ queue.

Citation

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Gagan L. Choudhury. David M. Lucantoni. Ward Whitt. "Multidimensional Transform Inversion with Applications to the Transient M/G/1 Queue." Ann. Appl. Probab. 4 (3) 719 - 740, August, 1994. https://doi.org/10.1214/aoap/1177004968

Information

Published: August, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0808.65140
MathSciNet: MR1284982
Digital Object Identifier: 10.1214/aoap/1177004968

Subjects:
Primary: 65D30
Secondary: 60K25

Keywords: Fourier transforms , Fourier-series method , generating functions , Laplace transforms , M/G/1 queue , multidimensional transforms , Numerical transform inversion , Poisson summation formula , transient distributions

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.4 • No. 3 • August, 1994
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