An Introduction to Numerical Transform Inversion and Its Application to Probability Models

Abstract

Numerical transform inversion has an odd place in computational probability. Historically, transforms were exploited extensively for solving queueing and related probability models, but only rarely was numerical inversion attempted. The model descriptions were usually left in the form of transforms. Vivid examples are the queueing books by Takács [Takács, 1962] and Cohen [Cohen, 1982]. When possible, probability distributions were calculated analytically by inverting transforms, e.g., by using tables of transform pairs. Also, moments of probability distributions were computed analytically by differentiating the transforms and, occasionally, approximations were developed by applying asymptotic methods to transforms, but only rarely did anyone try to compute probability distributions by numerically inverting the available transforms. However, there were exceptions, such as the early paper by Gaver [Gaver, 1966]. (For more on the history of numerical transform inversion, see our earlier survey [Abate and Whitt, 1992a].) Hence, in the application of probability models to engineering, transforms became regarded more as mathematical toys than practical tools. Indeed, the conventional wisdom was that numerical transform inversion was very difficult. Even numerical analysts were often doubtful of the numerical stability of inversion algorithms. In queueing, both theorists and practitioners lamented about the “Laplace curtain” obscuring our understanding of system behavior.