Computing Laplace transforms for numerical inversion via continued fractions

It is often possible to effectively calculate probability density functions (pdfs) and cumulative distribution functions (cdfs) by numerically inverting Laplace transforms. However, to do so it is necessary to compute the Laplace transform values. Unfortunately, convenient explicit expressions for required transforms are often unavailable for component pdfs in a probability model. In that event, we show that it is sometimes possible to find continued-fraction representations for required Laplace transforms that can serve as a basis for computing the transform values needed in the inversion algorithm. This property is very likely to prevail for completely monotone pdfs, because their Laplace transforms have special continued fractions called S fractions, which have desirable convergence properties. We illustrate the approach by considering applications to compute first-passage-time cdfs in birth-and-death processes and various cdfs with non-exponential tails, which can be used to model service-time cdfs in queueing models. Included among these cdfs is the Pareto cdf.