Computing Laplace transforms for numerical inversion via continued fractions

It is often possible to effectively calculate probability density functions (pdf's) and cumulative distribution functions (cdf's) by numerically inver ting Laplace transforms. However, to do so it is necessary to compute the L aplace transform values, Unfortunately, convenient explicit expressions for required transforms are often unavailable for component pdf's in a probabi lity model. In that event, we show that it is sometimes possible to find co ntinued-fraction representations for required Laplace transforms that can s erve as a basis for computing the transform values needed In the inversion algorithm. This property is very likely to prevail for completely monotone pdf's, because their Laplace transforms have special continued fractions ca lled S fractions, which have desirable convergence properties. We illustrat e the approach by considering applications to compute first-passage-time cd f's in birth-and-death processes and various cdf's with non-exponential fai ls, which can be used to model service-time cdf's In queueing models. Inclu ded among these cdf's is the Pareto cdf.