Infinite-series Representations of Laplace Transforms of Probability Density Functions for Numerical Inversion

In order to numerically invert Laplace transforms to calculate probability density functions (pdf’s) and cumulative distribution functions (cdf’s) in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., of a waiting-time cdf) can be computed when the Laplace transform values of component pdf’s (e.g., of a service-time pdf) can be computed. However, there are few explicit expressions for Laplace transforms of component pdf’s available when the pdf does not have a pure exponential tail. In order to remedy this problem, we propose the construction of infiniteseries representations for Laplace transforms of pdf’s and show how they can be used to calculate transform values. We use the Laplace transforms of exponential pdf’s, Laguerre functions and Erlang pdf’s as basis elements in the series representations. We develop several specific parametric families of pdf’s in this infinite-series framework. We show how to determine the asymptotic form of the pdf from the series representation and how to truncate so as to preserve the asymptotic form for a time of interest.