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 (pdfs) and cumulative distribution functions (cdfs) 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 pdfs (e.g., of a service-time pdf) can be computed. However, there are few explicit expressions for Laplace transforms of component pdfs available when the pdf does not have a pure exponential tail. In order to remedy this problem, we propose the construction of infinite-series representations for Laplace transforms of pdfs and show how they can be used to calculate transform values. We use the Laplace transforms of exponential pdfs. Laguerre functions and Erlang pdfs as basis elements in the series representations. We develop several specific parametric families of pdfs 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.