Efficient computation of passage time densities and distributions in Markov chains using Laguerre method

The Laguerre method for the numerical inversion of Laplace transforms is a well known approach to the approximation of probability density functions (PDFs) and cumulative distribution functions (CDFs) of first passage times in Markov chains. Results are presented that relate the Laguerre generating functions and Laguerre coefficients of a PDF with those of the corresponding complementary CDF. This enables the ability to compute the PDF or CDF from the Laplace transform of either at the cost of computing only one set of Laguerre coefficients.