Numerical exploration of a bivariate Lindley process via the bivariate Laguerre transform

In a recent paper by Sumita and Kijima, the bivariate Laguerre transform was developed, which provides a systematic numerical tool for evaluating repeated combinations of bivariate continuum operations such as bivariate convolutions, marginal convolutions, double tail integration, partial differentiation and multiplication by bivariate polynomials. The formalism is an extension of the unvariate Laguerre transform developed by Keilson, Nunn and Sumita, using the product orthonormal basis generated from Laguerre functions. In this paper, the power of the procedure is demonstrated by studying numerically a bivariate Lindley process arising from certain queueing systems. Various descriptive distributions reflecting transient behavior of such queueing systems are explicitly evaluated via the bivariate Laguerre transform.