Approximating distributions and transient probabilities of Markov chains by Bernstein expolynomial functions

In this extended abstract we consider the use of Bernstein polynomials (BPs) for the approximation of distributions and transient probabilities of continuous time Markov chains (CTMCs). We show that while standard BPs are not appropriate to this purpose it is possible to derive from them a family of functions, called in the sequel Bernstein expolynomials (BEs), which enjoys those properties that are necessary for these approximations. For what concerns distribution fitting, BEs correspond to a subset of the family of matrix exponential distributions and hence they are of interest in the field of matrix analytic methods. For what concerns transient probabilities of CTMCs, BEs can provide closed-form approximations which are useful in the analysis of models where the process subordinated to a possibly non-Markovian period is described by a CTMC. The application of BEs for approximating both distributions and transient probabilities will be illustrated through several numerical examples.