Continuous approximations of discrete phase-type distributions and their applications to reliability models

Using Erlangization, we construct two types of continuous phase-type (PH) random variables that approximate discrete PH-random variables and finite discrete random variables. The key idea of the method is to use Erlang random variables to approximate constants. The approximations have (i) explicit closed form PH-representations; (ii) a small set of parameters, (iii) the same mean as the original random variable; (iv) moments and variances that can be arbitrarily close to that of the original random variable; (v) distribution functions that converge to that of the original random variable; and (vi) distribution functions that decreasing in convex order. The approximations are utilized in the analysis of two basic reliability structures: the serial model and the parallel model. Some bounds on such basic stochastic models are obtained. Numerical examples are presented to demonstrate the effectiveness of the approximations. (c) 2022 Elsevier B.V. All rights reserved.