The Number of Visits to a Subset of the State Space by an Irreducible Semi-Markov Process during a Finite Time Interval: The Probability Mass Function

Abstract

This chapter is concerned with a method for the numerical evaluation of the probability mass function of ${M_{{A_1}}}(t)$ under the semi-Markov assumption. The notation from Chapters 8 and 9 is retained here. The method considered here is based on explicit expressions for the Laplace transforms of the family of vector-valued functions defined in (9.12). In Section 10.1, which is devoted to the theory, the ‘generalised renewal argument’, already familiar in the semi-Markov context from the previous two chapters, is used to arrive at a set of recursive integral equations for the family of functions in (9.12). These equations are then solved in the Laplace transform domain. In Section 10.2, the method which will be used later for the numerical inversion of the Laplace transforms and its NAG implementation are discussed. In Section 10.3, two reliability examples are considered. The first one is the Markov model of a two-unit system in a fluctuating environment, formulated as Model 1 in Section 1.2.1. This example allows the proposed method to be assessed in the light of the results by the closed form expressions for the Markov case from Section 5.2.1. The second example is the semi-Markov model of a two- unit system of transformers, known from Section 1.2.2 as Model 7. The results for this case are validated by simulation. Section 10.4 is devoted to implementation issues. The language of implementation of the present method was FORTRAN 77 on the VAX mainframe using the commercially available, and in the U.K. very popular, numerical subroutine library NAG [NUM]. Some of the most important features of this library will be summarized. The alternative to using a software library is, of course, writing one’s own procedures. To corroborate MATLAB’s power and to discuss the notion and the use of what is termed a ‘function function’ we also provide a MATLAB implementation of the Laplace transform inversion algorithm used in this chapter; the MATLAB code will turn out to be very concise.