Age-usage semi-Markov models

This paper presents a non-homogeneous age-usage semi-Markov model with a measurable state space. Several probability functions useful to assess the system’s reliability are investigated. They satisfy the same family of equations we call indexed Markov renewal equations. Sufficient conditions to assure the existence and uniqueness of their solutions are provided. The numerical analysis of these equations is executed through the construction of a process discrete in time and space, which is shown to converge to the continuous one in the Skorohod topology. An algorithm useful for solving the discretized system of equations is presented by using a matrix representation.