Probabilistically-oriented analysis of a job execution in a semi-Markov environment

In this paper, a single machine's functioning process in a random environment is studied. The machine, which is subject to failures and renewals, executes stochastic jobs, with the lifetime of the machine and the renewal time being random variables with general distributions. Here, we present a novel, purely probabilistic approach to solve the stochastic system that describes the operation of the machine, without relying on Kolmogorov equations. The results are formulated as rigorous mathematical statements, thus offering a significant simplification in the reliability analysis of the considered stochastic system. These results, together with related developments in the literature, illustrate that for a broad family of semi–Markov reliability and queuing systems, purely probabilistic arguments based on renewal and regeneration properties could provide transparent and effective analytical tools. This approach can be applied, among many others, to semi–Markov models in which supplementary variables are also employed, including extensions of classical queuing systems (e.g., M/G/1 models with vacations, retrials, reneging, balking, or feedback) and repairable reliability systems with general lifetime and repair distributions. In conclusion, the proposed methodology provides an efficient probabilistically transparent framework for the transient analysis of semi–Markov systems; as a result, it may serve as a useful alternative or complement to classical analytical techniques with important practical implications in reliability and the queuing theory.