Semi-Markov failure rates processes
Let \(\{\lambda(u):u\geq 0\}\) be a semi-Markov process defined by a Markov renewal kernel. The author studies the survival function \(R\) defined as the expectation of \(\exp \left(-\int_0^t\lambda(u)du\right)\), for \(t \geq 0\). Note that, \(R(t)\) is considered to be the reliability of an object with a random failure rate. In the main theorem of the paper, it is proved that the associated conditional reliability functions satisfy a system of specific renewal equations. The solution of this system is unique in the class of measurable and uniformly bounded functions. This theorem is highlighted in the case of a finite or countable state space. Also, the particular case of a Furry-Yule process is treated. In this way, the author extends some results due to \textit{I. Kopocińska} and \textit{B. Kopociński} [Zastosow. Mat. 17, 5--14 (1980; Zbl 0471.60089)] and \textit{F. Grabski} [Appl. Math. Comput. 135, No.~1, 1--16 (2003; Zbl 1021.60070)].
scientific article; zbMATH DE number 5931348