Properties related with conditional expectation for a non-homogeneous Poisson Process

In this paper, we study a non-homogeneous Poisson Process that generalizes the well-known Jelinski-Moranda (J-M) model, originally proposed as a predictive model for software failures. The process under investigation consists of a fixed number of identical independent Poisson arrival processes, each operating from a common starting point until its first arrival occurs. While numerous studies in the field of software reliability are built upon the J-M model, there is a dearth of existing literature that examines the properties of this generalized version beyond software reliability. Considering this process is very common in real-world scenarios situated within stochastic environments, our study focuses solely on its theoretical exploration of general mathematical properties. First we formally define it in two ways, and clarify its relationships with established models such as continuous-time Markov Chain, Markov process and Markov Arrival Process by constructing it as a special case. Then, within the framework of Poisson process, we delve into an analysis of specific properties related to conditional expectations, with our primary contribution being in the computation of expected conditional arrival times.