Some New Concepts and Their Computational Formulae in Aggregated Stochastic Processes with Classifications Based on Sojourn Times

The system’s performance is one of most important issues in both theory and practice. The task of evaluation of system performance first needs a series of indexes which can describe the system’s performance properly and correctly. Different indexes provide different descriptions and result in different conclusions on system’s performance. On the other hand, although it has a lot of indexes for evaluation of system performance such as reliability, availability and safety and so forth, they still cannot meet the variety requirements on the evaluation of system performance. Thus it is an important work to introduce and develop some new indexes to measure the system’s performance. In this paper, two types of related probability measures, point-wise and interval-wise probabilities, including their concepts and computation formulae, are developed under an alternative renewal process and its derivative aggregated stochastic process with state classifications based on sojourn times. All limits of introduced new measures are discussed too, and the relationship among these new measures is studied. Finally, two special cases: constant and exponential, are discussed too, and some numerical examples are presented to illustrate the concepts intuitively in this work. The work for introduction of new concepts is motivated by some practical problems, specially in repairable systems. The research results can be used not only in reliability, but also may be used in finance, engineering, economy and other fields.