Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death

The paper proposes a method for constructing models based on the analysis of birth and death processes with linear growth in semimartingale terms. Based on this method, stochastic models of simple just-in-time systems (analyzed in the theory of productive systems) and windows of vulnerability (widely discussed in risk theory) are considered. The main results obtained in the work are presented in terms of the average values of the time during which the processes reach zero values. At the same time, they are considered and used in the study of assessment models for local times of the processes.
Here, simple Markov processes with a linear growth of intensities (perhaps, depending on time) are analyzed. At the same time, the obtained and used estimates are of theoretical interest. Thus, for example, the average value of the stopping time, at which the process reaches zero, depends on functions such as the harmonic number and the remainder term for the logarithmic function in the Taylor theorem.
As the main result, the method of mathematical modeling of just-in-time systems and windows of vulnerability is proposed. The semimartingale description method used here should be considered as the first step of such a modeling, since, being a trajectory method, it allows diffusion (including non-Markov processes) generalizations when constructing stochastic models of windows of vulnerability and just-in-time. In the theoretical part of the article, we formulate statements for the average values of the local time and the stopping times when the birth and death processes reach a given value. This allows us to uniformly present estimates for the models of the just-in-time system and for windows of vulnerability, the result for which is given in the form of a limit theorem. The main results are formulated as theorems and lemmas. The proofs use semimartingale methods.