Development and preliminary study of a stochastic semi-Markov model of continuous supply of product management under the condition of constant consumption

The paper deals with a discrete semi-Markov stochastic model describing the operation of a control system of continuous supply of product with constant consumption. The model is a couple of random processes $(x(t), \zeta(t))$ where the main process $x(t)$ describes the amount of stock in the system at time t and the accompanying random process is a semi-Markov process with a finite set of states. The optimal control problem is put in relation to the stationary indicators related to the accompanying process. This indicator is the average of the specific nature of the profits earned in the evolution of the initial inventory control system. An explicit analytical representation for the probability characteristics of semi-Markov models is obtained. In the future, the results will allow to find an explicit representation of the Quality Score and solving the problem of optimal control.