Asymptotic behavior of the average recovery cost in models of recovery processes

During the operation of rocket and space technology, electronic computing systems, power supply systems, heat supply systems, transport systems and many others, failures occur, there are threats of attacks, security threats and many other impacts that are random in nature and have a negative role in their work. Such impacts lead to restoration processes in which the operating time of the restored elements before their failure, the number of failures, the time and cost of restorations are random variables. In the theory of probability and in the mathematical theory of reliability, when studying restoration processes, the restoration function (the average value of the number of random failures) plays a special role. We especially note its importance in optimization problems when choosing a strategy for carrying out recovery processes. So one of the most important criteria for optimality is the average number of failures, the average cost of restoration, cost intensity, availability factor. We also note the problem of the need and timing of preventive restorations. Within the framework of the mathematical theory of reliability, models of restoration processes are considered taking into account the cost of restorations with varying distribution functions of the time to failure of the restored elements and the costs of restorations. For the models under consideration, a formula for the cost function (average cost of restorations) through the restoration functions of two general restoration processes is obtained, which allows proving theorems on the asymptotic behavior of the cost function, well known for the asymptotic behavior of the restoration function of the general restoration process, where the restoration time is not taken into account. The obtained asymptotic theorems for the average cost of restorations are generalized to the introduced alternating (when the random time of restorations is also taken into account) restoration process, taking into account the cost of restorations with changing distribution functions of the time to failure of the restored elements and the costs of their restorations.