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

Failures can occur during the operation of rocket and space technology, electronic computing systems, power supply systems, heat supply systems, transport systems and many others; there are threats of attacks, security threats and many other impacts that are random in nature and have got a negative role in their operation. Such impacts lead to recovery processes where the operating time of the recovered elements before their failure, the number of failures, the time and cost of recoveries are random variables. In the theory of probability and in the mathematical theory of reliability, when studying recovery processes, the recovery 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 optimality criteria is the average number of failures, the average cost of recovery, cost intensity, availability factor. We also outline the problem of the need and timing of preventive recoveries. Within the framework of the mathematical theory of reliability, models of recovery processes are considered taking into account the cost of recoveries with varying distribution functions of the time to failure of the recovered elements and the costs of recoveries. For the models under consideration, a formula for the cost function (average recovery cost) through the recovery functions of two general recovery processes is obtained, which allows to prove theorems on the asymptotic behavior of the cost function, well known for the asymptotic behavior of the recovery function of the general recovery process, where the recovery time is not taken into consideration. The obtained asymptotic theorems for the average cost of recoveries are generalized to the introduced alternating (when the random time of recoveries is also considered) recovery process, taking into account the cost of recoveries with changing distribution functions of the time to failure of the recovered elements and the costs of their recoveries.