On exponential convergence of availability factor

The availability factor is the probability that the system is working properly at a given moment of time. It’s estimation and calculation are one the most important tasks in a reliability theory. The calculating of the stationary value of the availability factor is not difficult for the most cases. However, in real applications it is necessary to know how soon the value of the availability factor becomes sufficiently close to its limiting value, i.e. it is important to know the rate of convergence of the availability factor to its stationary value. In cases where the distributions function of the operating and recovery times are exponential, the question of the rate of convergence of the availability factor is the question of the rate of convergence of the solution of the Kolmogorov – Chapman equations with constant coefficients. This problem is solvable by means of Laplace transform. The most processes describing the behavior of reliability systems are regenerative, and for them the type of rate of convergence of the distribution to a stationary one is known – exponential or polynomial. Thе strong upper bounds of the rate of the convergence for regenerative processes can be obtained using the Lorden's inequality, when the distribution of the length of a regeneration period is known. But for reliability systems, it is possible to take into account the specific features of the investigated regeneration period. This article shows a method for constructing a strong exponential bounds for convergence rate of the availability factor for one restorable element.