About Solutions of Renewal Equations and Determinations of Failure Moments of a System as Equilibrium Points

Abstract:- The results presented in this paper lead to solving main problems of the reliability theory: specifying the failure moments of a failure and determining the solutions of renewal equations. We analyze the following situations: 1. the system structure is not taken into consideration; 2. the system structure is known. In the first case we presume that the adopted efficiency function is the average operation time and, by using specific methods of the theory of games, we can prove that there is no equilibrium type solution, so the failure moment of the system cannot be precisely determined. By solving some specific problems, type maximum or minimax, we can only get the interval where the failure point of the system is found. The optimal problem type maximum is solved by specific methods from the theory of games while the optimal problem type minimax is solved by using the maximum principle of Pontriaghin. In the second case we start from the graph structure associated to a system with renewal operations and we build immediately the equation system with finite differences and the system of differential equations associated to this graph. Applying the Laplace transformation it is determined the system availabilities and unavailabilities caused by its subsystems. The failure moments of the system are determined as equilibrium points but the difficulties in calculations lead to obtaining only an approximate solution. Knowing the failure moments of the analyzed system lead to the reconsideration of the renewal policies of the system. Practically, there are determined the approximat