Numerical solution of the steady-state probability and reliability of a repairable system with three unites

In the present paper, a series-parallel repairable system with three unites was developed under some assumptions. By using the supplementary variables method, probability arguments and limiting transitions, the integro-differential equations governing the behavior of the system were obtained. Since some of the system equations have two hazard functions involved, the numerical simulation methods were used to analyze the reliability of the system. Firstly, combining the boundary conditions and characteristic curve method, the state equations were transformed into a set of integral equations. Secondly, the sequences of approximating functions were constructed for the integral equations which are analogous to the Gauss–Seidel or SOR iterative scheme for solving a system of linear equations. It was showed that the sequences of approximating functions converge pointwise to the solution of integral equations and they have continuously differential solutions. Finally, the numerical solutions of the integral equations, system equations and some reliability indices were presented by building a discretization of approximating sequence.