An explicit solution for a renewal process with waiting time and its variational principle

The forward and backward equations for the conditional probability density are derived for a reliability system consisting of a single component whose repair is subject to a delay time in providing a spare part but whose mean rate of repair is otherwise constant and whose time to failure is exponentially distributed. Exact solutions are quoted. These equations are then shown to be an adjoint pair that provide stationary conditions for a variational principle, in elementary form, from which all properties of the systems can be predicted with an accuracy greater than that implied by the trial functions or approximations used. A second or specific form of variational principle provides specific estimates to questions at hand. The second or adjoint field in the first elementary principle is the backward Kolmogorov solution and the in the specific form is the importance function, as used in nuclear reactor theory. The solutions are given for long-time and in a recurrence relation form valid for all times so that approximate solutions can be checked. Approximations suitable for variational trial functions are given. Two examples give the effect of a change of delay time for a steady state and an initial transient, respectively.