Linear Poisson checking schedules

The problem of minimizing expected cost until detection of failure is addressed here, using random checking schedules. Homogeneous Poisson checking processes are investigated first, leading to the explicit determination of the optimal solution in a simple nonlinear programming problem. Then the case of nonhomogeneous Poisson processes, with linear intensity, is considered. In this case, an approximation of the original problem is discussed. A unique optimal solution is proved to exist and it is then characterized, using the Kuhn-Tucker conditions.