A new approximation for the inspection period of systems with different failure rates

Consider a system whose working state is only known if it is inspected. Suppose that each inspection has a fixed cost and also suppose that there is a fixed cost for unit time of bad working. This paper presents a new approximate solution for the inspection period, which is a generalization of a previous one. It verifies that when the failure rate is constant (exponential distribution) then the approximation solution is a (nearly) optimal solution. The paper also analyses the validity of that new approximation when the failure rate is: (a) decreasing; (b) increasing; (c) first increasing and then decreasing; and, finally, (d) when the failure rate has a ‘bathtub’ shape. Generally, it verifies that in these cases this approximation solution is better than a previous one.