An exponential riddle

We present a problem of finding the optimum inspection procedure for a system whose time to failure is an exponential random variable. We characterize a simple and surprising solution to the problem. Consider a system which is subject to failure and whose time to failure may be characterized by an exponential random variable with mean 1/b. The system does not display any symptoms of failure unless it is inspected at a cost of I We assume that the inspections are imperfect in the sense that there is a positive probability q of not detecting the failure even if an inspection is held after failure. While in the failed state, the system incurs a cost at the rate of v per unit time until the detection of failure. The problem is to determine when to inspect the system so as to minimize the expected cost until detection of failure. Based on the memoryless property of the exponential random variable, do you think that the optimum inspection intervals should be equal? If your answer to this riddle is an unqualified 'Yes', please proceed with caution. A potential application of this model is in the area of quality control. A production process goes out of control and starts producing defective items after some random time units. This fact can be discovered only by an error-prone and costly inspection of the product quality. While the system is out of control, it incurs a steady cost of producing defective items. The problem is to find a minimum-cost inspection schedule for the detection of failure. For a comprehensive list of other potential applications for this and related models, see Luss [3]. A different version of this problem was solved by Barlow et al. [2]. The problem is a continuous-time Markovian decision process (MDP) with