Two-Stage Start-Up Demonstration Testing

Start-up demonstration tests and various extensions and generalizations of them (in order to accommodate dependence between the trials, to allow for corrective action to be taken once the equipment fails for the first time, etc.) have been discussed quite extensively in the literature. In this paper, we propose a start-up demonstration test to be performed in two stages which would facilitate an early rejection of a potentially bad equipment and would also enable the experimenter to place a more stringent requirement for acceptance upon observing a certain number of failures. Specifically, the decision procedure proposed is as follows. Perform start-up demonstration tests on the equipment under study consecutively and decide to: Accept the equipment (in the first stage) if a run of c1 successes occurs before d1 failures.Accept the equipment if no run of c1 successes occurs before d1 failures, but a run of c2 successes is observed before the next d2 failures.Reject the equipment if no run of c1 successes occurs before d1 failures and also no run of c1 successes occurs before the next d2 failures.We then derive the probability generating function of the waiting time for the termination of the start-up demonstration testing, and the mean of this waiting time. We also establish some recurrence relations satisfied by the probability mass function which will facilitate easy recursive computation of probabilities. We also discuss the distributions of some related random variables such as the numbers of successes and failures.