Markovian start-up demonstration tests with rejection of units upon observing d failures

By conditioning on the time of the first failure, several results are derived for demonstration tests of the start-up reliability of equipment. The start-ups are assumed to follow a Markov chain. The equipment is accepted if k consecutive successful start-ups occur before d failures. We compute the probability of the demonstration test ending with acceptance of the unit and the expectation and variance of the number of start-ups until termination of the test. We also give recursive formulas for computing the probability distribution of the number of start-ups and the probability of acceptance or rejection of the equipment in a specified number of trials. The conditional distribution of the number of start-ups in the test given that the unit is accepted or rejected is obtained as a by-product. Examples are given to illustrate the usefulness of these results.