Start-up demonstration tests based on consecutive successes and total failures

The start-up demonstration test that requires k consecutive successes for acceptance prior to f total failures and rejection otherwise is examined. A general Markov Chain approach is given that allows the user to easily calculate quantities such as expectation, variance, probability mass function, and the distribution of the test length, as well as the probability the equipment will be accepted. This methodology produces compact formulas that are readily implemented and understood by the practitioner and naturally extend to the non-i.i.d. case. Procedures are derived to give guidance for determining the appropriate startup demonstration test parameters k and f. Confidence statements for p, the reliability for the start-up demonstration test, based on maximum likelihood estimators, are calculated and investigated.