Bayesian reliability demonstration for failure-free periods

We study sample sizes for testing as required for Bayesian reliability demonstration in terms of failure-free periods after testing, under the assumption that tests lead to zero failures. For the process after testing, we consider both deterministic and random numbers of tasks, including tasks arriving as Poisson processes. It turns out that the deterministic case is worst in the sense that it requires most tasks to be tested. We consider such reliability demonstration for a single type of task, as well as for multiple types of tasks to be performed by one system. We also consider the situation, where tests of different types of tasks may have different costs, aiming at minimal expected total costs, assuming that failure in the process would be catastrophic, in the sense that the process would be discontinued. Generally, these inferences are very sensitive to the choice of prior distribution, so one must be very careful with interpretation of non-informativeness of priors.