First hitting time for renewal process with uncertain interarrival times and random rewards

Renewal reward processes measure the cumulative rewards up to a given time, and sometimes uncertainty and randomness exist at the same time due to the complexity of real world. This article considers the first hitting time for the renewal process with uncertain interarrival times and random rewards. Analytic representations and numerical algorithms for the chance distribution and expected value of first hitting time are derived. Numerical examples illustrate our methods in detail. Finally, the first hitting time is applied to modeling the shortage index for an uncertain random production risk process.