Computation of waiting time to successful experiment using age-dependent branching model

This paper considers the properties of the waiting time to survive forever of the supercritical age-dependent branching process modified with an immigration component. Conditioning on ultimate extinction of the process we analyze the conditional distribution of the cycle length and its expectation. We then derive and give estimates of the conditional expected total progeny of a cycle and its higher moments. In general, the model originates from the problem of estimating the waiting time to a successful experiment in industrial wastewater treatment by bacterial culture systems. Computer runs were made for two reproduction laws and different reproduction means.