Suppose we have some data where we see the age of each sample and whether each sample received treatment. Further, for those that received treatment some (but not all) of them have a time of treatment attached. We can posthoc compute the interval of time in which each sample could have received treatment.
We are interested in two population parameters: $\rho$, which is a "eligibility" proportion for receiving treatment and $\lambda$, which is the (exponential) rate that eligible people in the population may receive treatment. The reason to try to use a model is that for those who didn't receive the treatment, we can't tell if they were not eligible or simply unlucky to not receive treatment in the time they had. On the other hand, there are those that received treatment at a specific time, and those that simply have had the treatment sometime in the interval where they could have.
In some ways, this reminds me of a zero-inflated model, except instead of having some proportion of zeroes in the data, we now have some proportion of events that'll never fire. Is there a name or general strategy for this type of problem?