I am attempting to model the number of attempts required in order to make event $Z$ happen. Each day we make $k_i$ attempts with setting $X_i$ (a vector) in an effort to make $Z$ happen for object $i$, for $i$ from 1 to $m$. In other words, we are making multiple attempts with one setting (these settings affect the probability each attempt causes $Z$ to happen), then multiple attempts with another setting, et cetera. However, once we start making attempts with a new setting the previous set of attempts have no effect, because now we've given up on making $Z$ happen (or we've made it happen already) for object $i$ and are attempting to make $Z$ happen for object $j$.
The attempts have a cost and the events have a payoff. We are wanting to predict the number of attempts needed so that if a large number of attempts is predicted to be needed we can know not to make any attempts, because it won't be worth it.
My first thought was to just model this with Negative Binomial regression, and the target "count" would be number of attempts required. The problem is that most of the time, event $Z$ doesn't actually happen, so the count for that observation is missing (but known to be greater than $k_i$).