We consider an r component system having an arbitrary binary monotone structure function. We suppose that shocks occur according to a point process and that, independent of what has already occurred, each new shock is one of r different types, with respective probabilities p1, …, pr. We further suppose that there are given integers n1, …, nr such that component i fails (and remains failed) when there have been a total of ni type-i shocks. Letting L be the time at which the system fails, we are interested in using simulation to estimate E[L], E[L2], and P(L > t). We show how to efficiently accomplish this when the point process is (i) a Poisson, (ii) a renewal, and (iii) a Hawkes process.