Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.
