We consider the problem of number and weight distributions for breakage-mechanism branching processes whose break distributions are general. We derive a recursive relation between the expected empirical distribution after (n + 1) breaks and after n breaks, making use of length-biased sampling. Using this relation and the strong law of large numbers, we derive integro-differential equations for the asymptotic expected empirical distribution and its associated weight distribution. The mean of the asymptotic number distribution is derived using the integro-differential equation. We then provide approximate solutions to these equations and the moments of these approximations. Finally we apply these results to the case where the break distribution is a symmetric beta distribution.

A stochastic model for a periodic screening program is presented in which the natural history of a chronic disease is assumed to follow a progressive path from a preclinical state to a clinical state. The sampling of preclinical state sojourn times by screening examinations generates bounds on the preclinical state recurrence times. The distribution of the bounded forward recurrence time is derived and used to obtain the distribution and mean of the lead time, and relationships for calculating the proportion of preclinical cases detected. These expressions are derived in terms of the preclinical state sojourn-time distribution and adjustible parameters important in the design of a periodic screening program.