Some further results on the limiting proportion of types

For 1 less than or equal to k < infinity let t(k) denote the arrival time of some k-th unit into a system, and p(j) the probability that this unit is of the j-th "type", where Sigma(j greater than or equal to 1) p(j) = 1 Suppose each unit is endowed with a lifetime whose distribution is specific to its type, during which the unit is "active" and after which the unit is inactive. Under such conditions, the population of active units comprises eventually only a portion of those that have entered the system. Moreover, if some unit types have stochastically longer lifetimes than others, the proportion of the active population comprised of type j units may differ from p(j) in the limit. The behavior of this proportion as t --> infinity is considered in the case of fixed arrival times, arrival times that are partial sums of i.i.d. random variables, and arrival times that occur in accordance with a nonhomogeneous Poisson process.