Suppose t 1, t 2,… are the arrival times of units into a system. The kth entering unit, whose magnitude is X k and lifetime L k , is said to be ‘active’ at time t if I(t k < t k + L k ) = I k,t = 1. The size of the active population at time t is thus given by A t = ∑k≥1 I k,t . Let V t denote the vector whose coordinates are the magnitudes of the active units at time t, in their order of appearance in the system. For n ≥ 1, suppose λn is a measurable function on n-dimensional Euclidean space. Of interest is the weak limiting behaviour of the process λ*(t) whose value is λm (V t ) or 0, according to whether A t = m > 0 or A t = 0.