Suppose t1, t2,… are the arrival times of units into a system. The kth entering unit, whose magnitude is Xk and lifetime Lk, is said to be ‘active’ at time t if I(tk < tk + Lk) = Ik,t = 1. The size of the active population at time t is thus given by At = ∑k≥1Ik,t. Let Vt 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(Vt) or 0, according to whether At = m > 0 or At = 0.