Stochastic processes involving random deletion (Q2731154)

Consider a sequence of i.i.d. vectors \((X_1,L_1)\), \((X_2,L_2),\dots\), where \(L_1\geq 0\) almost surely. This paper focusses on a system in which a unit of magnitude \(X_k\) enters at time \(S_k\) and stays active until time \(S_k+ L_k\), where \(k\geq 1\) and \(S_1< S_2<\cdots\), and proceeds to investigate the finite time properties and limit behaviour of the empirical distribution of the magnitudes of the active units (and of the departed units) at time \(t\), as \(t\to\infty\). Strong and weak results are presented and then extended to systems having stages through which the units must pass. Some illustrative examples are also presented.