Random permutations generated by delay models and estimation of delay distributions

Abstract

Objects arrive at a system at times $U_1,U_2,\dots$ and leave at times $U_1+X_1,U_2+X_2,\dots$, where we assume that the arrivals are independent and uniformly distributed on the unit interval, that the delay times are independent with distribution function G, and that arrival and delay times are independent. Let ${\Pi }_n$ be the random permutation that connects the ranks of the first n arrivals and departures. We investigate the use of ${\Pi }_n$ for estimating G. We consider empirical copulas in the nonparametric and pattern frequencies in the parametric situation.