DFR (Decreasing Failure Rate) Property of First Passage Times and Its Preservation under Geometric Compounding.

Abstract : It is shown that if the transition kernel of a discrete time Markov chain with state space (0,1,...) is totally positive of order two (TP2), the first passage time from state 1 to state 0 has decreasing failure rate (DFR). This result is usd to show that (i) the sum of a geometric number (i.e., geometric compound) of i.i.d. DFR random variables is DFR and (ii) the number of customers served during a busy period in an M/G/1 queue with increasing failure rate service times is DFR. Recent results of Szekli (1986) and the closure property of i.i.d. DFR random variables under geometric compounding are combined to show that the stationary waiting time in a GI/G/1 (M/G/1) queue with DFR (increasing mean residual) service times is DFR. We also provide sufficient conditions on the inter-renewal times under which the renewal function is concave. These results shed some light on a conjecture of Brown (1981). (Author)