DFR Property of First-Passage Times and its Preservation Under Geometric Compounding

Abstract

It is shown that if a discrete-time Markov chain on the state space $\{0, 1,\ldots\}$ has a transition probability matrix $\mathbf{P}$ and a transition survival probability matrix $\mathbf{Q}$ which is totally positive of order two $(TP_2)$, where $Q(i, j) = \sum_{k\geq j}P(i, k)$, then the first-passage time from state 1 to state 0 has decreasing failure rate (DFR). This result is used to show that (i) the sum of a geometrically distributed 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 life) 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).