DFR property of first-passage times and its preservation under geometric compounding
If the transition survival probability matrix of a discrete-time Markov chain on the non-negative integers is \(TP_ 2\), then the author proves that the first passage time from state 1 to state 0 is DFR. The transition survival probability matrix \(Q:=\| Q(i,j)\|\) corresponding to a Markov matrix \(P:=\| P(i,j)\|\) is defined by \(Q(i,j):=\sum_{k\geq j}P(i,k)\). Related consequences/results derived are: (i) the DFR property is closed under geometric compounding, (ii) in an M/G/1 queue with IFR service times, the number of customers served during a busy period is DFR, (iii) the stationary waiting time in a GI/G/1 (M/G/1) queue with DFR (IMRL respectively) service times is DFR, (iv) sufficient conditions on inter-renewal times for a concave renewal function.
scientific article; zbMATH DE number 4036857