  In the present paper we consider general counting processes stopped at a
random time $T$, independent of the process. Provided that $T$ has the
decreasing failure rate (DFR) property, we give sufficient conditions on the
arrival times so that the number of events occurring before $T$ preserves the
DFR property of $T$. These conditions involve the study of the conditional
interarrival times. As a main application, we prove the DFR property in a
context of maintenance models in reliability, by the consideration of Kijima
type I virtual age models under quite general assumptions.
