  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 $T$. In particular, when the interarrival times are
independent, we consider applications concerning the DFR property of the
stationary number of customers waiting in queue for specific queuing models.
