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In the present paper we consider general counting processes stopped at arandom time $T$, independent of the process. Provided that $T$ has thedecreasing failure rate (DFR) property, we give sufficient conditions on thearrival times so that the number of events occurring before $T$ preserves theDFR property of $T$. In particular, when the interarrival times areindependent, we consider applications concerning the DFR property of thestationary number of customers waiting in queue for specific queuing models.

The DFR property for counting processes stopped at an independent random time