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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 $T$. These conditions involve the study of the conditionalinterarrival times. As a main application, we prove the DFR property in acontext of maintenance models in reliability, by the consideration of Kijimatype I virtual age models under quite general assumptions.

On the Decreasing Failure Rate property for general counting process. Results based on conditional interarrival times