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

TL;DR

This paper investigates how the decreasing failure rate (DFR) property of a random stopping time T influences the DFR property of the count of events in a process stopped at T, with applications to queueing models.

Contribution

It provides sufficient conditions under which the DFR property of T is preserved in the counting process stopped at T, especially with independent interarrival times.

Findings

DFR property of T can be preserved in the stopped counting process under certain conditions.

Applications to queueing models demonstrate the practical relevance of the theoretical results.

The paper extends understanding of the DFR property in stochastic processes with random stopping times.

Abstract

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.