On first-passage times in increasing Markov processes (Q1914303)

In the reliability theory, there exists in literature studies on increasing Markov processes. So, \textit{M. Brown} and \textit{N. R. Chaganty} [Ann. Probab. 11, 1000-1008 (1983; Zbl 0529.60069)] explained that if the transition is \(\text{TP}_2\), then the process is IFR, and if the transition matrix is stochastically monotone, then the process is IFRA. \textit{Shaked} and \textit{Shanthikumar} (1987) extended the same model for the IFRA case. \textit{I. Karasu} and \textit{S. Özekici} [J. Appl. Probab. 26, No. 4, 827-834 (1989; Zbl 0758.60095)] showed that if the potential matrix is stochastically monotone, then the process has new better than used in expectation (NBUE) first passage times. In this note, the authors use the same model as that of Karasu and Özekici. Also, the authors extend the existing literature to other reliability classes.