HAZARD RATE AND REVERSED HAZARD RATE MONOTONICITIES IN CONTINUOUS-TIME MARKOV-CHAINS

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (le ft) are permitted. If a Markov chain is skip-free both to the right an d to the left, it is called a birth-death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in t he sense of likelihood ratio ordering then it must be an (extended) bi rth-death process. This paper proves that if an irreducible Markov cha in in continuous time is monotone in the sense of hazard rate (reverse d hazard rate) ordering then it must be skip-free to the right (left). A birth-death process is then characterized as a continuous-time Mark ov chain that is monotone in the sense of both hazard rate and reverse d hazard rate orderings. As an application, the first-passage-time dis tributions of such Markov chains are also studied.