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 (left) are permitted. If a Markov chain is skip-free both to the right and 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 the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

Consider a file with n records denoted by R 1, R 2 , …, Rn . At each access of a record, the file has to be searched sequentially and it is assumed that the search cost is proportional to the number of probes needed to retrieve the records. The access probabilities p 1, p 2, …, pn are assumed to be unknown constants and accesses are assumed to be independent. A move-forward self-organizing rule moves a record accessed in the ith position to the li th position without changing the relative ordering of other records where li , = 1, li , < i for i = 2, …, n and l i+1≧li . A move-forward rule R is said to be ≦ another move-forward rule R′ if l′i ≦li for all i. It is shown that when p 2 = p 3 = ··· = p n , R ≦ R ′, implies cost R ≦ cost R′ . This is a generalization of some known results. A new consequence is that the move-up-k + 1 rule is more costly than the move-up-k rule and the move-to-front rule is the most costly of all move-forward rules.