DISTRIBUTIONS OF RUNS AND CONSECUTIVE SYSTEMS ON DIRECTED TREES

SIGEO AKI

Department of Mathematical Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

(Received March 27, 1997; revised November 10, 1997)

Abstract. In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutivek-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.

Key words and phrases: Probability generating function, discrete distributions, run, directed tree, graph, reliability, Markov tree, lifetime, consecutive system, order statistics.