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DISTRIBUTIONS OF RUNS AND CONSECUTIVE SYSTEMS ON

DIRECTED TREES

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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 consecutive-*k*-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.

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