Abstract
We analyze the PMC [10] model for fault tolerant systems by means of random directed graphs. Previous work has shown that the minimum in-degree of the testing digraph had to exceed the expected number of faulty units (vertices). We show, for much sparser digraphs than those described above, that the asymptotic probability of correct diagnosis of a faulty system tends to 1. Specifically we show that if our directed graph has n vertices, arc probability $p = {{(c\log n)} / n}$ and vertex failure probability $q < \frac{1}{2}$ then a simple algorithm will correctly diagnose the system “almost always” when ${{c > 1} / {(1 - q)}}$. However, when ${{c < 1} / {(1 - q)}}$ no algorithm can correctly diagnose the system. We present a similar analysis for a family of directed graphs which are not random. We conclude with analogous results for undirected graphs.
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