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<p><it>Abstract</it>—The tree quorum algorithm (TQA) uses a tree structure to generate intersecting (tree) quorums for distributed mutual exclusion. This paper analyzes the number of messages required to acquire a quorum in TQA. Let <it>i</it> be the depth of the complete binary tree used in TQA, and let <it>M</it><sub><it>i</it></sub> be the number of messages required to acquire a quorum or to determine that no quorum is accessible. We discuss <it>M</it><sub><it>i</it></sub> as a function of <it>i</it> and <it>p</it>, where <math><tmath>$p\left({1\over 2} < p < 1\right)$</tmath></math> is the probability that each site is operational. Let <it>C</it><sub><it>i</it></sub> denote the average number of sites in the quorum that TQA finds. The analysis shows that, although both <it>M</it><sub><it>i</it></sub> and <it>C</it><sub><it>i</it></sub> increase without bound as <it>i</it> increases, <it>M</it><sub><it>i</it></sub>/<it>C</it><sub><it>i</it></sub> approaches to <math><tmath>${1 + p \over p}$</tmath></math> as <it>i</it> increases. According to the result, an approximate close form for <it>M</it><sub><it>i</it></sub> is derived.</p>
Distributed mutual exclusion, tree quorum algorithm, quorum size, message complexity.

H. Chang and S. Yuan, "Message Complexity of the Tree Quorum Algorithm," in IEEE Transactions on Parallel & Distributed Systems, vol. 6, no. , pp. 887-890, 1995.
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