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<p>Suppose that a distributed system is modeled by an undirected graph G = (V,E), where V and E, respectively, are the sets of processes and communication links. Israeli and Jalfon proposed a simple self-stabilizing mutual exclusion algorithm: A token is circulated among the processes (i.e., vertices) and a process can access the critical section only when it holds the token. In order to guarantee equal access chance to all processes, the token circulation is desired to be fair in the sense that all processes have the same probability of holding the token. However, the Israeli-Jalfon token circulation scheme does not meet the requirement. This paper proposes a new scheme for making it fair. We evaluate the average of the longest waiting times in terms of the cover time and show an O({\rm deg}(G)n^2) upper bound on the cover time for our scheme, where n and {\rm deg}(G) are the number of processes and the maximum degree of G, respectively. The same (tight) upper bound is known for the Israeli-Jalfon scheme.</p>
distributed systems, self-stabilizing systems, random walk, token circulation, cover time

M. Yamashita, N. Okumoto, S. Ikeda and I. Kubo, "Fair Circulation of a Token," in IEEE Transactions on Parallel & Distributed Systems, vol. 13, no. , pp. 367-372, 2002.
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