This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Fair Circulation of a Token
April 2002 (vol. 13 no. 4)
pp. 367-372

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.

[1] R. Aleliunas, R.M Karp, R.J. Lipton, L. Lovaász, and C. Rackoff, “Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems,” Proc. 20th Ann. Symp. Foundations of Computer Science, pp. 218-223, 1979.
[2] G. Brightwell and P. Winkler, “Maximum Hitting Time for Random Walks on Graphs,” J. Random Structures and Algorithms, vol. 3, pp. 263-276, 1990.
[3] D. Coppersmith, P. Tetali, and P. Winkler, "Collisions among Random Walks on a Graph," SIAM J. Discrete Mathematics, vol. 6, no. 3, pp. 363-374, Aug. 1992.
[4] W. Feller, An Introduction to Probability Theory and its Applications, vol. 1-2.New York: Wiley series in Probability and Mathematical Statistics, 1966.
[5] F. Harary, Graph Theory, Reading, Mass.: Addison-Wesley, 1969.
[6] A. Israeli and M. Jalfon,“Token management schemes and random walks yield self-stabilizing mutual exclusion,” Proc. Ninth Ann. ACM Symp. Principles of Distributed Computation,Quebec City, Aug. 1990, pp. 119-132.
[7] L. Isaacson and W. Madsen, Markov Chains: Theory and Application, New York: Wiley series in probability and mathematical statistics, 1976.
[8] R. Motwani and P. Raghavan, Randomized Algorithms. Cambridge Univ. Press, 1995.
[9] J.R. Norris, Markov Chains, New York: Cambridge Univ. Press, 1997.
[10] M. Schneider, “Self-Stabilization,” ACM Computing Surveys, vol. 25, no. 1, pp. 45-67, Mar. 1993.

Index Terms:
distributed systems, self-stabilizing systems, random walk, token circulation, cover time
Citation:
Satoshi Ikeda, Izumi Kubo, Norihiro Okumoto, Masafumi Yamashita, "Fair Circulation of a Token," IEEE Transactions on Parallel and Distributed Systems, vol. 13, no. 4, pp. 367-372, April 2002, doi:10.1109/71.995817
Usage of this product signifies your acceptance of the Terms of Use.