Game Theoretic Analysis of Self-Stabilizing Systems on Arrays

• Published in: Journal of computer & systems sciences international Vol. 60; no. 2; pp. 227 - 238
• Main Authors: Shoja, E., Izadi, M.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-03-2021; Springer Nature B.V
• Subjects: Algorithms; Arrays; Control; Engineering; Game theory; Mechatronics; Robotics; Stabilization; State machines; Systems analysis; Systems Analysis and Operations Research; Upper bounds
• ISSN: 1064-2307; 1555-6530
• DOI: 10.1134/S1064230721020131

In 1973 E.W. Dijkstra introduced the notion of self-stabilization in the context of mutual exclusion. Considering the same problem on an array, we present a game theoretic analysis of self-stabilizing systems with three- ‎or four-state machines. We give a formalized definition of the problem as a game where each player’s strategy represents the state of its corresponding machine. For the three-state case, we prove the impossibility of any infinite self-stabilizing systems on an array. For the four-state case we consider two algorithms. For Ghosh’s solution [1] we prove the upper bound of (
n
– 1)(
n
– 3) steps and that this bound is tight. Also we present another four-state self-stabilizing system, ‎and prove that at most
n
2
–
5
n +
7 steps are required for the system to reach self-stabilization.