Resource bounds for self stabilizing message driven protocols

A self-stabilizing system is a distributed system which can be started in any possible global state. Once started the system regains its consistency by itself, without any kind of an outside intervention. The self-stabilization property is very useful for systems in which processors may crash and then recover spontaneously in an arbitrary state. When the intermediate period in between one recovery and the next crash is long enough the system stabilizes. Most of the work in this field assumes the communication model of shared variables. The paradigm of self-stabilization is a very general one and does not depend on the communication media used by the system's processors. A very natural subject would be to look at self stabilizing message passing systems. Surprisingly, there are very few papers which addressed this subject. The size of a configuration of a message passing system is the number of bits required to encode the configuration entirely. We prove a lower bound on the configuration size for protocols for a large class of tasks called weak exclusion. The weak exclusion class contains all non-trivial tasks which require continuous changes in the system's configuration; in particular this class includes both mutual exclusion and token passing. We show that the configuration size of any self stabilizing protocol which realizes any weak exclusion task is at least logarithmic in the number of steps executed by the protocol. We then present three self stabilizing message driven protocols for token passing. The rate of growth of configuration size for all three protocols matches the aforementioned lower bound. Our results have interesting interpretation in terms of automata theory.