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Many distributed scheduling algorithms have been developed and reported in the current literature. However, very few of them explicitly treat stability issues. This paper first discusses stability issues for distributed scheduling algorithms in general terms. Two very different distributed scheduling algorithms which contain explicit mechanisms for stability are then presented and evaluated with respect to individual specific stability issues. One of the algorithms is based on stochastic learning automata and the other on bidding. The results indicate how very specific the treatment of stability is to the algorithm and environment under consideration. [ABSTRACT FROM AUTHOR]