Not all fair probabilistic schedulers are equivalent

We propose a novel, generic definition of probabilistic schedulers for population protocols. We then identify the consistent probabilistic schedulers, and prove that any consistent scheduler that assigns a non-zero probability to any transition i → j, where i and j are configurations satisfying i ≠ j, is fair with probability 1. This is a new theoretical framework that aims to simplify proving specific probabilistic schedulers fair. In this paper we propose two new schedulers, the State Scheduler and the Transition Function Scheduler. Both possess the significant capability of being protocol-aware, i.e. they can assign transition probabilities based on information concerning the underlying protocol. By using our framework we prove that the proposed schedulers, and also the Random Scheduler that was defined by Angluin et al. [2], are all fair with probability 1. Finally, we define and study equivalence between schedulers w.r.t. performance and correctness and prove that there exist fair probabilistic schedulers that are not equivalent w.r.t. to performance and others that are not equivalent w.r.t. correctness. © 2009 Springer-Verlag.