On the Fairness of Probabilistic Schedulers for Population Protocols

We propose a novel, generic definition of probabilistic schedulers for population protocols. We design 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. We prove that the proposed schedulers, and also the Random Scheduler that was defined by Angluin et al. [1], are all fair with probability 1. We also 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. We implement our schedulers using a new tool for simulating population protocols and evaluate their performance from the viewpoint of experimental analysis and verification. We study three representative protocols to verify stability, and compare the experimental time to convergence with the known complexity bounds. We run our experiments from very small to extremely large populations (of up to 10⁸ agents). We get very promising results both of theoretical and practical interest.