Abstract: We propose a novel, generic definition of probabilisticschedulers for population protocols. We then identify the consistent probabilisticschedulers, and prove that any consistent scheduler that assigns a non-zero probability to any transition i->j, where i and j are configurations satisfying that i is not equal to j, is fair with probability 1. This is a new theoretical framework that aims to simplify proving specific probabilisticschedulers 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., are all fair with probability 1. We also define and study equivalence between schedulers w.r.t. performance (time equivalent schedulers) and correctness (computationally equivalent schedulers). Surprisingly, we prove the following.
1. The protocol-oblivious (or agnostic) Random Scheduler is not time equivalent to the State and Transition Function Schedulers, although all three are fair probabilisticschedulers (with probability 1). To prove the statement we study the performance of the One-Way Epidemic Protocol (OR Protocol) under these schedulers. To illustrate the unexpected performance variations of protocols under different fair probabilisticschedulers, we additionally modify the State Scheduler to obtain a fair probabilisticscheduler, called the Modified Scheduler, that may be adjusted to lead the One-Way Epidemic Protocol to arbitrarily bad performance.
2. The Random Scheduler is not computationally equivalent to the Transition Function Scheduler. To prove the statement we study the Majority Protocol w.r.t. correctness under the Transition Function Scheduler. It turns out that the minority may win with constant probability under the same initial margin for which the majority w.h.p. wins under the Random Scheduler (as proven by Angluin et al.).

Abstract: In the near future, it is reasonable to expect that new types of systems will appear, of massive scale that will operating in a constantly changing networked environment. We expect that most such systems will have the form of a large society of tiny networked artefacts. Angluin et al. introduced the notion of "Probabilistic Population Protocols'' (PPP) in order to model the behavior of such systems where extremely limited agents are represented as finite state machines that interact in pairs under the control of an adversary scheduler. We propose to study the dynamics of Probabilistic Population Protocols, via the differential equations approach. We provide a very general model that allows to examine the continuous dynamics of population protocols and we show that it includes the model of Angluin et. al., under certain conditions, with respect to the continuous dynamics of the two models. Our main proposal here is to exploit the powerful tools of continuous nonlinear dynamics in order to examine the behavior of such systems. We also provide a sufficient condition for stability.

Abstract: Angluin et al. [1] introduced the notion of ``Probabilistic Population Protocols'' where extremely limited agents are represented as finite state machines that interact in pairs under the control of an adversary scheduler. We provide a very general model that allows to examine the continuous dynamics of population protocols and we show that it includes the model of [1], under certain conditions, with respect to the continuous dynamics of the two models.