Separation of Circulating Tokens

Self-stabilizing distributed control is often modeled by token abstractions. A system with a single token may implement mutual exclusion; a system with multiple tokens may ensure that immediate neighbors do not simultaneously enjoy a privilege. For a cyber-physical system, tokens may represent physical objects whose movement is controlled. The problem studied in this paper is to ensure that a synchronous system with m circulating tokens has at least d distance between tokens. This problem is first considered in a ring where d is given whilst m and the ring size n are unknown. The protocol solving this problem can be uniform, with all processes running the same program, or it can be non-uniform, with some processes acting only as token relays. The protocol for this first problem is simple, and can be expressed with Petri net formalism. A second problem is to maximize d when m is given, and n is unknown. For the second problem, the paper presents a non-uniform protocol with a single corrective process.Comment: 22 pages, 7 figures, epsf and pstricks in LaTe

Randomized finite-state distributed algorithms as Markov chains

Distributed randomized algorithms, when they operate under a memoryless scheduler, behave as finite Markov chains: the probability at n-th step to go from a configuration x to another one y is a constant p that depends on x and y only. By Markov theory, we thus know that, no matter where the algorithm starts, the probability for the algorithm to be after n steps in a \recurrent " configuration tends to 1 as n tends to infinity. In terms of self-stabilization theory, this means that the set Rec of recurrent configurations is included into the set L of \legitimate " configurations. However in the literature, the convergence of self-stabilizing randomized algorithms is always proved in an elementary way, without explicitly resorting to results of Markov theory. This yields proofs longer and sometimes less formal than they could be. One of our goals in this paper is to explain convergence results of randomized distributed algorithms in terms of Markov chains theory. Our method relies on the existence of a non-increasing measure ' over the configurations of the distributed system. Classically, this measure counts the number of tokens of congurations. It also exploits a function D that expresses some distance between tokens, for a fixed number k of tokens. Our first result is to exhibit a sufficient condition Prop on ' and D which guarantees that, for memoryless schedulers, every recurrent con guration is legitimate. We extend this property Prop in order to handle arbitrary schedulers although they may induce non Markov chain behaviours. We then explain how Markov's notion of \lumping " naturally applies to measure D, and allows us to analyze the expected time of convergence of self-stabilizing algorithms. The method is illustrated on several examples of mutual exclusion algorithms (Herman, Israeli-Jalfon, Kakugawa-Yamashita)

Stargazin Modulation of AMPA Receptors

Fast excitatory synaptic signaling in the mammalian brain is mediated by AMPA-type ionotropic glutamate receptors. In neurons, AMPA receptors co-assemble with auxiliary proteins, such as stargazin, which can markedly alter receptor trafficking and gating. Here, we used luminescence resonance energy transfer measurements to map distances between the full-length, functional AMPA receptor and stargazin expressed in HEK293 cells and to determine the ensemble structural changes in the receptor due to stargazin. In addition, we used single-molecule fluorescence resonance energy transfer to study the structural and conformational distribution of the receptor and how this distribution is affected by stargazin. Our nanopositioning data place stargazin below the AMPA receptor ligand-binding domain, where it is well poised to act as a scaffold to facilitate the long-range conformational selection observations seen in single-molecule experiments. These data support a model of stargazin acting to stabilize or select conformational states that favor activation

Coupling and Self-stabilization

Abstract. A randomized self-stabilizing algorithm A is an algorithm that, whatever the initial configuration is, reaches a set L of legal configurations in finite time with probability 1. The proof of convergence towards L is generally done by exhibiting a potential function ϕ, which measures the “vertical ” distance of any configuration to L, such that ϕ decreases with non-null probability at each step of A. We propose here a method, based on the notion of coupling, which makes use of a “horizontal” distance δ between any pair of configurations, such that δ decreases in expectation at each step of A. In contrast with classical methods, our coupling method does not require the knowledge of L. In addition to the proof of convergence, the method allows us to assess the convergence rate according to two different measures. Proofs produced by the method are often simpler or give better upper bounds than their classical counterparts, as examplified here on Herman’s mutual exclusion and Iterated Prisoner’s Dilemma algorithms in the case of cyclic graphs.