Markov Chain Markov Field dynamics: Models and statistics

Abstract

This study deals with time dynamics of Markov fields defined on a finite set of sites with state space E, focussing on Markov Chain Markov Field (MCMF) evolution. Such a model is characterized by two families of potentials: the instantaneous interaction potentials, and the time delay potentials. Four models are specified: auto-exponential dynamics (E = R + ), auto-normal dynamics (E = R), auto-Poissonian dynamics (E = N) and auto-logistic dynamics (E qualitative and finite). Sufficient conditions ensuring ergodicity and strong law of large numbers are given by using a Lyapunov criterion of stability, and the conditional pseudo-likelihood statistics are summarized. We discuss the identification procedure of the two Markovian graphs and look for validation tests using martingale central limit theorems. An application to meteorological data illustrates such a modelling.

Performance analysis of MAC protocols in wireless line networks using statistical mechanics

“…Let Ω = E S denote the configuration space, i.e., the state space of the entire network. Let X i (t) ∈ E be the state of station i at time t, and X(t) = {X i (t)} i∈S ∈ Ω be the configuration, or the state of the network, at time t. We are interested in MAC protocols for which X = {X(t)} t∈N is a Markov Chain of Markov Field (MCMF) [6], i.e., a process for which…”

“…In [6] it is shown that a sufficient condition for this to occur is that the Markov chain is time-reversible and the transition probability is synchronous. We consider these two properties in the following.…”

“…Note that (4) is the probability that station i chooses state y i in the protocol described in Section III. From [5], [6], the Markov chain is time-reversible if…”

“…Therefore, by [6] the MAC protocol described in Section III admits a stationary distribution with Markov property if…”

“…The channel can be either a collision channel or a multipacket reception channel [4]. Our contributions include the following: 1) We use Markov Chain of Markov Fields (MCMF) [5], [6] to show that under mild conditions (reversible and synchronous, defined in Section IV) this type of MAC protocol leads to a stationary state such that only short-range interactions exist in the stationary state, which implies the validity of using the Ising model to represent a wireless network. 2) We analytically derive the throughput and transmission probability of a wireless network using techniques in statistical mechanics, explicitly demonstrating the effect of interactions between neighboring stations on these performance measures.…”