A Bayesian framework for distributed estimation of arrival rates in asynchronous networks

Abstract: In this paper we consider a network of agents monitoring a spatially distributed arrival process. Each node measures the number of arrivals seen at its monitoring point in a given time-interval with the objective of estimating the unknown local arrival rate. We propose an asynchronous distributed approach based on a Bayesian model with unknown hyperparameter, where each node computes the minimum mean square error (MMSE) estimator of its local arrival rate in a distributed way. As a result, the estimation at each node "optimally" fuses the information from the whole network through a distributed optimization algorithm. Moreover, we propose an ad-hoc distributed estimator, based on a consensus algorithm for time-varying and directed graphs, which exhibits reduced complexity and exponential convergence. We analyze the performance of the proposed distributed estimators, showing that they: (i) are reliable even in presence of limited local data, and (ii) improve the estimation accuracy compared to the purely decentralized setup. Finally, we provide a statistical characterization of the proposed estimators. In particular, for the ad-hoc estimator, we show that as the number of nodes goes to infinity its mean square error converges to the optimal one. Numerical Monte Carlo simulations confirm the theoretical characterization and highlight the appealing performances of the estimators.