  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.
