Optimal threshold strategies for remote estimation over the collision channel with communication costs

Consider a remote estimation system consisting of a sensor observing a Gaussian random variable and a remote estimator connected over a wireless network modeled by a collision channel. At every time step, the sensor makes a measurement and decides whether to communicate it or not over the network. The goal is to minimize a performance index consisting of an expected squared estimation error plus a communication cost. Existing results show that the optimal communication policies for this problem are deterministic threshold strategies. We prove the existence and characterize the optimal thresholds as symmetric or asymmetric, depending on the parameters that define the problem. We show that locally optimal solutions can be computed using a Lloyd-Max type algorithm and we establish its global convergence.