Optimal Risk-Sensitive Scheduling Policies for Remote Estimation of Autoregressive Markov Processes

We consider a remote estimation setup, where data packets containing sensor observations are transmitted over a Gilbert-Elliot channel to a remote estimator, and design scheduling policies that minimize a risk-sensitive cost, which is equal to the expected value of the exponential of the cumulative cost incurred during a finite horizon, that is the sum of the cumulative transmission power consumed, and the cumulative squared estimation error. More specifically, consider a sensor that observes a discrete-time autoregressive Markov process, and at each time decides whether or not to transmit its observations to a remote estimator using an unreliable wireless communication channel after encoding these observations into data packets. Modeling the communication channel as a Gilbert-Elliot channel allows us to take into account the temporal correlations in its fading. We pose this dynamic optimization problem as a Markov decision process (MDP), and show that there exists an optimal policy that has a threshold structure, i.e., at each time t it transmits only when the current channel state is good, and the magnitude of the current “error” exceeds a certain threshold.