Minimization of Age of Incorrect Estimates of Autoregressive Markov Processes

We consider a source that sends freshness-sensitive status updates about an auto-regressive Markov process to a monitor, in a time-slotted system. The source samples the random process at the start of every slot and decides whether to transmit the sample or not. The transmission of a sample incurs a fixed cost. When a monitor receives a sample, its information about the source is perfect. However, when no samples are received, it estimates the realization of the process based on previously received samples. We adopt a metric referred to as the age of incorrect estimates (AoIE), defined as the product of an estimation error, E, and, v, the time elapsed since the latest time at which the monitor had a sufficiently correct estimate. We formulate an optimization problem to decide when a source must transmit a packet for minimizing the long-term average expected weighted sum of the AoIE and the transmission cost. We cast this problem as a Markov decision process and prove that the optimal policy is a threshold-type policy, in which, for a fixed v, there exists a threshold on E beyond which it is optimal to transmit, and vice versa. Using numerical simulations, we illustrate this threshold structure of the optimal policy. We also consider a simple periodic policy in which the information packets are transmitted periodically, after every fixed number of slots, irrespective of the realizations of E and v, and numerically show that its performance is significantly worse than that of the optimal threshold-type policy.