Analysis of the Real-Time Monitoring Error for the Ornstein–Uhlenbeck Process in the D/GI/1 Queue

We analyze the mean squared error (MSE) in real-time monitoring of an Ornstein-Uhlenbeck (OU) process with observational noise, using multiple past samples to improve estimation. Assuming a D/GI/1 queueing model for system delay, we show that the time-averaged MSE decomposes into two components: (i) a term involving the Laplace-Stieltjes transform (LST) of the age of information (AoI), and (ii) a term determined by the number of observationsmand the parameters of the OU process and noise. We further derive a closed-form expression for this latter term and investigate its limiting behavior.