Linear Estimators for a Class of Stationary Queueing Processes

We find the minimum-variance linear estimator for the expected value of a stationary stochastic process, observed over a finite time interval, whose covariance function is a sum of decaying exponentials. Such processes and estimators include many of the common measurements on Markovian queueing systems, such as the number of busy servers, all servers busy, number in queue, etc. We derive an explicit form for the best linear estimator, and investigate its properties. Next, we compare it to the standard estimator which assigns equal weights to all the observations, and show that asymptotically both behave in the same way. We conclude that in many practical cases, using the optimal weights will reduce slightly the estimator's variance.