Learning traffic correlations in multi-class queueing systems by sampling queue lengths, with routing applications

We consider a system of parallel single-server queues. Work of different classes arrives as correlated Gaussian processes with known drifts but unknown covariance matrix Σ, and it is deterministically routed to the different queues according to some routing matrix. We first provide a procedure to estimate Σ based on the empirical large deviations behavior of the individual queues, for a finite set of routing matrices, and prove that the resulting estimate is asymptotically consistent under minimal technical conditions. We also introduce a more efficient procedure to estimate Σ based on the empirical large deviation behavior of linear combinations of queues, for a single routing matrix, and prove that the resulting estimate is asymptotically consistent under some technical conditions. We establish, however, that in specific cases the latter approach cannot be used due to an inherent loss of information produced by the dynamics of the queues. Finally, given a well-behaved cost function on the steady-state marginal queue lengths, we show how our procedures can be used to obtain an asymptotically consistent estimator for the cost under any routing matrix, and identify an optimal one.