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We consider a multidimentional Brownian control problem (BCP) with modeluncertainty that formally emerges from a multiclass M/M/1 queueing controlproblem under heavy-traffic with model uncertainty. The BCP is formulated as amultidimensional stochastic differential game with two players: a minimizerthat has an equivalent role to the decision maker in the queueing controlproblem and a maximizer whose role is to set up the uncertainty of the model.The dynamics are driven by a Brownian motion. We show that a state-spacecollapse propery holds. That is, the multidimensional BCP can be reduced to aone-dimensional BCP with model uncertainty that also takes the form of atwo-player stochastic differential game. Then, the value function of both ofthe games is characterized as the unique solution to a free-boundary problemfrom which we extract equilibria for both games. Finally, we analyze thedependence of the value function and the equilibria on the ambiguityparameters.

Brownian control problems for a multiclass M/M/1 queueing problem with model uncertainty