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We consider a critically-loaded multiclass queueing control problem withmodel uncertainty. The model consists of $I$ types of customers and a singleserver. At any time instant, a decision-maker (DM) allocates the server'seffort to the customers. The DM's goal is to minimize a convex holding costthat accounts for the ambiguity with respect to the model, i.e., the arrivaland service rates. For this, we consider an adversary player whose role is tochoose the worst-case scenario. Specifically, we assume that the DM has areference probability model in mind and that the cost function is formulated bythe supremum over equivalent admissible probability measures to the referencemeasure with two components, the first is the expected holding cost, and thesecond one is a penalty for the adversary player for deviating from thereference model. The penalty term is formulated by a general divergencemeasure.  We show that although that under the equivalent admissible measures thecritically-load condition might be violated, the generalized $c\mu$ rule isasymptotically optimal for this problem.

Asymptotic optimality of the generalized $c\mu$ rule under model uncertainty