We consider a critically-loaded multiclass queueing control problem with model uncertainty. The model consists of $I$ types of customers and a single server. At any time instant, a decision-maker (DM) allocates the server’s effort to the customers. The DM’s goal is to minimize a convex holding cost that accounts for the ambiguity with respect to the model, i.e., the arrival and service rates. For this, we consider an adversary player whose role is to choose the worst-case scenario. Specifically, we assume that the DM has a reference probability model in mind and that the cost function is formulated by the supremum over equivalent admissible probability measures to the reference measure with two components, the first is the expected holding cost, and the second one is a penalty for the adversary player for deviating from the reference model. The penalty term is formulated by a general divergence measure.

We show that although that under the equivalent admissible measures the critically-load condition might be violated, the generalized $c\mu$ rule is asymptotically optimal for this problem.

我们考虑具有模型不确定性的临界负荷多类排队控制问题。该模型包括$\mathrm{一世}$客户类型和单个服务器。在任何时候，决策者（DM）都会将服务器的工作分配给客户。DM的目标是最大程度地减少因模型不明确而导致的凸形持有成本，即到达率和服务费率。为此，我们考虑一个对手角色，其角色是选择最坏的情况。具体而言，我们假设决策模型考虑了参考概率模型，并且成本函数由参考度量的等价容许概率度量的上限值组成，其中包括两个组成部分，第一个是预期持有成本，第二个是对对手玩家偏离参考模型的处罚。惩罚期限由一般分歧度量表述。

我们表明，尽管在等效的允许措施下，可能会违反临界载荷条件， $C\mu$ 规则对于这个问题是渐近最优的。