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We propose a model uncertainty approach to heavy traffic asymptotics thatallows for a high level of uncertainty. That is, the uncertainty classes ofunderlying distributions accommodate disturbances that are of order 1 at theusual diffusion scale, as opposed to asymptotically vanishing disturbancesstudied previously in relation to heavy traffic. A main advantage of theapproach is that the invariance principle underlying diffusion limits makes itpossible to define uncertainty classes in terms of the first two moments only.The model we consider is a single server queue with multiple job types. Theproblem is formulated as a zero sum stochastic game played between the systemcontroller, who determines scheduling and attempts to minimize an expectedlinear holding cost, and an adversary, who dynamically controls the servicetime distributions of arriving jobs, and attempts to maximize the cost. Theheavy traffic asymptotics of the game are fully solved. It is shown that anasymptotically optimal policy for the system controller is to prioritizeaccording to an index rule and for the adversary it is to select distributionsbased on the system's current workload. The workload-to-distribution feedbackmapping is determined by an HJB equation, which also characterizes the game'slimit value. Unlike in the vast majority of results in the heavy traffictheory, and as a direct consequence of the diffusive size disturbances, thelimiting dynamics under asymptotically optimal play are captured by astochastic differential equation where both the drift and the diffusioncoefficients may be discontinuous.

Scheduling in the high uncertainty heavy traffic regime