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This paper studies a 2-class, 2-server parallel server system under therecently introduced extended heavy traffic condition, which states that theunderlying 'static allocation' linear program (LP) is critical, but does notrequire that it has a unique solution. The main result is the construction ofpolicies that asymptotically achieve a lower bound, proved in [1], on anexpected discounted linear combination of diffusion-scaled queue lengths, andare therefore asymptotically optimal (AO).  Each extreme point solution to the LP determines a control mode, i.e., a setof activities (class--server pairs) that are operational.  When there are multiple solutions, these modes can be selected dynamically.It is shown that the number of modes required for AO is either one or two. Inthe latter case there is a switching point in the (normalized) workload domain,characterized in terms of a free boundary problem.  Our policies are defined by identifying pairs of elementary policies andswitching between them at this switching point.  They provide the first example in the heavy traffic literature where weaklimits under an AO policy are given by a diffusion process where both the driftand diffusion coefficients are discontinuous.

Asymptotic optimality of switched control policies in a simple parallel server system under an extended heavy traffic condition