This paper concerns a dynamic scheduling problem for a queueing
system that has two streams of arrivals to infinite capacity buffers and two
(nonidentical) servers working in parallel. One server can only process jobs
from one buffer. The service time distribution may depend on the buffer being
served and the server providing the service. The system manager dynamically
schedules waiting jobs onto available servers. We consider a parameter regime
in which the system satisfies both a heavy traffic condition and a resource
pooling condition. Our cost function is a mean cumulative discounted cost of
holding jobs in the system, where the (undiscounted) cost per unit time is a
linear function of normalized (with heavy traffic scaling) queue length. We
first review the analytic solution of the Brownian control problem (formal
heavy traffic approximation) for this system. We "interpret" this solution by
proposing a threshold control policy for use in the original parallel server
system. We show that this policy is asymptotically optimal in the heavy traffic
limit and the limiting cost is the same as the optimal cost in the Brownian
control problem. The techniques developed here are expected to be useful for
analyzing the performance of threshold-type policies in more complex
multiserver systems.
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