For a multiclass G/G/1 queue with finite buffers, admission and scheduling control, and holding and rejection costs, we construct a policy that is asymptotically optimal in the heavy traffic limit. The policy is specified in terms of a single parameter which constitutes the free boundary point from the Harrison-Taksar free boundary problem, but otherwise depends explicitly on the problem data. The $c\mu$ priority rule is also used by the policy, but in a way that is novel, and, in particular, different than that used in problems with infinite buffers. We also address an analogous problem where buffer constraints are replaced by throughput time constraints.
"An asymptotic optimality result for the multiclass queue with finite buffers in heavy traffic." Stoch. Syst. 4 (2) 556 - 603, 2014. https://doi.org/10.1214/13-SSY124