Optimal use of excess capacity in two interconnected queues

We consider a two-stage tandem queueing network where jobs from station 1 join station 2 with a certain probability. Each job incurs a linear holding cost, different for each station. Each station is attended by a dedicated server, and there is an additional server that is either constrained to serve in station 1 or can serve in both stations. Assuming no switching or other operating costs for the additional server, we seek an allocation strategy that minimizes expected holding costs. For a clearing system we show that the optimal policy is characterized by a switching curve for which we provide a lower bound on its slope. We also specify a subset of the state space where the optimal policy can be explicitly determined