Optimal pricing and capacity sizing for the GI / GI / 1 queue

We consider how to jointly set the upfront, or static, price and capacity in order to maximize the steady-state expected profit in a GI/GI/1 queue with a high rate of prospective customer arrivals. We show that the optimal price and service rate for the GI/GI/1 queue are such that the system operates in heavy traffic. Then, since it is not possible to provide analytic expressions for the exact optimal price and service rate, we use the solution to an approximating (static) diffusion control problem to propose a price and service rate that we can prove is asymptotically optimal.