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We consider a model for optimal pricing and capacity for a service facility. The problem is formulated as one of optimal design of a single-server queueing system, in which the design variables are the service rate and the arrival rate (equivalently, the price charged for admission). The model is a variant of one introduced by Dewan and Mendelson. We allow for an upper bound on the arrival rate and consider slightly more general user value functions. We show that an optimal solution may not lie in the interior of the feasible region and thus may not be characterized by the first-order differential conditions. Moreover, the first-order conditions typically have several solutions, some of which may be relative minima and produce a negative value of the objective function (customer value minus the sum of expected delay cost and capacity cost per unit time). We also examine the stability of the equilibrium arrival rate and the convergence of a dynamic adaptive algorithm for finding the optimal service rate, in the context of a model in which the distribution of customers' value of service is uniform. We show that the equilibrium arrival rate is stable if and only if the service rate is above a threshold value, which depends on the price charged for admission and the parameters of the uniform distribution of value of service. The dynamic, adaptive algorithm always converges to a relative maximum of the objective function if the service rate can be adjusted every time the arrival rate changes. Otherwise, the algorithm will start to diverge if and when the service rate ever falls below the threshold value associated with stability of the equilibrium arrival rate.
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