Optimal state‐dependent pricing policies for a class of stochastic multiunit service systems

This paper models a k‐unit service system (e.g., a repair, maintenance, or rental facility) with Poisson arrivals, exponential service times, and no queue. If we denote the number of units that are busy as the state of the system, the state‐dependent pricing model formalizes the intuitive notion that when most units are idle, the price (i.e., the service charge per unit time) should be low, and when most units are busy, the price should be higher than the average. A computationally efficient algorithm based on a nonlinear programming formulation of the problem is provided for determination of the optimal state‐dependent prices. The procedure ultimately reduces to the search on a single variable in an interval to determine the unique intersection point of a concave increasing function and a linear decreasing function. The algorithm takes, on the average, only about 1/2 second per problem on the IBM 360/65 (FORTRAN G Compiler). A discrete optimal‐control approach to the problem is shown to result in essentially the same procedure as the nonlinear‐programming formulation. Several properties of the optimal state‐dependent prices are given. Comparisons of the optimal values of the objective function for the state‐dependent and state‐independent pricing policies show that the former is on the average, only about 0.7% better than the latter, which may explain partly why state‐dependent pricing is not prevalent in many service systems. Potential generalizations of the model are discussed.