A queuing-based model for optimal dimension of service firms

This paper examines a stochastic model to determine optimal pricing, waiting time, output, and sizing decisions for service firms which compete on time in an uncertain environment. Sizing decisions concern optimal service capacity and maximum physical waiting room (with a given probability). Customers are sensitive to money price and expected waiting time. The firm, modeled as an M/M/1 queue, is assumed to be a full price taker as a result of the stochastic version of the perfect competition assumption. Firm costs function includes fixed and variable costs. We initially fix the service rate and obtain a closed-form expression for the waiting time, price, output and waiting room which lead to maximize the net revenue. Subsequently, we obtain first-order conditions for the profit maximizing service capacity. We find that an optimal capacity may exist when service capacity costs are strictly convex. When capacity costs are linear or concave, an optimal capacity does not exist, and it is possible to obtain higher expected profits by increasing service capacity, as the time competition principle states. However, in the linear case we find a failure of that principle, which is, there is a limit in speeding up processes.