Revenue Maximization in Service Systems with Heterogeneous Customers
  In this paper, we consider revenue maximization problem for a two server system in the presence of heterogeneous customers. We assume that the customers differ in their cost for unit delay and this is modeled as a continuous random variable with a distribution $F.$ We also assume that each server charges an admission price to each customer that decide to join its queue. We first consider the monopoly problem where both the servers belong to a single operator. The heterogeneity of the customer makes the analysis of the problem difficult. The difficulty lies in the inability to characterize the equilibrium queue arrival rates as a function of the admission prices. We provide an equivalent formulation with the queue arrival rates as the optimization variable simplifying the analysis for revenue rate maximization for the monopoly. We then consider the duopoly problem where each server competes with the other server to maximize its revenue rate. For the duopoly problem, the interest is to obtain the set of admission prices satisfying the Nash equilibrium conditions. While the problem is in general difficult to analyze, we consider the special case when the two servers are identical. For such a duopoly system, we obtain the necessary condition for existence of symmetric Nash equilibrium of the admission prices. The knowledge of the distribution $F$ characterizing the heterogeneity of the customers is necessary to solve the monopoly and the duopoly problem. However, for most practical scenarios, the functional form of $F$ may not be known to the system operator and in such cases, the revenue maximizing prices cannot be determined. In the last part of the paper, we provide a simple method to estimate the distribution $F$ by suitably varying the admission prices. We illustrate the method with some numerical examples. 
Tejas Bodas, D. Manjunath
cs.GT