  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.
