Augmenting Revenue Maximization Policies for Facilities where Customers Wait for Service

In this paper, we study the dynamic pricing problem of a multi-server facility that processes requests from several customer classes on a first come first served basis. We assume an arrival belongs to one of a finite number of customer classes and that each class is distinguished by known and arbitrary service valuations and arbitrary but non-decreasing waiting cost functions. We model the facility as an M/M/S/I queue and use the theory of Markov Decision Processes to identify dynamic pricing strategies that maximize the revenues obtained from customers who are assumed to be acting individually to maximize their utility. The key to our approach is recognizing, in context of our model, that the maximum revenue obtainable from a service facility is bounded by the maximum collective benefits less waiting costs that customers could receive in such a facility which is achieved through state dependent social (welfare) optimization. For our model, we show that this upper limit can be achieved with state dependent revenue maximization when it is possible to exactly charge customers the benefit they receive less the waiting costs they incur. This in turn can be accomplished when it is possible to identify the group of customers to which each customer belongs, where customers in each group have the same benefit and waiting cost function. We further demonstrate that some or most of this additional revenue can also be achieved even when customers in each group only have similar benefits and waiting cost functions. Finally, we illustrate that the revenue maximizing pricing policy might actually increase arrival rates as system ocuupancy increases, by admitting customer classes that were not admitted in previous states.