Optimal Pricing in Finite Server Systems

We consider a system of K servers, where customers arrive according to a Poisson process, and have independent and identical (i.i.d.) exponential service times and i.i. d. valuations of the service. We consider the setting where customers leave when they find all servers busy. Service provider announces a price to an incoming customer, depending on the number of busy servers. An incoming arrival enters the system if its valuation exceeds the price. We find the optimal state dependent pricing, that maximizes the revenue rate for the service provider.