The price of anarchy in loss systems

Assume a multi-server memoryless loss system. Each server is associated with a service rate and a value of service. Customers from a common Poisson arrival process are routed to the servers in an unobservable way, where the goal is to maximize the long-run expected reward per customer (which is the service value times the probability that the customer is not blocked). We first solve this problem under two criteria: social optimization and Nash equilibrium. Our main result is that the price of anarchy, defined as the ratio between the expected gain under the two criteria, is bounded by 2. We also show, via examples, that this bound is tight for any number of servers.