Bertrand equilibria and efficiency in markets for congestible network services

This paper is motivated by study of the economics of Quality of Service (QoS) of congestible services. We introduce a queueing game framework to study such problems. We consider multiple competing providers, each offering a queued service. Users are sensitive to both access price and expected delay, and pick providers with the smallest price plus delay cost. We study equilibrium of the pricing (Bertrand) game between the congestible network service providers. We establish the existence of a Nash equilibrium under some natural assumptions. We then consider a setting with multiple classes of differentiated service. Differentiated Services (DiffServ) technologies of the Internet that can provide QoS guarantees have failed to catch on, primarily due to economic impediments. Each provider is now modeled as operating a multi-class queue. We provide sufficient conditions for the existence of a Nash equilibrium in the Bertrand (pricing) game between the providers. We characterize the inefficiency (price of anarchy) due to strategic pricing to be 2/3. Surprisingly, the price of anarchy for the multi-class setting is the same as for the single-class setting.