On the Downs–Thomson Paradox in a Self-Financing Two-Tier Queuing System

We model a two-tier queuing system with free and toll service options as two parallel M/M/1 servers. We solve for the welfare-maximizing toll service capacity and toll subject to the constraint that the toll service cover its costs. If the free and toll services are both used in equilibrium, a larger free-service capacity implies longer expected waiting time for the free service and lower welfare: an analogue to the Downs–Thomson paradox in transportation economics. The paradox is caused by the presence of scale economies in the toll service combined with the requirement that it be self-financing.