We envision that in the near future, just as Infrastructure-as-a-Service (IaaS), radios and radio resources in a wireless network can also be provisioned as a service to Mobile Virtual Network Operators (MVNOs), which we refer to as Radio-as-a-Service (RaaS). In this paper, we present a novel auction-based model to enable fair pricing and fair resource allocation according to real-time needs of MVNOs for RaaS. Based on the proposed model, we study the auction mechanism design with the objective of maximizing social welfare. First, we present an Integer Linear Programming (ILP) based auction mechanism for obtaining optimal social welfare. To reduce time complexity, we present a polynomial-time greedy mechanism for the RaaS auction. Both methods have been formally shown to be truthful and individually rational. Extensive simulation results show that the proposed greedy auction mechanism can quickly produce close-to-optimal solutions.