In a centralized radio access network (RAN), the signals from multiple radio access points (RAPs) are centrally processed in a data center. A centralized RAN enables advanced interference coordination strategies while leveraging the elastic provisioning of data processing resources. It is particularly well suited for dense deployments, such as within a large building where the RAPs are connected via fiber and where many cells are underutilized. This paper considers the computational requirements of a centralized RAN with the goal of illuminating the benefits of pooling computational resources. A new analytical framework is proposed for quantifying the computational load associated with the centralized processing of uplink signals in the presence of block Rayleigh fading, a distance-dependent path loss, and fractional power control. Several new performance metrics are defined, including the computational outage probability, the outage complexity, the computational gain, the computational diversity, and the complexity-rate tradeoff. The validity of the analytical framework is confirmed by numerically comparing it with a simulator compliant with the 3GPP LTE standard. Using the developed metrics, it is shown that centralizing computing resources provides a higher net throughput per computational resource as compared with local processing.