A large call center faces a great amount of daily traffic. Since the rate of incoming calls varies over time, a call center may become overloaded during the peak hours of a day. In addition, many service-oriented call centers are intentionally operated in an overloaded regime so as to save staffing costs. Queues with many parallel servers are building blocks for modeling call centers operations. In this paper, we use GI/GI/n+GI queues to model a call center in the overloaded regime. Using this model, we obtain useful performance formulas, including the distributions of the steady-state queue length and virtual waiting time, for service level estimation in efficiency-driven call centers. The proposed diffusion model is justified by limit theorems for overloaded queues. We prove the convergence of queue length and virtual waiting time processes when they scaled in both space and time.