The ability to deliver acceptable levels of quality of service is crucial for cloud systems, and this requires performance as well as availability analysis. Existing modeling attempts mainly focus on pure performance analysis; however, the software and hardware components of cloud infrastructures may have limited reliability. In this study, analytical models are presented for performability evaluation of cloud centers. A novel approximate solution approach is introduced which allows consideration of large numbers of servers. The challenges for analytical modeling of cloud systems mentioned in the literature are considered. The analytical models and solutions, therefore, are capable of considering large numbers of facility nodes typically up to orders of hundreds or thousands, and able to incorporate various traffic loads while evaluating quality of service for cloud centers together with server availabilities. The results obtained from the analytical models are presented comparatively with the results obtained from discrete event simulations for validation.