Cloud computing offers on-demand availability of computing resources over the Internet. κ-times anonymous authentication (κ-TAA) is an attractive approach to construct pricing schemes, providing access controllability, user anonymity and public traceability. In κ-TAA schemes, authenticated users are permitted to anonymously access services from a provider at most κ times, while the ones whose the number of access times exceeds κ can be publicly traced. That is, κ-TAA schemes offer a prepaid plan that charges users based on the amount of access times. Alternatively, pay-as-you-go (PAYG) is a pricing strategy that allows users to be charged based on the amount of usage, reducing the costs on unnecessary resources. Adopting κ-TAA schemes to PAYG model, the access bound κ is decided by the prepayment amount and the service usage is tracked by the number of access times. However, this approach is impractical, since existing κ-TAA schemes only allow an one-time access in an authentication. This work aims to bridge this gap in the literature by designing an efficient and secure authentication system for PAYG cloud computing, supporting flexible access controllability, user anonymity and public traceability.