Cloud Radio Access Network (C-RAN) is considered as one of the most promising candidate for Fifth Generation (5G) wireless communication system. However, C-RAN imposes stringent requirements on fronthaul in terms of capacity which in turn becomes a bottleneck for deployment of cloud-RAN in dense networks. Compressed CPRI and functional split are two primary solutions to overcome this. However, this makes Remote Radio Unit (RRU) more complex and the computational resources in RRU are localized and thus cannot be shared with other RRUs when not in use. In this paper, we introduce the concept of Variable Rate Fronthaul (VRF) for C-RAN. This scheme operates on a CPRI type of interface (e.g., one that transmits I/Q data samples) with the novelty of dynamically changing the cell bandwidth, and consequently the fronthaul data rates, depending on the cell load, with the support of a Software Defined Network (SDN) controller. This allows for a more efficient transport of C-RAN cells data over a shared backhaul. We first propose a mathematical analysis of the VRF performance using a queuing theory approach based on the Markov model. We then provide the results of our simulation framework both for validation and in support of the mathematical analysis. Our results show that the proposed VRF scheme provides significantly lower blocking probability over a shared backhaul than standard CPRI.