Cloud Radio Access Networks (C-RANs) are considered one of the most promising candidates for implementing 5G mobile communication systems. C-RAN enables centralisation of baseband processing, enabling advanced coordination between base stations, such as coordinated multi-point and inter-cell interference cancellation. In addition, it allows pooling of resources across several cells, providing statistical multiplexing gains of computing resources. However, the link between the remote radio unit and the baseband unit requires high transmission capacity, making the fronthaul link a potential bottleneck for future dense cell deployments. One of the current solutions to this issue is to compress the fronthaul transmission rate. A second under standardisation is to adopt a different functional split that can reduce the transmission capacity requirement. While this solution decreases the capacity requirements on the transport link, it decentralises some of the computational resources, requiring a more complex remote radio unit (e.g., compared with a CPRI type of solution), whose resources cannot be utilised by other cells when not in use. It is thus expected that in the future, multiple solutions (different functional splits and CPRI) will coexist.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.