In the emerging Big Data scenario, distributed File systems (DFSs) are used for storing and accessing information in a scalable manner. Many cloud computing systems use DFS as the main storage component. The Big Data applications de-ployed in cloud computing systems more frequently perform read operations and less frequently the write operations. So, improving the performance of read access has become an im-portant research issue in DFS. In the literature, many client side caching with appropriate pre fetching techniques are proposed for improving the performance read access in the DFS. A speculation-based approach which uses client side caching is also proposed in the literature for improving the performance of read access in the DFS. In this paper, we have proposed a new read algorithm for the DFS based on anticipated parallel processing. We have evaluated the per- formance of the proposed algorithm using mathematical and simulation methods and the results indicate that the pro-posed algorithm performs better than the speculation-based algorithm proposed in the literature.