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Distributed Computing System (DCS) has become very popular for its high fault-tolerance, potential for parallel processing, and better reliability performance. One of the important issues in the design of the DCS is the reliability performance. Distributed Program Reliability (DPR) has to be addressed to obtain this reliability measure. An efficient network topology is quite important for the distributed computing system. For example, the ring network has been widely used in current distributed system design. In this paper, we focus on DCS with ring topologies. We propose polynomial-time algorithms to analyze the DPR of the dual ring topology and show that solving the DPR problem on a ring of trees topology is NP-hard.