Modeling Common Cause Failures in Systems with Triple Modular Redundancy and Repair

Triple modular redundancy (TMR) is commonly employed to increase the reliability and mean time to failure (MTTF) of a system. This improvement can be shown by using a continuous time Markov chain. However, typical Markov chain models do not model common cause failures (CCF), which is a singular event that simultaneously causes failure in multiple redundant modules. This paper introduces a new Markov chain to model CCF in TMR with repair systems. This new model is compared to the idealized models of TMR with repair without CCF. The fundamental limitations that CCF imposes on the system are shown and discussed. In a motivating example, it is seen that CCF imposes a limitation of $51\times$ on the reliability improvement in a system with TMR and repair compared to a simplex system, (i.e., without TMR). A case study is also presented where the likelihood of CCF is reduced by a factor of $18\times$ using various mitigation techniques. Reducing the CCF compounds the reliability improvement of TMR with repair and leads to a overall system reliability improvement of $10,000\times$ compared to the simplex system as supported by the proposed model.