Exact Reliability Model for a Mixed Redundant System with Heterogeneous Components and Component Sequencing

A mixed redundancy, in which some components of a subsystem operate under active redundancy while the others wait in cold standby, has recently been shown to achieve a higher system reliability than the traditional active and standby strategies within equivalent resources. Existing reliability models for the strategy, however, either provide only a lower bound of the subsystem reliability or restrict the components to be identical with exponential or Erlang lifetimes. This study proposes an exact reliability model for a mixed redundant system composed of heterogeneous components. The time to failure of every candidate component is described by a generalized phase-type distribution (PHD), and a structured continuous-time Markov chain (CTMC) integrates the active redundant module, the standby components—installed in a specified sequence recorded by an ordered component-index set—and an imperfect fault detector/switch. Because the model yields the infinitesimal generator of the subsystem lifetime, it provides not only the exact reliability but also the hazard function and the moments of the system lifetime. Building on a pre-computed database that enumerates every feasible subsystem structure, the redundancy allocation problem determining the component types, their number and sequence, and the number of active redundancies is formulated as a compact binary integer linear program and solved to optimality. Benchmark experiments show that the previous approximate function underestimates the achievable design by an average maximum possible improvement (MPI) of 15.6%, and that permitting heterogeneous components raises the optimal system reliability by a further 7.3% on average—up to 18.7% as the resource budget grows.