Static random-access memory-based field-programmable gate arrays are increasingly being used for on-board processors in space missions. However, they are very susceptible to single event upsets that can generate on-board processor system malfunction or system failures in space radiation environments. This paper presents an on-board processor system adopting Triple Modular Redundancy with the concept of mitigation windows and external scrubber, and then suggests a mathematical model that predicts the on-board processor system failure rate by only using the information of system configuration resources. Our mathematical derivation can estimate on-board processor system reliability as a function of the single event upset rate, the number of mitigation windows, and on-board processor shield thickness. In addition, a guideline of the on-board processor system design is provided for achieving good single event upset mitigation capability and system reliability.