Reliability Analysis of a Hardware–Software Series Repairable System with Multiple Vacations of a Repairman

This paper develops a reliability analysis model for a class of computer systems composed of hardware and software in series, considering a repairman taking multiple vacations. The system follows a series failure rule: hardware can be repaired to be as good as new after failure; software undergoes minor repairs to maintain operability after the first N−1 failures with an increasing failure rate, and is overhauled to be as good as new with cycle reset after the N-th failure. Based on the principle of probability conservation and the supplementary variable method, the state probability evolution equations of the system are derived. A Banach space is constructed, and a linear operator is defined, whose denseness, dissipativity, and closedness are verified. It has been proven that the operator generates a positive contractive C0-semigroup, thus rigorously establishing the well-posedness of the model and the existence of a unique positive dynamic solution. Further spectral analysis verifies that zero belongs to the continuous spectrum rather than the point spectrum of the system operator. This indicates that the investigated system admits no time-invariant constant steady-state probability distribution, and only presents slowly decaying quasi-stationary dynamic behavior. The results can provide theoretical support for the reliability design and maintenance strategy optimization of hardware–software series repairable systems.