| Reparable system is one of the most important systems discussed in the reliability theory, and also the primary subject studied in reliability mathematics. This paper mainly investigates the exponential stability of the reparable systems described by generalized Markov models, which are formulated by supplementary variable technique. It is significant for the reliability of the systems. Many scholars have done a lot of work about the well-posedness and asymptotic stability of the systems. However, the exponential stability of systems is no well solved. In this paper, under the reasonable assumption that the mean of the reparable rate exists, we prove the solution of the system, with or without integral boundary, is exponentially stable. Taking a further step, we could consider the numerical simulation and optimal control. In a more detailed description, we investigate the well-posedness and exponential stability of a two identical unit parallel system with a warm standby. We also investigate the exponential stability of the system consisting of a reliable machine, an unreliable machine and a storage buffer, which has integral boundary. For the former system, we show that there exists a unique nonnegative time-dependent solution of the model by using the Volterra operator equation. Besides, both of the system operators can generate a positive contraction C0 -semigroup in L1 space, so the solutions of the systems are nonnegative with probability character. Moreover, 0 is shown to be the unique spectral point on the imaginary axis. In terms of the spectral properties, the C0 -semigroup can be proved to be quasi-compactness and 0 is an isolated eigenvalue with algebra multiplicity one. At last, by proving the irreducibility of the C0 -semigroup, we obtain that the time-dependent solution of the system converges to the steady-state solution exponentially. At the same time, we also prove that the positive contraction C0 -semigroup generated by M/G/1 queueing model operator is irreducible.
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