On a dynamical system for a reliability model

摘要

By studying the spectrum on the imaginary axis of the operator corresponding to the system consisting of a reliable machine, an unreliable machine and a storage buffer with infinite capacity described by an infinite-dimensional non-local (pseudo-differential) dynamical system, we prove that the time-dependent solution of the system converges strongly to its steady-state solution as time tends to infinity and we show that our study provides the optimal result on convergence. Then by discussing the monotonicity of the corresponding C_0-semigroup, we obtain the asymptotic behavior of the reliability indices for the system.