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This paper solves the computation of the reliability of Markov systems. The solution is especially useful for systems where the number of states is large. Not only is the mean time-to-failure computed but an analytic expression is given for the probability of the system's being in a certain state as a function of time. The solution is derived by solving the eigenvalues and eigenvectors of the transitionrate matrix of the Markov system. In the solution, numerical algorithms have been chosen to keep the computer cost down. The method uses just a fraction of the time that, for example, the Runge-Kutta method uses.