In this paper we consider the problem of computing the stationary distribution of nearly completely decomposable Markov processes, a well-established area in the classical theory of Markov processes with broad applications in the design, modeling, analysis and optimization of computer systems and applications. We devise a general mathematical framework of numerical solution methods that exploits forms of mixed-precision computation to significantly reduce computation times and that exploits forms of iterative approximate computing approaches to mitigate the impact of inaccurate computations, further reduce computation times, and ensure convergence. Then we derive a mathematical analysis that establishes theoretical properties of our general algorithmic framework including results on approximation errors, convergence behaviors, and other algorithmic characteristics. Numerical experiments demonstrate that our general algorithmic framework provides significant improvements in computation times over the most-efficient existing numerical methods.