  A divide-and-conquer approach to analyzing Markov chains (MCs) is not
utilized as widely as it could be, despite its potential benefits. One primary
reason for this is the fact that most MC decomposition approaches involve a
complex and inflexible methodology: decomposed subchains must be disjoint,
transition rates of these decomposed subchains must be altered in a way
tailored to the particular MC model, and the procedure to aggregate suchains
needs to incorporate a nonlinear normalization constraint, complicating the
analytical expression of performance measures. In contrast, we propose a
versatile yet simple decomposition method for continuous time MCs based on the
total expectation theorem. Leveraging the properties of this theorem, our
method has great flexibility in the choice of subchains, and the procedure to
obtain expected values of interest is simply a linear summation of subchains'
properties, which is not affected by the normalization constraint. We prove
that to maintain the correct distribution of decomposed subchains one may use
our novel termination scheme, a modification of transition rates, that ensures
partial flow conservation at boundary states. This termination scheme is
applicable to MCs with any structure, since the scheme depends only on the
boundary-state distribution, not on the structure of the MCs. To demonstrate
the generality and capability of our method, we analytically solve various
models, such as a congestion-based staffing queue and a Markov-modulated
Mt/Mt/1 queue. As not all systems admit an analytical solution, we complement
this analysis with numerical studies of MCs with various sizes using the
algorithm based on our method.
