The two basic hypotheses of Markov chains and the four basic hypotheses of B-Matrix chains proposed are carefully examined and compared to find a way to improve Markov chainsin certain specific situations . Our objective is to allow Markov chains to handle boundary conditions and / orsource / sink term in addition to ensuring the stability and convergence of the Markov series. Moreover, the numerical analysis of the eigenvalues of the proposed matrix B and of its eigenvectors validates the proposed following principle: [For positive symmetric physical power matrices, the sum of their eigenvalues is equal to the eigenvalue of their sum of power series]. This principle facilitates the search for summation solutions of many infinite algebraic power series such as