Convergence of a stationary iteration method for linear systems

Question

Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, strictly diagonally dominant $M$-matrix. So I establish a stationary iteration scheme as follows, $$Sx^{(k+1)} = Tx^{(k)} + b.$$ According to numerical results, it seems that this iterative scheme is always convergent. Is it rational from theoretical analysis? So, can we prove the convergence of this iterative scheme? i.e., (we need to show $\rho(S^{-1}T) < 1$, where $\rho(\cdot)$ is the spectral radius.