  We consider the problem of computing the minimal nonnegative solution $G$ of
the nonlinear matrix equation $X=\sum_{i=-1}^\infty A_iX^{i+1}$ where $A_i$,
for $i\ge -1$, are nonnegative square matrices such that $\sum_{i=-1}^\infty
A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type
Markov chains, since the matrix $G$ provides probabilistic measures of
interest. A new family of fixed point iterations for the numerical computation
of $G$, that includes the classical iterations, is introduced. A detailed
convergence analysis proves that the iterations in the new class converge
faster than the classical iterations. Numerical experiments confirm the
effectiveness of our extension.
