  We show that one can approximate the least fixed point solution for a
multivariate system of monotone probabilistic polynomial equations in time
polynomial in both the encoding size of the system of equations and in
log(1/\epsilon), where \epsilon > 0 is the desired additive error bound of the
solution. (The model of computation is the standard Turing machine model.)
  We use this result to resolve several open problems regarding the
computational complexity of computing key quantities associated with some
classic and heavily studied stochastic processes, including multi-type
branching processes and stochastic context-free grammars.
