Although stochastic user equilibrium (SUE) problem has been studied extensively in the past decades, the solution convergence of SUE is generally quite slow because of the use of the method of successive averages (MSA), in which the auxiliary flow pattern generated at each iteration contributes equally to the final solution. Realizing that the auxiliary flow pattern is in fact approaching to the solution point when the iteration number is large, in this paper, we introduce the method of successive weighted averages (MSWA) that includes a new step size sequence giving higher weights to the auxiliary flow patterns from the later iterations. We further develop a self-regulated averaging method, in which the step sizes are varying, rather than fixed, depending on the distance between intermediate solution and auxiliary point. The proposed step size sequences in both MSWA and self-regulated averaging method satisfy the Blum theorem, which guarantees the convergence of SUE problem. Computational results demonstrate that convergence speeds of MSWA and self-regulated averaging method are much faster than those of MSA and the speedup factors are in a manner of magnitude for high accuracy solutions. Besides SUE problem, the proposed methods can also be applied to other fixed-point problems where MSA is applicable, which have wide-range applications in the area of transportation networks.