This paper addresses the problem of approximately computing the Lyapunov exponent of stochastic max-plus linear systems. Our approach allows for an efficient simulation of bounds for the Lyapunov exponent. We provide sufficient conditions for the convergence of the bounds. In particular, a perfect sampling scheme for the Lyapunov exponent is established. We illustrate the effectiveness of our bounds with an application to (real-life) railway systems. © 2010 Elsevier B.V. All rights reserved.