The problem of routing customers to parallel heterogeneous service stations in such a way as to optimise a queueing system’s performance is known to be one for which optimal policies are difficult to characterise. The application of dynamic programming is impractical in problems of realistic size, and there is a need for effective heuristics to be developed. Some possible approaches to the problem include the development of indices for the stations similar to the Gittins indices for multi-armed bandit problems, and simulation-based methods including those based on artificial neural networks.