Stochastic model predictive control (MPC) strategies can provide guarantees of stability and constraint satisfaction, but their online computation can be formidable. This difficulty is avoided in the current technical note through the use of tubes of fixed cross section and variable scaling. A model describing the evolution of predicted tube scalings facilitates the computation of stochastic tubes; furthermore this procedure can be performed offline. The resulting MPC scheme has a low online computational load even for long prediction horizons, thus allowing for performance improvements. The efficacy of the approach is illustrated by numerical examples.