We analyze steady rolling of a cylinder on a Winkler foundation governed by a rate-and-state friction law. A full iterative solution resolves the stick–slip partition and the evolution of the state variable across the footprint. In parallel, a simplified averaging method evaluates the law at a representative slip rate and enforces consistency at the stick–slip boundary, yielding closed-form expressions for the global response. Both approaches reproduce the hallmarks of creepage curves — universal initial slope, peak traction, post-peak softening, and the jump at loss of stick — while clarifying how they depend on renewal length and reference speed in velocity-strengthening and velocity-weakening regimes. The averaging method matches the iterative solution at negligible cost and fits experimental force–creepage data across multiple speeds with a single parameter set, providing a fast, physically transparent tool for rolling analysis, while retaining the flexibility of rate-and-state friction to include memory and environmental effects.