Summary: The work function algorithm (WFA) is an on-line algorithm that has been studied mostly in connection with the \(k\)-server problem, but can actually be used on a wide variety of on-line problems. Despite being a simple algorithm, WFA has proven to be difficult to analyze, and until recently few interesting results were known. We analyze the performance of the generalized work function algorithm, denoted \(\alpha\)-WFA, for on-line traversal of layered graphs. We show that for layered graphs of width \(k\) and any \(\alpha> 1\), \(\alpha\)-WFA achieves competitive ratio \((\alpha+ 1)(2\alpha/(\alpha- 1))^{k- 1}-\alpha\). This gives an \(O(k2^k)\)-competitive ratio for appropriate choice of \(\alpha\), improving the previous upper bound of \(O(k^3 2^k)\).