Summary: We study the relationship between the distribution of the supremum functional \(M_X=\sup_{0\leq t<\infty} (X(t)-\beta t)\) for a process \(X\) with stationary, but not necessarily independent increments, and the limiting distribution of an appropriately normalized stationary waiting time for G/G/1 queues in heavy traffic. As a by-product we obtain explicit expressions for the distribution of \(M_X\) in several special cases of Lévy processes.