A second-order heavy traffic approximation for the stationary waiting-time d.f. G for GI/G/1 queues is derived, the first-order term of which is Kingman's (1961), (1962a), (1965) exponential approximation. On the way to this result there are others of independent interest, such as a convolution equation relating this waiting time d.f. G with the d.f. of a related ladder height, an integral equation for G and some stochastic bounds for G. The main result requires a particular type of functional convergence that may also be of interest.
