For the GI/G/1 queueing model with traffic load a<1, service time distribution B(t) and interarrival time distribution A(t), whenever for t1B(t)c(t/β)ν+O(eδt),c>0,1<ν<2,δ>0, and 0tμdA(t)< for μ>ν, (1a)1ν1w converges in distribution for a1. Here w is distributed as the stationary waiting time distribution. The L.-S. transform of the limiting distribution is derived and an asymptotic series for its tail probabilities is obtained. The theorem actually proved in the text concerns a slightly more general asymptotic behavior of 1B(t), t, than mentioned above.