We consider the stationary waiting-time distribution W in GI/G/1. KINGMAN [4] used two integral inequalities to obtain estimations for W above and below. But he gave only one solution explicitely for each of them. We show that the inequalities possess a great methodological importance. By means of them we first get a result of DALEY-MORAN [1] which permits to find further solutions. But the very point is a methed resting on the inequalities to improve every solutions successively. Examples for the refinement of KINGMANS estimations are given in (1.11)-(1.13). - The method can be carried over to a generalization of GI/G/1 in which a custumer who finds the server idle is not served immediately but only after a randam "warming- up time".