Investigations of motorist delays at the intersection of a major and minor road have involved variations of one or the other of two merging strategies: (i) Miller's (Miller, A. J. 1961. A queueing model for road traffic flow. J. R. Statist. Soc. B23 64–75.) model that assumes that a minor-road motorist makes consecutive decisions consistently, and (ii) that of Weiss and Maradudin’ (Weiss, G. H., A. A. Maradudin. 1962. Some problems in traffic delay. Opns. Res.10 74–104.) that assumes that a different, independent decision is made for each headway. In this paper we show that results for both models may be obtained easily, using the results of queuing theory, if certain reasonable assumptions are made. Simple expressions for the Laplace-Stieltjes transforms and means of the delays are derived in the case of independent major-road headways with a common, general distribution. For exponential headways, a numerical study shows the mean Miller delay to increase with the variation in the merge times, while the mean Weiss-Maradudin delay decreases. To illustrate the application of the expressions obtained, we investigate the reduction in the delay achieved by inserting a traffic island between two streams of major-road traffic.