Mean waiting-time approximations are derived for a single-server multi-queue system with nonexhaustive cyclic service. Nonzero switchover times of the server between consecutive queues are assumed. The main tool used in the derivation is a pseudo-conservation law recently found by Watson (1984). The approximation is simpler and, as extensive simulations show, more accurate than existing approximations. Moreover, it gives very good insight into the qualitative behaviour of cyclic-service queueing systems. © 1987.