We consider inference for queues based on inter-departure time data. Calculating the likelihood for such models is difficult, as the likelihood involves summing up over the (exponentially-large) space of realisations of the arrival process. We demonstrate how a likelihood recursion can be used to calculate this likelihood efficiently for the specific cases of M/G/1 and Er/G/1 queues. We compare the sampling properties of the mles to the sampling properties of estimators, based on indirect inference, which have previously been suggested for this problem.