This paper studies a queue with a finite buffer modelled as a semi-Markov process of consecutive visits to the empty buffer and the full buffer state. The martingale description of queues allows application of the optional sampling theorem to obtain the joint moment generating function of the length of an interval and the number of customers served in an interval. This leads to an explicit solution for an M/M/l/B queue and for a special case of a queue with constant service times. Extensions to priority queues and applications to control of queues are briefly discussed.