Summary: We study an M/G/1 multi-queueing system consisting of M finite capacity queues, at which customers arrive according to independent Poisson processes. The customers require service times according to a queue- dependent general distribution. Each queue has a different priority. The queues are attended by a single server according to their priority and are served in a non-preemptive way. If there are no customers present, the server takes repeated vacations. The length of each vacation is a random variable with a general distribution function. We derive steady state formulas for the queue length distribution and the Laplace transform of the queueing time distribution for each queue.