We investigate a single-server composite priority queue with batch inputs where an arriving batch is composed of multi-class customers. The main motivation for studying this type of priority is its potential applicability to communication switching systems. The Laplace-Stieltjes transform of the waiting time distribution is derived via busy-cycle analysis. The conservation law and waiting time characterization for a general class of composite priority queues are also presented. The results in the paper cover previous results as special cases.