Constant bit rate (CBR) traffic is expected to be an important source of traffic in high-speed networks. Such sources may have stringent delay and loss requirements and in many cases, they should be delivered exactly as they were generated. A simple delay priority scheme will bound the cell delay and jitter for CBR flows, so that in the network switches, CBR traffic will only compete with other CBR traffic in the networks. In this paper, we will consider a slotted queue at a typical (intermediate) node in such an environment. The cell arrival process of each source is characterized by a dispersed periodic arrival process (DPAP), the cells of which are assumed to arrive periodically and distribute randomly within the period. We provide an exact analysis of buffer fill and delay distribution using Ballot Theorems. Another generalized source model, the dispersed batch periodic arrival process (DBPAP) is also investigated. A DBPAP source is assumed to have a number of batches of cell arrivals and the batches position randomly in the period. This work shows that the traditional Markovian approximation models may grossly overestimate the buffer fill and delay, unless the number of sources is very large.