In this paper, we analyze the individual sojourn delay experienced by cells from each virtual channel (VC) passing through an ATM switch port. Traffic from each VC is described by a four-parameter Markov-Modulated Bernoulli process (MMBP). A switch port is assumed to receive traffic from a group of heterogeneous MMBPs and the queueing behavior is modelled by a H-MMBPs/D/1/ queue. Two service disciplines are considered: FIFO and priority. Closed-form formulas of overall as well as individual sojourn delays for both service disciplines are obtained. Although approximation is inevitable in our analysis, the accuracy is good when compare with computer simulations. As a result we provide an efficient tool to estimate cell delay for each individual VC before it is established. Our result can be applied to network resource decision or control problems such as call admission control and routing.