In this paper we study the busy period of an MEP/MEP/1 system, where both the arrival and the service processes can be serially correlated Matrix Exponential Processes. A dynamic programming algorithm is given to compute the probabilities for serving n customers in a busy period and expressions for the first two moments are derived. We study both the effect of correlation in the arrival and service processes and the squared coefficient of variation on these probabilities. The solutions give us qualitative insights into the nature of the busy period of the MEP/MEP/1 system. The resulting algorithms are easily programmable and efficient.