We consider a finite-buffer single-server queue with batch Markovian arrival process. In the case of finite-buffer batch arrival queue, there are different customer rejection/acceptance strategies such as partial batch rejection, total batch rejection, and total batch acceptance policy. We consider partial batch rejection strategy throughout our paper. We obtain queue length distributions at various epochs such as pre-arrival, arbitrary, and post-departure as well as some important performance measures, like probability of loss for the first, an arbitrary, and the last customer of a batch, mean queue lengths, and mean waiting times. The corresponding queueing model under single and multiple vacation policy has also been investigated. Some numerical results have been presented in the form of tables by considering phase-type and Pareto service time distributions. The proposed analysis is based on the successive substitutions in the Markov chain equations of the queue-length distribution at an embedded post-departure epoch of a customer. We also establish relationships among the queue-length distributions at post-departure, arbitrary, and pre-arrival epochs using the classical argument based on Markov renewal theory and semi-Markov processes. Such queueing systems find applications in the performance evaluation of teletraffic part in 4G A11-IP networks.