In this paper, we present and solve a discrete-time G[X]/D/1-S queueing system with a finite queue size, and batch arrivals with a general batch-size distribution. The motivation for this model arises from performance modeling of a statistical multiplexer with synchronous transmission of fixed-size data-units in synchronous time slots. The arrival process to the multiplexer, for example, may originate from a number of independent sources with packets of variable lengths. Hence, a packet arrival corresponds to an arrival of a batch of data-units. Different performance measures such as percentage of packet loss and data-unit loss are considered under two different admission policies of packets into the queue.