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In this paper, we propose an approximation for the loss probability, PL (x), in a finite buffer system with buffer size x. Our study is motivated by the case of a high-speed network where a large number of sources are expected to be multiplexed. Hence, by appealing to Central Limit Theorem type of arguments, we model the input process as a general Gaussian process. Our result is obtained by making a simple mapping from the tail probability in an infinite buffer system to the loss probability in a finite buffer system. We also provide a strong asymptotic relationship between our approximation and the actual loss probability for a fairly large class of Gaussian input processes. We derive some interesting asymptotic properties of our approximation and illustrate its effectiveness via a detailed numerical investigation.