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The literature on queueing systems with finite buffers addresses mostly asymptotic performance metrics on an aggregate flow, and/or generally relies on a convenient, but provably inaccurate, approximation of the loss probability by the overflow probability in an infinite size buffer. This paper addresses non-asymptotic per-flow metrics in a multi-flow queueing system with finite buffer and FIFO scheduling. The analysis dispenses with the above approximation, and lends itself to several interesting insights on the impact of finite buffers on per-flow metrics. Counterintuitively, the per-flow delay distribution is not monotonous in the buffer size, and such an effect is especially visible in high burstiness regimes. Another observation is that buffer dimensioning becomes insensitive to the type of SLA constraint, e.g., fixed violation probability on either loss or delay, in high multiplexing regimes. In the particular case of aggregate scheduling, the results on the aggregate input flow significantly improve upon existing results by capturing the manifestation of bufferless multiplexing in regimes with many flows.