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We exhibit upper and lower bounds on the pairwise error probability of the exponential-server timing channel in terms of an appropriately-defined distance between codewords. We show that this distance plays a crucial role in determining the reliability function at low rates. In particular, by lower bounding the minimum distance of good-low rates codes, we provide an improved lower bound on the reliability function of the channel at rate zero. This improved bound proves that at low rates, the exponential-server timing channel is strictly more reliable than the related Poisson channel with zero dark current, answering an open question posed by Arikan. Some remarks are also made about using the results of this paper to prove an improved upper bound on the reliability function at rate zero.