We determine the reliability exponent E(R) of the Anantharam-Verdú exponential server timing channel with service rate μ for all rates R between a critical rate R c = (μ/4) log 2 and the channel capacity C = e -1μ. For rates between 0 and R c, we provide a random-coding lower bound E r(R) and a sphere-packing upper bound E sp(R) on E(R). We also determine that the cutoff rate R o for this channel equals μ/4, thus answering a question posed by Sundaresan and Verdú. An interesting aspect of our results is that the lower bound E r (R) for the reliability exponent of the timing channel coincides with Wyner's reliability exponent for the photon-counting channel with no dark current and with peak power constraint μ. Whether the reliability exponents of the two channels are actually equal everywhere remains open. This shows that the exponential server timing channel is at least as reliable as this type of a photon-counting channel for all rates.