In this paper, we study the performance properties of the phase II exponential chart with an unknown in-control (IC) rate parameter, used to monitor high-yield processes. The average time to signal (ATS) is used as the chart performance criterion instead of the usual average run length (ARL). Based on the IC conditional ATS (CATS) distribution, we examine the properties of both the equal-tailed and the ATS-unbiased exponential charts with estimated control limits and adjust the limits so that a nominal IC ATS performance is achieved. Two perspectives are investigated: the unconditional, under which the average of IC CATS distribution is set equal to a nominal ATS and, the conditional so that the IC CATS is set to or exceed a nominal ATS with a high probability. It is shown that the ATS-unbiased exponential chart under the conditional perspective has a better IC performance.