Our interest was to evaluate changes in fixation duration as a function of time-on-task (TOT) during a random saccade task. We employed a large, publicly available dataset. The frequency histogram of fixation durations was multimodal and modelled as a Gaussian mixture. We found five fixation types. The ``ideal'' response would be a single accurate saccade after each target movement, with a typical saccade latency of 200-250 msec, followed by a long fixation (> 800 msec) until the next target jump. We found fixations like this, but they comprised only 10% of all fixations and were the first fixation after target movement only 23.4% of the time. More frequently (57.4% of the time), the first fixation after target movement was short (117.7 msec mean) and was commonly followed by a corrective saccade. Across the entire 100 sec of the task, median total fixation duration decreased. This decrease was approximated with a power law fit with R^2=0.94. A detailed examination of the frequency of each of our five fixation types over time on task (TOT) revealed that the three shortest duration fixation types became more and more frequent with TOT whereas the two longest fixations became less and less frequent. In all cases, the changes over TOT followed power law relationships, with R^2 values between 0.73 and 0.93. We concluded that, over the 100 second duration of our task, long fixations are common in the first 15 to 22 seconds but become less common after that. Short fixations are relatively uncommon in the first 15 to 22 seconds but become more and more common as the task progressed. Apparently. the ability to produce an ideal response, although somewhat likely in the first 22 seconds, rapidly declines. This might be related to a noted decline in saccade accuracy over time.