Prediction of drowsiness based on an objective measure is demanded in machine and vehicle operations in which human errors may cause fatal accidents. Currently we focused on the pupil of the eye which is controlled by the autonomic nervous system, and easily observable non-invasively from the outside of the body. We employed uneventful driving simulation to induce drowsiness of human subjects, and an anti-saccade task to evaluate their cognitive and motor performance. First we confirmed that pupil diameter fluctuates with large amplitude at low frequencies when the subject is aware of his/her drowsiness as reported previously. During this period, the reaction time of anti-saccade initiation was elongated and varied. We then found that prior to this fluctuation, pupil diameter decreases gradually in most subjects, and they were not aware of sleepiness during this period. We conclude that this monotonic gradual miosis can be a reliable premonitor of drowsiness.