Sleepiness is not simply the result of being awake for a long time. Two independent drives combine to produce it.
The first is a circadian rhythm: an internally generated cycle of roughly 24 hours that governs the timing of alertness and sleepiness. Its neural basis is the suprachiasmatic nucleus (SCN), a small cluster of neurons in the hypothalamus that acts as the body's master clock. The SCN keeps its own near-24-hour rhythm even without external cues, and light exposure adjusts, or entrains, its phase so that the internal cycle stays aligned with day and night. Because this drive is timed rather than accumulated, it produces a predictable afternoon dip in alertness and a period of high sleepiness in the biological night, regardless of how rested you are.
The second is homeostatic sleep pressure: a need for sleep that grows the longer you stay awake and falls during sleep. A major contributor is adenosine, a byproduct of neuronal activity that accumulates in the brain during wakefulness and inhibits the arousal systems that keep you alert. The longer you are awake, the more adenosine builds, and the stronger the pressure to sleep becomes.
The two-process model formalizes this. Process C is the circadian drive, an oscillating signal that rises and falls across the day. Process S is the homeostatic drive, which rises monotonically with time awake and declines during sleep. What you actually experience as sleepiness is the combined output of the two. That is why you can feel drowsy in the mid-afternoon after a full night's sleep, and why staying awake all night does not make you feel uniformly worse each hour: the circadian drive can temporarily oppose the rising homeostatic pressure.