It might be better to take a statistical approach from a real world study rather than starting with a hypothetical equation and trying to make the results fit. You'd need to assess an enormous number of people for your experiment (and the results) to be credible. There would also be the issue of "too many variables" and inquantifiable factors, such as medications, stress levels, demographics, recent exercise, hydration, blood-sugar concentration, illnesses, head trauma, childhood factors, discomfort, noise, body temperature, ambient temperature, gender, age, normal sleep patterns, work patterns, smells in room, blood pressure, bladder capacitance.
Not that I'm trying to put you off. Go for it.