Sleeping problem index

Imagine a world that you don't have to work, no appointment. There is no sleeping problem, right?

Think about Sunday or holiday, you can sleep when ever you like.

So, the sleeping problem is come from your work, somebody require you to be some places on certain time.

And usually, job require a regular schedule. Thus, in my point of view sleeping problem is the mis-match of the schedule and your sleeping pattern.

For simplicity, we can defined a naive sleeping problem index to indicate how serious is that mis-match.

P= (wake up time - scheduled time) /12.

Today, my P = 0.5 , very serious.

For little complicated definition, a mis-match should be in term of "phase shift".

for normal people, there sleeping pattern can be well described by a periodic function, say, a square pulse function, with duty (sleep-on time) 8 hours and 24 hours a cycle.

Then, the mis-matching is the synchronization of this function and working schedule.

Lets put little more math on it. The sleeping function is:

S= 0, for 24>t>8, 1 for 8>t>0=24

The working function is

W= 1, for 18>t>8, 0 for 8>t>18

Then it is perfect match. And the sleeping problem index is zero.

However, If the sleeping pattern becomes

S= 0, for 24+a>t>8+a, 1 for 8+a>t>a

Then, the mis-matching hour "a" is related to your sleeping problem.

Since the clock run 24 hours a cycle. The worst case is a=12. Thus, we should divided 12 for simplicity. Or if you like, we can compare your sleeping problem in different planet, which 1 day can be any hours.

Of course, this is a very simple model. The sleeping and working function can be variated. And in principle, the mis-matching can tell you how bad your are, coz mis-matching hours, is not quite comparable.

Why? Think about an extreme case, the working pattern is 1 hour work, the next hour off, an repeat. And your sleeping pattern is same. Is the 1 hour mis-match is the same as the normal case?