Lateness theory

I need help (for once!) with something:

I have really annoyed lately at how people are always late, and tried to apply some economic thought to why people are late.

My situation is basically that there are 4 in my house, and we car pool to work. We can't leave until everyone is ready, and we all want to sleep as long as possible. We have to be at work by 8.30, to miss traffic, and we aim to leave at 8.00. Problem is, if anyone is ready early (mostly because he got up earlier), they have to wait until the rest is ready and thus could have slept longer. What I noticed was how everybody was getting up later and later, and we constantly left the house later than 8.00. First week it was about 8.05, then 8.10, and third week 8.15. So I designed a matrix for what would happen if it was assumed that everybody was aiming to be on time (on time meaning 0 minutes late/early), but hade a deviation of +-5 minutes :pcguru:

. What I noticed was that the mean time of departure is increased depending on how many people you have to wait for, and how high the standard deviations are (see spreadsheet), as we had to wait for the last guy to get ready independantly of how early anyone was.

first of all, can someone mathematically inclined design a formula for how the mean changes depending on how many people you have, and how big the standard deviation is? I.e. (Mean leaving time)=f(Standard deviation, number of people), what is the function?

Secondly, if we assume that people adapt and adjust the time they aim to be ready depending on when the left previously (if we left early, maybe aim to leave at 7.55 instead the next day, and if we were late aim for 8.05) By using the fact that the mean leaving time is positive in the set scenario, it seems to explain why we continously leave the house late and later. So I was just wondering if someone could critise/comment on this method and conclusion, I think this might be heading somewhere no economist have been before :biggrin:

And lastly, if anyone knows any reading on this it would be greatly appreciated.

Rep to anyone who can even understand what I am trying to say :rolleyes: