If milk induces sleep, it is not because of its effect on skin temperature (in the case of warm milk) or tryptohpan. 

> consuming milk as such is unlikely to induce sleep, as the amount of tryptophan present in milk is too small. Milk coupled with a carbohydrate food at bedtime can be a potential sleep [aid](http://www.dairynutrition.ca/facts-fallacies/health-concerns/sleep-and-tryptophan).
>
> having a glass of warm milk is unlikely to raise skin temperature enough to have any [effect](http://scienceline.org/2006/10/ask-klein-milk/)