http://www.diagnose-me.com/treat/T138573.html
says:
Some 1000 to 2000mg of L-tryptophan are needed in order to increase blood levels sufficiently to induce sleep. However, the lowest dose (often as little as 500mg) that works as a sleep aid should be maintained to continue benefits. The dose may be repeated if one wakes during the night. Doctors often recommend starting with 1gm of L-tryptophan 30-45 minutes before bed, which will reduce the time it takes to fall asleep. If 1000mg is insufficient, the dose may be increased by 500mg each night until either the desired effects are achieved or a total of 3000mg is reached.