What a strange coincidence. I was watching a 1951 movie about Tyrone Power going back from 20th century New York to 18 century London (a bit like a Connecticut Yankee in King Arthur's court) while being asked about a script about time. Weird indeed!
Sorry, I did not look at the script in any detail, I just noticed that it is very complicated and it seems you also do some scripting in MySQL which I try to avoid. At this point I don't understand the procedure you are trying to set up. It seems that people sign in for a time slot. If this is a world wide script, I would set up the time slots in Universal Time or GMT (essentially the same thing). People signing in would have to pick their time zone, could be from a drop down menu. Then they can use local time because the time zone gives the difference to GMT including day light saving and all that jazz.
Probably the best course is to throw out the old idea and start coding afresh. Knowing the time zones, php makes calculating times very easy if the whole thing if based on GMT.
For example, 4 PM GMT for someone at GMT +10 would be
4+12+10 = 26
26 = tomorrow (24) + 2
Tokyo is GMT +9, it is already Monday for them:
Monday, 29 April 2013, 04:07:13 JST
(I looked it up in http://www.timeanddate.com/worldclock/city.html?n=248 )