15 minutes three times a day is a rule of thumb that may not apply to you at all.
You're looking to irrigate often enough, at a great enough volume, to provide an appropriate average level of moisture in your root zone. Prob gonna be a whole lot less, esp for young plants which don't have a lot of leaf area transpiring water.
Fer instance, my rockwool drip gets irrigated for 27 seconds every hour and 15 minutes that the lights are on. ( I have cool timers).
If you're using a cycle timer to control the pump, just plug it into the same power strip as the lighting system- it'll cycle on first.
I don't like standard timers for drip systems- the granularity is too coarse. I'm a " lots of small feeds instead of waterlogging with a few big ones" type of guy. YMMV

Basically, you have to play with it until you get a pattern that keeps your media moist but not too wet, by adjusting how long and how often pump is on. Easier with a quicker draining medium.

You may or may not need to irrigate a little while lights are off, depending on night humidity- I never have. YMMV

Your lights are only on six hours a day?