What's your minimum sleep requirement before exercising?

I had planned on today being my long run day with a goal of four miles. I typically run on Mondays starting at 730am.

My son had a nightmare at four, and I fell asleep at 1130. I can't fall back asleep so I'm thinking about just heading out soon to run. But I'm already psyching myself out because I'm tired and feel like I'm going to run 1/2 a mile before giving up.

On a different note - anyone have good diet tips for helping with sleep? I've found that limiting carbs helps me sleep better throughout the night but makes it more difficult to actually fall asleep. Eating carbs makes it easy to fall asleep but can lead to me waking up in the middle of the night with blood sugar issues.