Yesterday I brewed my second batch at my new place/house. First time I used the hose that was on the faucet for my chill water (one of those coiled/springy type hoses) that has an ID in the 3/8" range (judging by the OD, it could be smaller though). I had my other hose attached to that, but since the first step was a smaller ID, it was restricting/reducing the flow a lot. Chill time was significantly longer than I had experienced before. This is with my larger Duda Diesel plate chiller.
Between that brew day and yesterday, I purchased a 3/4" ID hose (long enough to go from the faucet to where the plate chiller was set up). Chill time was less than half from the previous batch. Plus, I was able to get the wort to a cooler temperature than before. On the first batch I had to stop at 68F with the chiller (used a swamp cooler to get down to where I needed it before pitching the yeast). This time I got it to 64F, which I was more comfortable with (for the yeast strain used, this is a good temp). I also purchased a brass shutoff valve that goes on either the faucet or end of the hose. I placed that one the end of the hose, so that I could turn the water on/off right there (didn't need to go over to the faucet to do this
Of course, I pitched my yeast starter slurry into the batch last night and it was showing activity within a few hours. About 6-8 hours later, it's REALLY active (Star San foam in the airlock).
Also, since I'm fermenting in the basement, which is in the upper 50's today (about 58 currently) the fermenting beer has actually gone down a few degrees as it's fermenting.
Moral of the story... If you're not pleased/happy with how long it takes to chill your wort down, or it's taking longer than you would expect it to with your chiller, look at the cooling water hose ID. If possible, use at least a 5/8" ID hose for your chill water source. Especially if your water source is from a faucet outside. I know some use adapters on kitchen faucets, where you're limited by other factors. Still, get the largest ID hose there that you can.
Side note: the larger ID hose is a bit tougher to coil up than the smaller ID hoses. But, IMO, it's well worth it. Faster chill times are more important than spending a few extra minutes to coil up a 50' 3/4" ID hose.