Should There Be A Turn Only Green Arrow?

As you drive around Billings, I'm sure you've run into several intersections without traffic lights that you feel could use a traffic light to help the flow of traffic. One such light would be the light by MetraPark.  As you are coming out of  the Heights and want to turn left into MetraPark, I noticed that there isn't a green arrow for those in the turn left only lane. After sitting through three green lights today (five O'clock traffic) I pondered the thought of why have they never put a green arrow on this light.

With so much stuff going on at MetraPark, like concerts, trade shows and this week the kick-off of Montana Fair, it would seem to me that it would be a no brainer to have a green arrow for that light. You have a turn only lane, so why not have the green arrow to help the flow of traffic?  I'm no engineer, and I'm sure that this question has been pondered and researched before. It just seems to make sense to me.  What are your thoughts?