Certainly Chris Johnson's contract and body of work will earn him the majority of snaps in the 2013 season, but what kind of action can we expect from newly added RB Shonn Greene? The way I see it there are three different models for how Mike Munchak and Dowell Loggains might use him based on his performance in mini-camp and OTAs.
Option 1 for a bad offseason: Use him like Javon Ringer in 2010
In 2010 Ringer had only 51 carries for about 250 yards behing a bad OL, and was primarily a backup that relieved CJ after he had taken a beating or when he was exhausted. If Greene has a bad offseason and looks like he can't carry a load I think either Jalen Parmele or Darius Reynaud would also see 50+ carries in this scenario given how much it looks like the Titans plan to run this season.
Option 2 for a middling offseason: Use him like Lendale White in 2009
Lendale White was the perfect three-yards-and-a-cloud-of-dust running back in 2009. He had 64 carries and was the clear short down yardage back. If Shonn Greene shows promise but doesn't wow the Titans coaching staff then they will be comfortable creating a role for him as a short-yardage back to grind between the tackles.
Option 3 for a great offseason: Use him like Lendale White in 2008
There was a time when the Titans used to give running backs entire drives before they would pull that running back out. During that time it was more of a running back A/running back B situation for the first half and then they would favor the RB that had more success in the first half. If Greene can show that he can handle a long season and contribute in every phase of the offense. If he can wow coaches during camp he can prove worthy of Lendale White's 200 carry season.
It would be best for the Titans if the third option worked out but that is a very optimistic take, it is more likely that the second option will occur. Either way as long as the Titans don't resort to option one then it will be a successful signing in my opinion.