I believe Southwest uses a system called SAAS, which they call "a Sabre Product" though it is actually based on the old Braniff Cowboy system and modified in-house. I remember talk of them switching to Sabre proper, but it's been years since I've been in that industry. Sabre I believe was made by some IBM engineers originally.

Some Relevant Links if you want to dig in, working with these systems was a lot of fun (I worked at several airlines):

Sabre: https://en.wikipedia.org/wiki/Sabre_(computer_system)

Global Distribution System: https://en.wikipedia.org/wiki/Global_distribution_system

Sabre Company site: http://www.sabreairlinesolutions.com/home/