Every RPG out there has an initiative system. It is a necessary evil when it comes to playing games like Table Top RPGs. Necessary because when things get hectic in the game, there needs to be a system that figures out who can talk and whose actions resolve when. Evil because by their nature initiative systems take what is a very exciting and chaotic sequence of events and slows it down while chunking it into very orderly segments or bites.
However, while every RPG comes with a pre-packaged initiative system, I can't think of a single game where how the initiative system works is of fundamental importance to the system. And with the exception of 7th Sea 2nd Edition I have yet to see a game with an initiative system that can't be taken and used in other games. I could be off on both of those - my experience is limited - but in most games - especially the mainstream ones - you can change initiative systems with relative ease.
That said, let's talk about initiative systems and why you may want to change from the default in your game.