Unlike other Martial Arts, BJJ Belt promotions do not use tick boxes that can allow you to get promoted quickly through the belt rankings (Think Taekwondo or Karate). Therefore there really is no easy way to fast track promotions.
On average, it takes around 2 years. But it could take a lot longer, or a bit quicker depending how much you are training. Your Professor will know, and understand you, where you are at regarding your overall potential. They will know when to promote you. You will be ready when they say you are. Asking to be promoted can often lead to the professor stalling the promotion as you have obviously not learnt to be humble. Brazilian JiuJitsu is a Martial Art that, whilst ranking is important, the longer you are a participant, the more you realise that the whole thing is about self growth. The belt colour matters less and less. (Obviously you respect everyone where they are at, and a promotion is a seriously important event!)
Well known navy seal and American decorated soldier Jocko Willink is often looked up to in the BJJ world. His overall attitude to life is something to take note of, and his idea of extreme ownership influences many in and out of the BJJ world.I will add more info and background on Jocko over time, but in the meantime go google him, and check out his Wikipedia page.
This interview on the BJJ belt promotion process goes a long way to explore when and why you should be promoted.