So I did a couple of posts about promotions but didn't mention anything about my own basis for promoting people. As this varies between every instructor it's definitely worth a mention.
First off, I have only given out two purple belts so far and nothing higher. I can't promote anyone to black belt for another 6 years and don't plan on promoting anyone to brown belt any time soon either, so I'll just look at blue and purple...
For both blue and purple, the most important aspect is always grappling performance; how does someone perform against their peers and a wider range of opposition? I judge this in a variety of situations... sparring in training, jiu-jitsu competition and even MMA. My judgement of performance isn't only based on results though, no doubt it's important to be able to get the better of someone in any sort of fight but I would never promote someone just because they can beat people using strength, they have to be using technique.
With that in mind, performance against peers is what I look at most closely. How somebody does against people of similar size, age and experience is a great indicator of ability. But also important is how they do against bigger/stronger opponents... and for that I definitely just differently depending on belt. For purple belt I would expect to see not only better technical ability but also a broader understanding of different tactical approaches to take against different opponents. If someone can beat similar sized opponents but then get smashed by anyone bigger because they are trying to triangle 130kg monsters, they are not due a purple belt imo.
Within the performance aspect there are also other factors... does someone just use the same techniques all the time? A limited range of techniques could get someone a blue belt, but for a purple I would definitely expect to see them using and experimenting with a wide variety of options, especially when rolling with white belts. For a blue belt someone might be good at triangles but not have that many options off it, that's cool... but if I am considering someone for purple belt I would expect them to have options for every triangle escape or defence. Can someone fight from every position? Being able to pass guard and tap every white belt is great, but if they end up on the bottom and are lost they are not ready to be promoted. For blue belt I would expect someone to have options from every position, for purple belt I would expect them to have a range of high percentage options from every position. Cardio is another issue of performance... being able to go flat out and smash any white belt in one round is no good if that person is then gassed and unable to roll the next round. No belt is going to be given to someone who doesn't roll every round.
There are then factors outside of performance... how does someone behave towards other students? I have no problem with people smashing in sparring, but if someone takes it to the level of bullying they show they are not mature enough to be given a higher grade. If they are the sort of person who tries to teach every new student an americana from mount during drilling/rolling then they are not going to be promoted. If they ignore advice and always think they know better, no promotion. If they are lazy and skip drilling, no promotion. If they have a generally bad attitude (talking while I'm teaching, whinging about stuff or whatever), no promotion.
I always consider time spent training in promoting too. I don't really have any minimums, and I definitely don't have timescales where I automatically start to consider promotion. But, if someone is performing to a standard I consider worthy of blue or purple, and they've got their in a short amount of time, I would expect them to have a history of competition success. If not there is no need to get promoted quickly.
And that takes us back to competition... I see competition success as an indicator I need to evaluate that student for promotion, but by no means is it a guarantee of promotion. Someone might win numerous competitions but not qualify for promotion due to any other reasons I've covered above.
Other people promote based on different factors though, or give different weighting to the factors I have spoken about... and after posting this I'll probably think of something I've forgotten about. Whatever, it's a bit of insight as to how someone promotes people... just keep training, keep trying to learn new techniques, compete, think about your training and how to train better... promotions will come in time.
EDIT: something I forgot to mention originally is what happens when people have time off, or haven't been training regularly. Again, in these instances I will not promote anyone. If someone has been off for months injured, it sucks for them but I will not promote them (not even a stripe) as soon as they get back. They will need to get back into regular training for at least a few months before any promotion. People may think this is harsh, but people I promote represent me from that point on. I don't want to give a belt to someone the first day they come back from a lay off, no matter why, in case I never see them again.
A similar thing applies to anyone who joins me from a different team or with existing grappling ability. No matter what their level I would not promote them until they have trained with me for a long time. I know instructors out there who promote people within weeks or months of them joining their team (or even at the first session!!). All this does is encourage people to "belt shop" moving from one instructor to another, often getting them in for high cost seminars, in order to quickly gain belt ranks. Nobody will ever achieve that through me.