Well, there are two things which determine the promotion rate - attrition and corps growth. We can do basic analysis of each of these separately, and then look at combining them:
1. Assume the officer corps is in a steady state. Officers are being introduced as fast as they are attrited by age, accidents, medical conditions, and lack of requirement. In this case, the promotion rate will be based entirely on the average lifespan of officers. Let's say it's 20 years. In this case, assuming that attrition is constant at all ranks (a bad assumption, but I'm doing this on a piece of paper for fun), then each rank will lose 5% of its strength every year to attrition. Which means that (5/3)% of officers of each rank will be promoted each year to the next rank up, plus those to replace losses of officers at that rank to the next rank up, and so on. In practice, this will increase the percentage loss for promotions to 2.5% in the lower ranks. So you'll see 7.5% turnover in each rank each year, and a 1 in 40 yearly chance of promotion for officers, or 1 per 480 officers per month. Doubling the lifespan of the officers to 40 years would drop this to 1 per 960 officers per month, but the number of officers in the corps for a given number of academies would double.
In practice, you're going to see quite a few officers attrited out at 5 years or so due to non-assignment (because they just didn't have the skills) while others can easily pass 40 years. I don't have the time to put together a better analysis of this case.
2. Assuming that there's no attrition, and promotion only comes from corps growth. In this case, there's a total of 1 promotion per 2 officers added. (One-third of officer additions cause a promotion from O-1 to O-2, but 1/3rd of those cause a promotion from O-2 to O-3, and so on. Those of you who had calculus just realize that we're doing a sum of a series, and wincing. The sum comes to 1/2.) In this case, the total size of the officer corps is irrelevant.
That said, I'm in favor of adding medical technology to the tech tree. If nothing else, it would make biology researchers more useful.