I was in your situation when I first started BJJ. I had no grappling experience whatsoever and I think it took me about 6 months before I ever got a submission. To be honest, it really depends on who you train with. At that BJJ dojo everyone there had at least a year's worth of experience on me and no new guys ever joined. However, when I would go to my college's judo club I could tap a lot of people out on the ground (after I was thrown of course
).
Now I attend a different BJJ dojo and have close to 2 years of total BJJ experience. I'm still a white belt but now I can tap most of the other white belts and some blues belts at my school.
Experience and mat time is what really counts. I'm sure right now the blue belts at your school seem godly, but if you give yourself a year, you'll be giving them a run for their money.