The rental queue works like first-comes first serve basis. And the first-come means whoever registered first on the rental queue site. The main queue is maintained by Stockholm State. There is no way for anyone to pay more money and get ahead in queue.
Apart from that, there are always new buildings which have no queue but rent is more (40 or 50% more than usual. Anyone who is willing to pay that amount can get an apartment within 2 months time frame.
Then there is a 3rd black market. A lot of people sublet there for shorter and some for longer terms.
Having said that all, I've thought about this problem but couldn't come to the prefect solution. When state controls the rents & the process renting, it gives guarantee to low-income families that they can also live in nice neighborhoods.