Two old methods - smoking and shaking
Here are two methods of introducing queens that were taught to me by old beekeepers when I started beekeeping. They probably didn't know the principles of what they were doing, but I suppose by trial and error they found ways of doing things that worked.
I have used both these methods on many occasions and they are as successful as any I have used. I have no doubt they will be frowned upon by some who think they know more than others - probably because they aren't in the books they have read!
For this method to be successful the colony needs to be queenless for at least 4 days. It will probably help if the queen to be introduced has been without food for half an hour or so.
The colony needs to be queenless. I have done it soon after removing the queen, but I think it better to wait a couple of hours after removing the queen, so the bees are frantic. It may also be better if the queen is hungry.Put a sloping board in front of the hive in the same way you would when hiving a swarm. The longer the board, the better.
I understand that some beekeepers may not wish to try these methods, but I find they work. If you are a bit doubtful, then try with a queen you wish to cull.