When breaking glass tubing or rod be sure to pull it apart as you snap it.
For flame-polishing, a laboratory Bunsen burner flame is about as large and as hot as you want to go because with more heat things will happen too quickly and over too large an area. The flame is soft and easily deflected by a draught eg if someone is walking about nearby. This would not be good for the glass or for your hand.
Rotate the tube end two or three inches above the top of the flame to warm it up slowly. There is a real possibility of a little bit of hot glass flying off, so wear eye protection. Make sure that you can see the end face that you want to melt.
Gradually lower the end – only the end – into the hot spot in the flame. As soon as you see the very end beginning to soften and smooth out return it to the hot air above the flame, still rotating, and let it sit there for half a minute or so then gradually lift it out of the hot air. The actual softening of the end will take only about 5seconds depending on the flame source. If you leave it 'cooking' for too long it will soften too much and melt back and develop a fat ring around the end, inside and out, which will somehow have to be ground off if you want a consistent o.d. Or perhaps you use forgiving O-rings?
Don't hurry the slow cooling: just letting it quench in air straight from the flame will introduce invisible strains in it and make it more sensitive to shock and liable to crack later.
This gentle heating and cooling are not quite so important if it is Pyrex-type glass, but then that wouldn't easily soften in a bunsen.