And likewise, the ants will herd the queen to ensure she stays put. Bees are the same way, it's hardwired instead of one controlling the other. Swarming, when a queen leaves the hive, is a joint "decision" caused by hivewide pheromones as a result of overcrowding. New queens are produced by workers when a queen leaves or dies, and the virgin queens will fight to the death, with the winner flying off, mating with drones to receive a lifetime supply of bee jizz, and flying back to serve the colony by producing new eggs for a few years.
Can't recall if the bees will ever kill off an aging queen or not, but it wouldn't surprise me, since they're capable of producing new ones as long as they have eggs that are less than 3 days old, since the introduction of royal jelly will affect development in a way that produces a new queen rather than a sterile female worker.
Bee sexuality is actually incredibly fucked up. Males, or drones, are produced simply by unfertilized eggs. The queen will select whether to lay a fertilized or unfertilized egg based on the honeycomb - drones are larger and require larger cells; you can actually force drone production by providing a baseboard with larger cell sizes (plastic sheet with raised honeycomb pattern that the bees build off of). If a queen dies and there are no suitable eggs for a new queen, worker bees will actually develop to lay drone eggs themselves, but will be incapable of laying fertilized worker eggs to continue the colony, as they have never mated and are incapable of mating. The hive will eventually die, as there's no new source of worker bees.
Now, as far as slavery and control and whatnot, here's another weird tidbit. You can introduce a new queen to a hive that has lost theirs. You can even separate and kill the queen if she's too old. Monitor the hive and destroy any queen cells you find for a few days, and then you can introduce a new queen manually (usually done to change breeds). But the bees will attack foreign queens, so you use a little wooden box with screening on the sides and a candy plug. There's a queen and a few attendant bees from the queen's hive. When the bees pick up queen pheromones in a queenless hive they'll initially be hostile, but can't attack through the screening. Over the course of the couple days that it takes them to eat away the sugar plug, the queen will adapt to their pheromones and go native, or the bees will get used to the queen's pheromones and turn friendly. When that happens, they'll accept her as a new queen. If you try to introduce a new queen to a hive that already has one, though, the old queen will seek her out and kill her.
So in the end it's far more complex than just one controlling the other. The system is best thought of as a true collective, where the queen is nothing but another asset to the hive as a whole.
t. had a dozen hives when I was a kid