It exists as a dome on the edge of Hell. The area leading to it is a blasted plain, which would be faintly radioactive if it existed on the corporeal plane. There is actually a well-traveled road to one of the sally ports that exist along the edge of the dome: Superiors and tour buses from Shal-Mari use this route.

Inside, one finds a warren of passageways and buildings and barracks, colored in a garish array of colors. The level of repair depends on security clearance: Infrared regions are about to fall apart (but never quite do), while the elite Ultraviolet sectors are marvels of high-tech beauty. The sound of various weapons discharging is so ubiquitous that the inhabitants get fairly disturbed when it stops.

Life for an Infrared in Alpha Complex isn't actually bad, for Hell. They wander to their jobs, 'eat' bland food, and generally spend their existence under the influence of massive amounts of mood regulators and tranquilizers. Not an exciting life, but it beats getting regularly eviscerated. They exist as mere filler. The Computer extracts Essence from them (via the drug treatment) almost as an afterthought.

Demons have a much more exciting life. They make up the entirety of security clearances from Red to Ultraviolet, and run Alpha Complex (as much as anything can be said to run Alpha Complex). New demons are made Troubleshooters, of Red clearance.

It should be remembered that inhabitants of Alpha Complex believe that they are living humans (except for the Lilim, who keep their heads down, keep each other out of trouble out of sheer enlightened self interest, and make a mint herding around outside tourists). Due to the nature of the Principality, celestial combat is unknown. This means that all the shooting, burning, stabbing and nuking doesn't actually kill anyone: in self-defense, the inhabitants have deluded themselves that they are all members of identical clone families. When someone 'dies', everyone ignores the 'corpse' until it gets up again, then acts as if he or she was a replacement. When someone finally 'dies' a sixth time, that someone goes into a coma… and wakes up a whole new clone.

Somewhere, in all of this, is the Computer. Precisely where is not known to even those of Ultraviolet clearance (the most powerful Servitors of Paranoia, who have realized what's going on. They each control an individual part of the Principality). By now, it might have merged itself with its Principality. Or it might be ensconced in the vessel of a cleaning robot. It's not telling.