whatOne server, random arrivals, random service. The knob is utilization, rho: the fraction of the server's capacity the arrivals consume. Each waiting job is one stroke; the circle is the server, its inner arc the job draining now; the trace top right is every finished job's time in system. Time runs eightfold, so the averages can converge while you watch.

whyBecause the wait explodes nonlinearly. The law says the average line is rho over one minus rho, so nudging ninety to ninety-five percent, with the server still not full, roughly doubles the queue. The hyperbola bottom right collects this queue's own time-averaged ticks at every utilization visited, and they land on the curve. Every on-call rotation and hospital hallway learns this law the hard way.

howAn M M 1 queue: Poisson arrivals, exponential service, one server, drawn from one seeded random stream, so the same seed replays the same rush. Click sends twenty jobs at once; near saturation the backlog lingers long after. The chart never forgets: each utilization bin keeps averaging for as long as you have ever sat there.