Two elevators serving up-traffic

A model is described for the motion of two elevators serving traffic only from a single lobby floor to multiple higher floors. Over a certain range of passenger arrival rates, the average round trip time of an elevator can be approximated by a linear function of the number of passengers carried which, in turn, is proportional to the time since the last elevator departure from the lobby (the headway). A pair of deterministic equations for the means of two state variables describes the approximate evolution of the system behavior. These equations lead to a very strange chaotic type of motion in the two-dimensional state space. For moderately light traffic the two elevators tend to travel close together, but for heavy traffic the headways tend to be nearly uniformly distributed over the mean round trip time of a single elevator.