Steady-states and kinetics of ordering in bus-route models: connection with the Nagel-Schreckenberg model

A Bus Route Model (BRM) can be defined on a one-dimensional lattice; where buses are represented by "particles" that are driven forward from one site to the next with each site representing a bus stop. We replace the random s equential updating rules in an earlier BRM by parallel updating rules. In o rder to elucidate the connection between the BRM with parallel updating (BR MPU) and the Nagel-Schreckenberg (NaSch) model, rye propose two alternative extensions of the NaSch model with space-/time-dependent hopping rates. Ap proximating the BRMPU as a generalization of the NaSch model, we calculate analytically the steady-state distribution of the time headways (TH) which are defined as the time intervals between the departures (or arrivals) of t wo successive particles (i.e., buses) recorded by a detector placed at a fi xed site (i.e., bus stop) on the model route. We compare these TH distribut ions with the corresponding results of our computer simulations of the BRMP U, as well as with the data from the simulation of the two extended NaSch m odels. We also investigate interesting kinetic properties exhibited by the BRMPU during its time evolution from random initial states towards its stea dy-states.