Summary: A totally asymmetric exclusion process with \(N\) particles on a periodic one-dimensional lattice of \(L\) sites is considered where particles can move one or two sites per infinitesimal timestep. An exact analysis for \(N=2\) and a mean-field theory in comparison with simulations show even/odd oscillations in the headway distribution of particles. The expression ‘headway’ is understood as the number of empty sites in front of a particle. Oscillations become maximal if particles only move at their maximum possible speed. A phase transition separates two density profiles around a generated perturbation that plays the role of a defect. The matrix-product ansatz is generalized to obtain the exact solution for finite \(N\) and \(L\). Thermodynamically, the headway distribution yields the mean-field result as \(N\to\infty \) while it is not described generally by a product measure.