Time-headway distribution for random-sequential-update TASEP with periodic and open boundaries

The temporal-headway distribution for Totally Asymmetric Simple Exclusion Process (TASEP) with random-sequential update is investigated. Considering the stationary/steady state of the process, exact formula for the step-headway distribution is derived for conditions when the stationary measure is Bernoulli, i.e., for periodic boundaries and for open boundaries with entering boundary rate α and leaving boundary rate β satisfying α+β=1. The step-headway formula for general values of boundary rates is calculated numerically by means of the matrix product ansatz.The formula is applicable mainly for the model defined on finite small lattice representing short segment of complex network. In this case the dependency of the motion of individual particles is noticeable and cannot be neglected. The finite lattice results are compared to continuous time distribution obtained by means of the large L limit.It can be observed that the scaled distribution converges quite fast to continuous time distribution. However, in the case of rather small lattice the distribution significantly differs from the limiting one. Moreover, in the case of Bernoulli stationary measure, the distribution is not dependent on the position of the reference site on the lattice. Considering general values of boundary parameters, the shape of the distribution is influenced by the density profile of the process near boundaries. This influence vanishes with increasing lattice size.