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In this paper, the stability of the uniform solutions is analysed for microscopic flow models in interaction with predecessors. We calculate general conditions for the linear stability on the ring geometry and explore the results with particular pedestrian and car-following models based on relaxation processes. The uniform solutions are stable if the relaxation times are sufficiently small. However the stability condition strongly depends on the type of models. The analysis is focused on the relevance of the number of predecessors K in the dynamics. Unexpected non-monotonic relations between K and the stability are presented. Classes of models for which increasing the number of predecessors in interaction does not yield an improvement of the stability, or for which the stability condition converges as K increases (i.e. implicit finite interaction range) are identified. Furthermore, we point out that increasing the interaction range tends to generate characteristic wavelengths in the system when unstable. [ABSTRACT FROM AUTHOR]

A class of microscopic stochastic models is proposed to describe 1D pedestrian trajectories obtained in laboratory experiments. The class contains continuous first-order models that are based on statistically calibrated optimal velocity functions. More specifically, we consider a model with an additive white noise and another one where the noise is determined by the inertial Ornstein–Uhlenbeck process. Simulation results show that both stochastic models give a good description of the characteristic relation between speed and spacing (fundamental diagram) and its variability. However, only the inertial noise model can reproduce the observed stop-and-go waves, bimodal speed distributions, and non-zero speed or spacing autocorrelations. This allows us to identify minimal microscopic stochastic mechanisms for the emergence of stable traffic waves. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional diffusion in a bounded domain with stochastic resetting. We start our analysis by presenting a method to derive the master equation for different resetting mechanisms. In the next step we compute the non-equilibrium steady state for a special case of this differential equation. Then we consider the existence of an absorbing point in the system and calculate the mean time to absorption of the diffusive particle by the target. Numerical and analytical calculations of the optimal resetting rate reveal a second order phase transition. Finally we discuss different special cases of the presented problem. [ABSTRACT FROM AUTHOR]