Steady state speed distribution analysis for a combined cellular automaton traffic model

Cellular Automaton (CA) based traffic flow models have been extensively studied due to their effectiveness and simplicity in recent years. This paper develops a discrete time Markov chain (DTMC) analytical framework for a Nagel--Schreckenberg and Fukui--Ishibashi combined CA model (W$^2$H traffic flow model) from microscopic point of view to capture the macroscopic steady state speed distributions. The inter-vehicle spacing Markov chain and the steady state speed Markov chain are proved to be irreducible and ergodic. The theoretical speed probability distributions depending on the traffic density and stochastic delay probability are in good accordance with numerical simulations. The derived fundamental diagram of the average speed from theoretical speed distributions is equivalent to the results in the previous work.