Parallel TASEP on a ring: blockage problem and non-analyticity of the current

The Totally Asymmetric Simple Exclusion Process (TASEP) is an important example of a particle system driven by an irreversible Markov chain. Assume $$n$$ particles are placed on a ring of length $$2n$$, at each step particles free to move in the clockwise direction occupy the free site independently with probability $$p$$. In this parallel framework I will do a simple yet rigorous derivation of the chain stationary measure. I will next consider the blockage problem (a.k.a. slow bond problem), deriving the exact expression of the current for an arbitrary blockage intensity $$\varepsilon$$ in the case of the so-called rule-184 cellular automaton, i.e. a parallel TASEP with $$p=1$$. Finally, I will discuss through numerical experiments the conjecture that for parallel updates other than rule-184 the current may be non-analytic in the blockage intensity around the value $$\varepsilon=0$$.