We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a back-pressure (BP) algorithm (also often called MaxWeight). There are N sites (with N finite or infinite), each may contain at most c particles, \(1 \le c < \infty \) . New particles enter the system at the left-most site 1 as a Poisson process of rate \(\alpha \le 1\) , unless site 1 has c particles. Particles (if any) are removed from the right-most site N as a Poisson process of rate \(\beta \le 1\) . The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site n to \(n+1\) at epochs of a rate 1 Poisson process, as long as the former site has strictly more particles than the latter. When \(c=1\) , this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as \(N\rightarrow \infty \) . In particular, we prove that interesting non-trivial phase transitions take place in a system with \(c>1\) . For example, if \(c>1\) and \(1/2 \le \beta \le 1\) , the maximum limiting flux 1/4 is achieved as long as \(\alpha \ge \alpha _c^*\) , where \(\alpha _c^* < 1/2\) is some non-trivial threshold. (For the standard TASEP the threshold is 1/2.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.
