Inspired by the huge relevance of simple exclusion processes models as biological traffic models or vehicular traffic as well, we will focus our work on the Totally Asymmetric Simple Exclusion Process (TASEP). The latter is defined on a one dimensional lattice, where each site could be occupied or not by only one particle. The particles can hop from left to right with a given hopping rate provided that the lattice site on the right is empty. There is moreover a specified entry rate in the first lattice site where particles can enter in the system and exit rate from the last one where particles can exit the system.