The Asymmetric Exclusion Process represents a fundamental family of models in out of equilibrium statistical physics, for which applications are found in a large variety of contexts, ranging from biology (e.g. the motion of ribosomes along mRNA or molecular motors on intracellular filaments) to electron hopping transport with applied voltage and vehicular highway traffic. The specific model we want to consider for this work is the Totally Asymmetric Simple Exclusion Process (TASEP). It is defined on a one dimensional lattice, where particles can hop only rightward, from each node to the next one, provided the latter is empty. Despite its simplicity, it can offer a very interesting series of results, depending on the characteristics with which we equip it.