This master thesis examines nonequilibrium, one-dimensional traffic phenomena modeled as lattice gas systems, where particles move on a discrete lattice. Movement follows a continuous time Poisson process. The focus is on the Totally Asymmetric Simple Exclusion Process (TASEP), where particles hop in one direction only if the landing site is empty, and a derived model, pausing-TASEP (pTASEP), where particles can stochastically pause and resume motion. The work is divided into two parts: approximate mean-field theories and exact combinatorial results. The thesis first explores pTASEP as a model for translationally-inhibited protein synthesis, where ribosomes on mRNA (the lattice) face pauses due to antibiotics.