Motor proteins, also known as biological molecular motors, play important roles in supporting and maintaining various processes inside the cell. Molecular motors perform tasks such as copying and repairing nucleic acids, intracellular transportation, transfer of genetic information, synthesis of proteins and nucleic acids, muscle functioning, cell motility, and signaling. Motor proteins use the hydrolysis of adenosine triphosphate (ATP) or other similar molecules to convert the chemical energy into mechanical work; the energy supports specific functions of the given molecular motor. Motor proteins have been extensively studied in the past decades, both experimentally and theoretically; however, their dynamics is not fully understood, especially when considering their collective behavior. It is widely accepted that the majority of motor proteins functions in groups, and interactions between individual molecules determine the cooperative behavior of molecular motors. These interactions have been measured for kinesin motor proteins, although the results are contradictory. The importance of interactions for motor proteins led to multiple theoretical investigations that aimed to understand the role of interactions in the collective dynamics. Most of them utilized totally asymmetric simple exclusion processes (TASEPs), which are non-equilibrium multi-particle models that have been widely employed to analyze various dynamic processes in Chemistry, Physics, and Biology. We developed a theoretical framework to investigate the collective dynamics of molecular motors, and were able to create a more fundamental approach to describe these interactions. Simultaneously, we also included other features into the model trying to have a more realistic description of biological transporters. We included irreversible dissociations, variable particle size, as well as the relaxation time analysis of the system. Our theoretical calculations generally agree well with our extensive Monte Carlo simulations results, suggesting that our approach is good to describe the main features of the processes that such enzymatic molecules undergo.