Finite Automata (FA) and Pushdown Automata (PDA) is the computational model used in developing systems of different complexities. FA works well with static states and simple transitions which can be applied to elevator controls with simple operations such as moving floors and controlling door operations. However, being memoryless, it would not be able to satisfy dynamic requests, prioritize actions to be performed, or go back to floors that were skipped. PDA solves this problem by using a stack to store and process floor requests hierarchically, thereby making the system capable of dealing with nested and complex situations. This paper compares these two models based on parameters such as memory usage, design complexity, real-world applicability, and computational efficiency. FA is found to be perfectly suited for simple state-driven operations with low computational overhead while PDA excels at managing advanced functionalities at the cost of increased complexity. This result gives insight into trade-offs between simplicity and capability for the selection of a proper computational model for an elevator system based on its operation demands.