The goal of this paper is to introduce bounded rational behaviors in a competitive queuing system. Furthermore, we propose a realistic queuing model for two last-mile delivery services in which consumers are in competition. This work is derived from a real-world e-commerce application. We study the problem using a game theoretical point of view: the e-consumers are interacting through the last-mile delivery service system creating congestion for each other. Specifically, we focus our analysis on several equilibrium concepts from congestion/routing games: Wardrop and Logit equilibria. The difference in these equilibrium concepts is on the rationality level of players in the game. We are able to prove the existence and uniqueness of both equilibria. We compare them through a new metric called the Price of Rationality and we also compare each one to the social optimum solution through the Price of Anarchy. Some numerical results are presented in order to illustrate the theoretical results obtained.