In classical queueing systems, customers wait in line to receive service. However, some customers prefer to join multiple queues simultaneously for the same service to obtain it as quickly as possible. Once a customer receives service from one of the queues, they immediately leave the others, thereby minimizing their waiting time. Based on this concept, this paper considers two $M/M/1$ queueing models with arrival and service rates $(lambda_1, mu_1)$ for the first model and $(lambda_2, mu_2)$ for the second. The minimum average waiting time in these two models is estimated using Bayesian, $E$-Bayesian, and Hierarchical Bayesian approaches under the General Entropy Loss Function. Furthermore, Monte Carlo simulations and a real dataset are employed to evaluate and identify the most appropriate estimator.