In ‎this ‎article, ‎queunig ‎model ‎‎$‎M/M/1‎$ ‎is ‎Considered, ‎in ‎which ‎the ‎innterarrival ‎of ‎customers ‎have ‎an ‎exponenial ‎disributon ‎with ‎the ‎parameter ‎‎$‎lambda‎$ ‎and ‎the ‎service ‎times‎ ‎have ‎an ‎exponenial ‎disributon with the ‎parameter ‎‎$‎mu‎$ ‎and ‎are ‎independent ‎of ‎the ‎interarrival ‎times.‎ ‎it ‎is ‎also ‎assumed ‎that ‎the ‎system ‎is ‎active ‎until ‎‎$‎T‎$‎. Then, under this stopping time Bayesian, ‎$‎E‎$‎-Bayesian and hierarchical Bayesian estima‎‏‎tion‎s of the traffic intensity parameter of this queuing model are obtained under the general entropy loss function and considering the gamma and erlang prior distributions for parameters ‎$‎lambda‎$ ‎and ‎‎$‎mu‎$‎, respicctively. Then, using numerical analysis and based on a new index, Bayesian, ‎$‎E‎$‎-Bayesian and hierarchical Bayesian estima‎‏‎tion‎s are compared.

‎The cost function and the system ‎s‎tationary‏‎ probability are two key criteria in the design of queuing systems. In this paper, the aim is to design a single server queuing models with infinite capacity, where the service times in the first model and the interarrival times in the second model are assumed to have an Erlang distribution. For this purpose, a new index based on the cost function and the system reliability probability is introduced, the larger of which indicates the optimality of the model. Several numerical examples and an applied example are presented to explain the computational details of the proposed method.