Optimizing the Estimation of the Traffic Intensity Parameter of Queuing Model ‎$‎M/M/1‎$ under Bayeian, E-Bayesian and Hierarchical Bayesian estimation Methods and based on Stationary Probability and System Cost ‎Function

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.