E-Bayesian and hierarchical Bayesian estimation of performance measures in Geo/Geo/1 queueing system

This paper develops Bayesian, E-Bayesian, and hierarchical Bayesian estimators for key performance measures of a discrete-time Geo/Geo/1 queueing system. Specifically, we focus on the probability that the server is busy, the average queue length, and the expected number of customers in the system. Bayesian estimators are derived under the squared error loss function using conjugate Beta priors, leading to closed-form expressions. To account for uncertainty in hyperparameter specification, E-Bayesian estimators are obtained by averaging the Bayes estimators over suitable prior distributions on the hyperparameters, while hierarchical Bayesian estimators are constructed by introducing hyperpriors directly into the model. Analytical results establish ordering relationships among the E-Bayesian estimators corresponding to different hyperpriors and demonstrate their asymptotic equivalence. Moreover, the E-Bayesian and hierarchical Bayesian estimators are shown to be asymptotically equivalent as the sample size increases. A comprehensive Monte Carlo simulation study is conducted to examine the finite-sample performance of the proposed estimators. The results indicate that the E-Bayesian estimators generally achieve smaller standard errors than their Bayesian and hierarchical Bayesian counterparts, particularly for moderate sample sizes.