This paper investigates Bayesian estimation of the performance measures of a machine inference queuing model under a geometric distribution-based framework, assuming the system operates in steady state. The primary parameters of interest are the traffic intensity ρ and the expected number of customers in the system Ls. Four prior distributions are considered: a Beta prior of the second kind (informative, natural conjugate), a Gamma prior (informative, non-conjugate), and Jeffreys' prior (non-informative). Bayesian estimators for ρ and Ls are derived under two loss functions—the Squared Error Loss Function (SELF) and the Precautionary Loss Function (PLF)—together with their corresponding posterior risks. Equal-tailed credible intervals for ρ are obtained from all four posterior distributions. Predictive distributions for future observations are derived under each prior, enabling posterior model comparison via Bayes factors following Jeffreys' (1998) decision rule. Relative efficiencies are assessed using mean squared error (MSE) via Monte Carlo simulation across different sample sizes and traffic intensities. Results indicate that all estimators converge to the true value as n increases, that the informative Beta prior consistently yields lower posterior risk when hyperparameters are well-specified, and that PLF estimators are uniformly larger than their SELF counterparts, providing a conservative guard against underestimation.