Bayesian analysis on single server Markovian queueing model with impatient customers

This paper focuses on Bayesian inference of an M/M/1 queuing model with balking, a phenomenon in which customers choose not to join a queue due to long waiting line. In this paper, the balking probability is considered as a function of number of customers and their impatience level. The degree of impatience of customers plays a crucial role effecting the balking probability. Higher threshold of impatience implies that customers are more sensitive to queue length, i.e. they are less willing to join a queue when the queue is even slightly longer. Conversely, lower threshold of impatience indicates that customers will less balk. In this scenario, there is a higher probability that customers will opt to join the queue, even when it extends to a considerable length. This paper provides the Bayesian estimates for traffic intensity (ρ), employing various prior distributions such as beta, truncated gamma, and uniform prior distributions under a squared error loss function. Through the sampling importance re-sampling (SIR) technique, we obtained the posterior estimates, risk, and credible intervals that showcase the effectiveness of our methodology. Furthermore, simulation studies demonstrate the convergence of estimators, and our findings are further validated through analysis of real-life data.