Bayesian Estimation of an M/M/𝑅 Queue with Heterogeneous Servers Using Markov Chain Monte Carlo Method

Abstract In this paper, we consider the Bayesian inference of M/M/𝑅 queue with 𝑅 heterogeneous servers with service rates μ 1 , μ 2 , … , μ R \mu_{1},\mu_{2},\ldots,\mu_{R} , where μ 1 > μ 2 > ⋯ > μ R \mu_{1}>\mu_{2}>\cdots>\mu_{R} . Assuming multivariate gamma prior distribution for service rates and gamma prior distribution for arrival rate 𝜆, we derive the conditional posterior densities of mean arrival rate and mean service rates. We apply the Markov chain Monte Carlo method and compute the Bayes estimates and credible interval for the M/M/3 queue, as a particular case of the M/M/𝑅 queue under squared error loss function, entropy loss function and linex loss function corresponding to a different set of hyperparameters.