A priori inverse gamma distribution in Bayesian queuing models

The Bayesian approach to construction of the queuing model $M\vert M\vert 1\vert 0$ is considered. Under the condition of the arrival and service rates' uncertainty, load parameters' characteristics for large groups of service systems or one system with varying parameters of functioning are studied. It is assumed that the a priori distributions of the model's main parameters are known. The paper extends the work devoted to the study of Bayesian queuing and reliability models. Assuming that one of the a priori distributions of the arrival and service rates is the inverse-gamma distribution and the other is the Fr$\acute{e}$chet distribution, the density, distribution function, and moments of the traffic intensity are calculated. The results are formulated in terms of a gamma-exponential function and can be used to study the relationship between two independent inverse gamma distributed random variables in various applied problems.