Gamma-Weibull a priori distributions in Bayesian queuing models

This paper considers the Bayesian approach to the queueing theory and reliability theory. Within the framework of the Bayesian approach, it is assumed that the key parameters of classical systems, such as the input flow intensity and the service intensity, are random variables with known a priori distributions. It is reasonable to use the Bayesian approach in the studying of the systems which are of the same type as well as in the studying of one system where changes of its characteristics happen at unpredictable moments of time. The randomization of the system's key parameters leads to the randomization of the system's characteristics, for instance, the traffic intensity. In the paper, the analytical results for the traffic intensity probability characteristics in the case of the gamma and Weibull a priori distributions of $M/M/1/0$ system's parameters are presented. The obtained results are formulated using the gamma-exponential function.