On the M/G/1 Bernoulli feedback queue with multi-class customers

We consider an M/G/1 queue with Bernoulli feedback policy and multi-class customers. We assume that different classes of customers have different arrival rates, service time distributions and feedback parameters. We obtain the joint probability generating function of system size of each class and also obtain the Laplace-Stieltjes transform of the total response time of a customer in each class. Using these results we obtain the moments of system size of each class and total response time of a customer in each class. Scope and purpose We obtain the joint distribution of system size of each class and distribution of total response time of a customer in each class for M/G/1 Bernoulli feedback queue with multi-class customers which has been an open problem since Takacs, Bell System Technical Journal 1963;42:509-19, analyzed the classical M/G/1 Bernoulli feedback queue with single-class customers. We also obtain the mean and higher moments of the system size of each class and total response time for a customer in each class.