Communications of the
Korean Mathematical Society CKMS

We consider M${}_1$,M${}_2$/M/1 retrial queues with two classes of customers in which the service rates depend on the total number of the customers served since the beginning of the current busy period. In the case that arriving customers are blocked due to the channel being busy, the class 1 customers are queued in the priority group and are served as soon as the channel is free, whereas the class 2 customers enter the retrial group in order to try service again after a random amount of time. For the first $N$ $(N\ge 1)$ exceptional services model which is a special case of our model, we derive the joint generating function of the numbers of customers in the two groups. When $N=1$ i.e., the first exceptional service model, we obtain the joint generating function explicitly and if the arrival rate of class 2 customers is 0, we show that the results for our model coincide with known results for the M/M/1 queues with smart machine.

Keywords: retrial queue, smart machine, balance equations

MSC numbers: Primary 60K25, Secondary 68M20