Two-Way Communication M/M/1 Retrial Queue with Server-Orbit Interaction

This paper considers an M/M/1 retrial queue where the server not only receives incoming calls but in idle time makes outgoing calls of two types whose durations follow the same exponential distribution. The outgoing calls of type 1 are directed to the customers in orbit while the outgoing calls of type 2 are directed to the customers outside the orbit. Using the generating function approach, we derive explicit expressions and recursive formulas for the stationary joint distribution of the number of customers in the orbit and the server state as well as for the partial factorial moments. A closed form formula for the mean number of customers in the orbit is also obtained.