Two Way Communication Retrial Queues with Balanced Call Blending

In call centers, call blending consists in the mixing of incoming and outgoing call activity. Artalejo and Phung-Duc recently provided an apt model for such a setting, with a two way communication retrial queue. However, by assuming a classical (proportional) retrial rate for the incoming calls, the outgoing call activity is largely blocked when many incoming calls are in orbit, which may be unwanted, especially when outgoing calls are vital to the service offered. In this paper, we assume a balanced way of call blending, through a retrial queue with constant retrial rate for incoming calls. For the single server case (one operator), a generating functions approach enables deriving explicit formulas for the joint stationary distribution of the number of incoming calls and the system state, and also for the factorial moments. This is complemented with a stability analysis, expressions for performance measures, and also recursive formulas, allowing reliable numerical calculation. For the multiserver case (multiple operators), we provide a quasi-birth-and-death process formulation, enabling deriving a sufficient and necessary condition for stability in this case, as well as a numerical recipe to obtain the stationary distribution.