Retrial queues with balanced call blending: analysis of single-server and multiserver model

Abstract

In call centers, call blending consists in the mixing of incoming and outgoing call activity, according to some call blending balance. Recently, Artalejo and Phung-Duc have developed 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 short- term blending balance is heavily impacted by the number of incoming calls, which may be undesired, especially when the balance between incoming and outgoing calls is vital to the service offered. In this contribution, we consider an alternative to classical 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 to derive 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. A correlation study enables to study the system's short-term blending balance, allowing to compare it to that of the system with classical retrial rate. For the multiserver case (multiple operators), we provide a quasi-birth-and-death process formulation, enabling to derive a sufficient and necessary condition for stability in this case (in a simple form), a numerical recipe to obtain the stationary distribution, and a cost model.