Scaling limits for single server retrial queues with two-way communication

Abstract

This paper studies M/G/1 retrial queues in which there are two arrival flows, i.e., incoming calls made by regular customers and outgoing calls made by the server in idle time. The stationary analysis of this system has been carried out in a recent paper by Artalejo and Phung-Duc (Appl Math Model 37(4):1811–1822, 2013). In this paper, we obtain a decomposition property where we prove that the queue length is decomposed into the sum of three independent random variables with clear physical meaning. We then derive scaling limits for the queue length distribution under some extreme conditions (i) heavy traffic, (ii) slow retrials and (iii) instantaneous connection to outgoing calls. Furthermore, we also investigate the convergence of our model to that without outgoing calls.