Call centers with redirected customers

We consider a queueing system where some of the customers are redirected from one queue to the other. We distinguish between those who are redirected and those who exit the system after being served by their service rate. Although it is assumed that all external arrived and services are Markovian this is not a Markovian system as the inter-departure of the redirected type from one queue (hence the inter-arrival to the other) is a non-Markovian. In fact, it is not even a renewal process. Using a novel approach, we obtain an almost identical PH representation of the inter-departure process under equilibrium. We then provide a tool in which we assess whether or not such systems can be analyzed when treating the inter-departure as if it was a renewal process. Finally, we analyze such queueing systems and provide intriguing observations and counter-intuitive results