Blended call center with idling times during the call service

We consider a blended call center with calls arriving over time and an infinitely backlogged amount of outbound jobs. Inbound calls have a non-preemptive priority over outbound jobs. The inbound call service is characterized by three successive stages where the second one is a break; i.e., there is no required interaction between the customer and the agent for a non-negligible duration. This leads to a new opportunity to efficiently split the agent time between inbound calls and outbound jobs. We focus on the optimization of the outbound job routing to agents. The objective is to maximize the expected throughput of outbound jobs subject to a constraint on the inbound call waiting time. We develop a general framework with two parameters for the outbound job routing to agents. One parameter controls the routing between calls and the other performs the control inside a call. We then derive structural results with regard to the optimization problem and numerically illustrate them. Various guidelines to call center managers are provided. In particular, we prove for the optimal routing that at least one of the two outbound job routing parameters has an extreme value.