Imbricating tasks in a multichannel contact center

Motivated by the call center practice, we consider a blended call center with calls arriving over time and an infinitely backlogged queue of emails. Calls have a non-preemptive priority over emails. The 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 nonnegligible duration. This leads to a new opportunity to efficiently split the agent time between calls and emails. We focus on the optimization of the email routing to agents. Our objective is to maximize the throughput of emails subject to a constraint on the call waiting time. We develop a general framework with two parameters for the email routing to agents. One parameter controls the routing between calls, and the other does the control inside a call. In particular, we prove for the optimal routing that all the time at least one of the two email routing parameters has an extreme value.