In a call center, there is a natural trade-off between minimizing customer wait time and fairly dividing the workload amongst the agents of different skill levels. The relevant control is the routing policy; that is, the decision concerning which agent should handle an arriving call when more than one agent is available. We formulate an optimization problem for a call center with two heterogeneous agent pools, one that handles calls at a faster speed than the others, and a single customer class. The objective is to minimize the steady-state expected customer wait time subject to a "fairness" constraint on the workload division.

The optimization problem we formulate is difficult to solve exactly. Therefore, we solve the diffusion control problem that arises in the many-server heavy-traffic QED limiting regime. The resulting routing policy is a threshold policy that prioritizes faster agents when the number of customers in the system exceeds some threshold level and otherwise prioritizes slower agents. We prove our proposed threshold routing policy is near-optimal as the number of agents increases, and the system's load approaches its maximum processing capacity. We further show simulation results that evidence that our proposed threshold routing policy outperforms a common routing policy used in call centers (that routes to the agent that has been idle the longest) in terms of the stead-state expected customer waiting time for identical desired workload divisions.

-- This is joint work with Mor Armony