Fair dynamic routing in large-scale heterogeneous-server systems: Technical appendix

In a call center, there is a natural trade-off between minimizing customer wait time and fairly dividing the workload amongst 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 other, and a single customer class. The objective is to minimize 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 num-ber 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 fur-ther show simulation results that evidence that our proposed threshold routing policy outper-forms a common routing policy used in call centers (that routes to the agent that has been idle the longest) in terms of the steady-state expected customer waiting time for identical desired workload divisions