Dynamic Skill-Based Routing for Call Centers with Transfers: A Value-Based Framework

We consider a multi-class, multi-pool queueing model of a telephone call center in which a service completion does not always end the call. The caller may be transferred to another agent for a second service, so the assignment of an agent to a call affects not only the caller's waiting time but also whether the call is resolved and how much further work it creates. We model these transfers as Markovian routing from original to rework classes, and the system manager dynamically assigns agent pools to classes so as to maximize the value generated by service, net of transfer penalties and waiting and abandonment costs, over a finite horizon. Focusing on the Halfin-Whitt heavy-traffic regime, we derive an approximating Brownian control problem. Its Hamilton-Jacobi-Bellman equation is a semilinear partial differential equation whose dimension equals the number of classes. Building on the deep splitting method of Beck et al. (2021), we develop a simulation-based computational method that approximates the value function and its gradient by neural networks, and we use it to construct an implementable routing policy for the original (prelimit) system. We calibrate the model to publicly available data from a large U.S. bank call center. In all test problems we consider, the proposed policy outperforms the best available benchmark policy we could find.