Performance Evaluation in Small Hierarchical Call Centres in a Transient Regime: A Case Study

This paper addresses a fundamental and practically significant problem in call centre operations, determining optimal call allocation policies and evaluating their performance so as to meet client service targets while minimising staffing costs. Motivated by a problem presented by an industry partner, we examine a real-world setting involving a relatively small call centre with hierarchical structure among agents. It is natural to model the operation of such a centre as a continuous-time Markov chain. To gain insight into the structure of optimal policies, we first (i) apply backward induction based on Bellman’s equation to a finite-horizon discrete-time model, and (ii) derive stationary policies for an infinite-horizon continuous-time model with discounting. Subsequently, we evaluate the performance of these policies in the original finite-horizon continuous time setting by computing the expected number of abandonments and the waiting time distributions of customers. This is achieved using first-step analysis combined with Laplace transform methods. The effectiveness of the proposed approach is illustrated through numerical examples.