The impact of customers? patience on delay and abandonment: some empirically-driven experiments with the M/M/ n + G queue

Abstract

Our research is motivated by a phenomenon that has been observed in telephone call center data: a clear linear relation between the probability to abandon and average waiting time. Such a relation is theoretically justifiable when customers’ patience is memoryless, but it lacks an explanation in general. We thus analyze its robustness within the framework of the M/M/n + G queue, which gives rise to further theory and empirically-driven experiments. In the theoretical part of the paper, we establish order relations for performance measures of the M/M/n + G queues, and some light-traffic results. In particular, we prove that, with \(\lambda, \mu, n\), and average patience time fixed, deterministic patience minimizes the probability to abandon and maximizes the average wait in queue. In the experimental part, we describe the behavior of M/M/n + G performance measures for different patience distributions. The findings are then related to our theoretical results and some observed real-data phenomena. In particular, clear non-linear relations (convex, concave and mixed) emerge between the probability to abandon and average wait. However, when restricted over low to moderate abandonment rates, approximate linearity prevails, as observed in practice.