Designing a Call Center with Impatient Customers

TLDR

Traditional call‑center models such as M/M/N, Erlang C, and Erlang B assume no customer abandonment, overlooking the common real‑world phenomenon of impatient callers leaving before service. The study analyzes the M/M/N + M abandonment model with exponentially distributed patience and unlimited waiting capacity. We develop an exact analytical method for the M/M/N + M model and then perform an asymptotic analysis for large call centers, using the results to derive practical approximations and design rules. The analysis shows that the M/M/N + M model yields actionable insights, with asymptotic approximations providing accurate performance estimates and confirming that diffusion‑based rules of thumb are useful for managing large call centers.

Abstract

The most common model to support workforce management of telephone call centers is the M/M/N/B model, in particular its special cases M/M/N (Erlang C, which models out busy signals) and M/M/N/N (Erlang B, disallowing waiting). All of these models lack a central prevalent feature, namely, that impatient customers might decide to leave (abandon) before their service begins. In this paper, we analyze the simplest abandonment model, in which customers' patience is exponentially distributed and the system's waiting capacity is unlimited (M/M/N + M). Such a model is both rich and analyzable enough to provide information that is practically important for call-center managers. We first outline a method for exact analysis of the M/M/N + M model, that while numerically tractable is not very insightful. We then proceed with an asymptotic analysis of the M/M/N + M model, in a regime that is appropriate for large call centers (many agents, high efficiency, high service level). Guided by the asymptotic behavior, we derive approximations for performance measures and propose “rules of thumb” for the design of large call centers. We thus add support to the growing acknowledgment that insights from diffusion approximations are directly applicable to management practice.