Cross-Selling in a Call Center with a Heterogeneous Customer Population

Abstract

A IntroductionThis is the technical appendix accompanying the paper, "Cross-Selling in a Call Center with a Heterogeneous Customer Population," [3]. The organization of this appendix is as follows: we begin in §B with the completion of the proof of Proposition 1, whose sketch was given in §A of [3]. We continue in §C with some preliminaries required for the performance analysis of (S)-(C).Specifically, we provide a sample-path construction that uses a collection of independent rate-1 Poisson processes. We also discuss some strong approximation tools. Finally, in §D we prove the main performance-analysis results which are used in the proof in §A of the main paper [3]. Some auxiliary results are proved in §E. B A Detailed Proof of Proposition 1This section is dedicated to the completion of the proof of Proposition 1 in [3]. Following our proof sketch in §A of the main paper [3] we show that there exists δ 0 ≥ 0 such that for any sequence of initial states {ξ n } ⊆ Ξ with |ξ n | → ∞:Whenever this holds we can always find a positive number K, such that for all |ξ| > K, (20) holds.Toward that end, let V (t) = j≥1 v j (t) be the amount of residual work in the system. Also, let † Columbia Business School, 4I Uris Hall, 3022 Broadway, NY, NY 10027. (ig2126@columbia.edu)

When Promotions Meet Operations: Cross-Selling and Its Effect on Call Center Performance

“…In a follow-up paper [14], the authors use the results of the current paper to study the impact of a heterogeneous pool of customers on the structure of asymptotically optimal staffing and control schemes. [14] also investigates the value of customer segmentation in such an environment.…”

“…This assumption is reasonable for systems in which waiting times are not too long and service quality is uniformly good. The independence assumption is relaxed in Gurvich et al [14] where a cruder form of analysis is performed.…”

“…Yet, operational aspects of cross-selling have so far attracted little attention in the literature. In particular, very few papers address the control problem of determining when to exercise cross-selling opportunities, and (to the best of our knowledge) this work (together with [14]) is the first to address the staffing problem of determining how many customer service representatives are needed. Those papers that have dealt with the control problem all illustrate that solving this problem is difficult, which could indeed be the reason why no simple solutions have been proposed so far.…”

“…Yet, because of the complexity involved in addressing both issues combined, they have been typically addressed separately in the literature. To our knowledge, this paper and its follow up paper [14] are the first to consider the staffing and dynamic control in a cross-selling environment jointly, in a single, common framework.…”

“…From a modeling perspective, this is the first paper (together with our followup paper [14]) to address the staffing and cross-selling control questions jointly in one single framework.…”