Motivated by call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called queue-and-idleness-ratio (QIR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified state-dependent proportion of the total queue length. An arriving customer is routed to the agent pool whose idleness most exceeds a specified state-dependent proportion of the total idleness. We identify regularity conditions on the network structure and system parameters under which QIR produces an important state-space collapse (SSC) result in the quality-and-efficiency-driven (QED) many-server heavy-traffic limiting regime. The SSC result is applied here to prove stochastic-process limits and in subsequent papers to solve important staffing and control problems for large-scale service systems.

In a recent paper we introduced the queue-and-idleness ratio (QIR) family of routing rules for many-server service systems with multiple customer classes and server pools. A newly available server serves the customer from the head of the queue of the class (from among those the server is eligible to serve) whose queue length most exceeds a specified proportion of the total queue length. Under fairly general conditions, QIR produces an important state-space collapse as the total arrival rate and the numbers of servers increase in a coordinated way. That state-space collapse was previously used to delicately balance service levels for the different customer classes. In this sequel, we show that a special version of QIR stochastically minimizes convex holding costs in a finite-horizon setting when the service rates are restricted to be pool dependent. Under additional regularity conditions, the special version of QIR reduces to a simple policy: linear costs produce a priority-type rule, in which the least-cost customers are given low priority. Strictly convex costs (plus other regularity conditions) produce a many-server analogue of the generalized-c\mu (Gc\mu ) rule, under which a newly available server selects a customer from the class experiencing the greatest marginal cost at that time.queues, many-server queues, heavy-traffic limits for queues, service systems, cost minimization in many-server queues, skill-based routing, generalized-c\mu rule, queue-and-idleness-ratio control

W e study large-scale service systems with multiple customer classes and many statistically identical servers.The following question is addressed: How many servers are required (staffing) and how does one match them with customers (control) to minimize staffing cost, subject to class-level quality-of-service constraints? We tackle this question by characterizing scheduling and staffing schemes that are asymptotically optimal in the limit, as system load grows to infinity. The asymptotic regimes considered are consistent with the efficiencydriven (ED), quality-driven (QD), and quality-and-efficiency-driven (QED) regimes, first introduced in the context of a single-class service system.Our main findings are as follows: (a) Decoupling of staffing and control, namely, (i) staffing disregards the multiclass nature of the system and is analogous to the staffing of a single-class system with the same aggregate demand and a single global quality-of-service constraint, and (ii) class-level service differentiation is obtained by using a simple idle-server-based threshold-priority (ITP) control (with state-independent thresholds); and (b) robustness of the staffing and control rules: our proposed single-class staffing (SCS) rule and ITP control are approximately optimal under various problem formulations and model assumptions. Particularly, although our solution is shown to be asymptotically optimal for large systems, we numerically demonstrate that it performs well also for relatively small systems.

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)

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