The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Call centers are an increasingly important part of today's business world, employing millions of agents across the globe and serving as a primary customer-facing channel for firms in many different industries. Call centers have been a fertile area for operations management researchers in several domains, including forecasting, capacity planning, queueing, and personnel scheduling. In addition, as telecommunications and information technology have advanced over the past several years, the operational challenges faced by call center managers have become more complicated. Issues associated with human resources management, sales, and marketing have also become increasingly relevant to call center operations and associated academic research.

In this paper, we provide a survey of the recent literature on call center operations management. Along with traditional research areas, we pay special attention to new management challenges that have been caused by emerging technologies, to behavioral issues associated with both call center agents and customers, and to the interface between call center operations and sales and marketing. We identify a handful of broad themes for future investigation while also pointing out several very specific research opportunities.

/pdf/the-modern-call-center-a-multi-disciplinary-perspective-on-2kcrgkbqth.pdf

The Modern Call Center: A Multi-Disciplinary Perspective on Operations Management Research

Many retailers have recently started to offer customers the option to buy online and pick up in store (BOPS). We study the impact of the BOPS initiative on store operations. We build a stylized model where a retailer operates both online and offline channels. Consumers strategically make channel choices. The BOPS option affects consumer choice in two ways: by providing real-time information about inventory availability and by reducing the hassle cost of shopping. We obtain three findings. First, not all products are well-suited for in-store pickup; specifically, it may not be profitable to implement BOPS on products that sell well in stores. Second, BOPS enables retailers to reach new customers, but for existing customers, the shift from online fulfillment to store fulfillment may decrease profit margins when the latter is less cost effective. Finally, in a decentralized retail system where store and online channels are managed separately, BOPS revenue can be shared across channels to alleviate incentive conflicts; it is rarely efficient to allocate all the revenue to a single channel.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2721474_code2314033.pdf?abstractid=2721474&mirid=1&type=2

Omnichannel Retail Operations with Buy-Online-and-Pickup-in-Store

: This report briefly summarizes research on the following topics: game theory and energy; scheduling of large research and development programs; bimatrix games; cost/benefit analyses; measures of worth of weapons systems; hybrid primal algorithm; branch and round algorithm. A listing of papers prepared and published is included. (Author)

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA049700&Location=U2&doc=GetTRDoc.pdf

Mathematics of Operations Research.

The subject of the present research is the M/M/n + G queue. This queue is characterized by Poisson arrivals at rate ?, exponential service times at rate ?, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents).

In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as ? and n increase indefinitely and ? held fixed: Efficiency-Driven (ED): $$n\ \approx \ (\lambda / \mu)\cdot (1 - \gamma),\gamma > 0,$$ 

Quality-Driven (QD): $$n \ \approx \ ( \lambda / \mu)\cdot (1 + \gamma),\gamma > 0$$ , and

Quality and Efficiency Driven (QED): $$ n \ \approx \ \lambda/ \mu+\beta \sqrt{\lambda/\mu},-\infty < \beta < \infty $$ .

In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate 1/ $$\sqrt{n}$$ ).

Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n + G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems.

Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes.

/pdf/call-centers-with-impatient-customers-many-server-5ggwfrjmmc.pdf

Call Centers with Impatient Customers: Many-Server Asymptotics of the M/M/n + G Queue

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.

/pdf/queue-and-idleness-ratio-controls-in-many-server-service-36cugsdbie.pdf

Queue-and-Idleness-Ratio Controls in Many-Server Service Systems

We 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 efficiency-driven (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.

/pdf/service-level-differentiation-in-call-centers-with-fully-3datf579ab.pdf

Service-Level Differentiation in Call Centers with Fully Flexible Servers

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-cμ (Gcμ) rule, under which a newly available server selects a customer from the class experiencing the greatest marginal cost at that time.

/pdf/scheduling-flexible-servers-with-convex-delay-costs-in-many-1vkhf8g1y6.pdf

Scheduling Flexible Servers with Convex Delay Costs in Many-Server Service Systems

Cross-selling is becoming an increasingly prevalent practice in call centers, due, in part, to its unique capability to allow firms to dynamically segment their callers and customize their product offerings accordingly. This paper considers a call center with cross-selling capability that serves a pool of customers that are differentiated in terms of their revenue potential and delay sensitivity. It studies the operational decisions of staffing, call routing, and cross-selling under various forms of customer segmentation. It derives near-optimal controls in each of the settings analyzed, and characterizes the impact of a more refined customer segmentation on the structure of these policies and the center's profitability.

/pdf/cross-selling-in-a-call-center-with-a-heterogeneous-customer-sjhuqq6xb1.pdf

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

We 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) in order to minimize cost or maximize profit, subject to quality of service (QoS) 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 main asymptotic regime considered is the many-server heavy-traffic Quality and Efficiency Driven (QED) regime. The Efficiency Driven (ED) regime is also studied. In the QED regime, which was formally introduced by Halfin and Whitt, a delicate balance is obtained between server efficiencies and quality of service. This balance is enabled by the economies of scale associated with the system size. Our main findings are: a) Decoupling of staffing and control, namely (i) Staffing disregards the multi-class nature of the system and is analogous to the staffing of a single class system with the same aggregate demand and the lowest priority class cost and QoS parameters, and (ii) Class level service differentiation is obtained by using a simple threshold-priority (TP) control (with state-independent thresholds), b) Robustness of the staffing and control rules: In the QED regime, our proposed Square-Root Safety (SRS) staffing rule and TP control are asymptotically optimal with respect to various problem formulations and model assumptions. c) The QED and ED regimes are obtained as solutions of the joint staffing and control problem rather than as assumptions.

Itay Gurvich

Papers

Queue-and-Idleness-Ratio Controls in Many-Server Service Systems

Service-Level Differentiation in Call Centers with Fully Flexible Servers

Scheduling Flexible Servers with Convex Delay Costs in Many-Server Service Systems

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

Staffing and Control of Large-Scale Service Systems with Multiple Customer Classes and Fully Flexible Servers