Noah Gans

Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective

A call center is a service network in which agents provide telephone-based services. Customers who seek these services are delayed in tele-queues. This article summarizes an analysis of a unique record of call center operations. The data comprise a complete operational history of a small banking call center, call by call, over a full year. Taking the perspective of queueing theory, we decompose the service process into three fundamental components: arrivals, customer patience, and service durations. Each component involves different basic mathematical structures and requires a different style of statistical analysis. Some of the key empirical results are sketched, along with descriptions of the varied techniques required. Several statistical techniques are developed for analysis of the basic components. One of these techniques is a test that a point process is a Poisson process. Another involves estimation of the mean function in a nonparametric regression with lognormal errors. A new graphical technique is introduced for nonparametric hazard rate estimation with censored data. Models are developed and implemented for forecasting of Poisson arrival rates. Finally, the article surveys how the characteristics deduced from the statistical analyses form the building blocks for theoretically interesting and practically useful mathematical models for call center operations.

A Call-Routing Problem with Service-Level Constraints

We consider a queueing system, commonly found in inbound telephone call centers, that processes two types of work. Type-H jobs arrive at rate ? H , are processed at rate I»i¾µ H, and are served first come, first served within class. Aservice-level constraint of the form E[delay] = a orP {delay = I²} = limits the delay in queue that these jobs may face. An infinite backlog of type-L jobs awaits processing at rate I»L , and there is no service-level constraint on this type of work. A pool ofc identical servers processes all jobs, and a system controller must maximize the rate at which type-L jobs are processed, subject to the service-level constraint placed on the type-H work.We formulate the problem as a constrained, average-cost Markov decision process and determine the structure of effective routing policies. When the expected service times of the two classes are the same, these policies are globally optimal, and the computation time required to find the optimal policy is about that required to calculate the normalizing constant for a simpleM/M/c system. When the expected service times of the two classes differ, the policies are optimal within the class of priority policies, and the determination of optimal policy parameters can be determined through the solution of a linear program with O( c3) variables and O( c2) constraints.

Economic Analysis of Simulation Selection Problems

Ranking and selection procedures are standard methods for selecting the best of a finite number of simulated design alternatives based on a desired level of statistical evidence for correct selection. But the link between statistical significance and financial significance is indirect, and there has been little or no research into it. This paper presents a new approach to the simulation selection problem, one that maximizes the expected net present value of decisions made when using stochastic simulation. We provide a framework for answering these managerial questions: When does a proposed system design, whose performance is unknown, merit the time and money needed to develop a simulation to infer its performance? For how long should the simulation analysis continue before a design is approved or rejected? We frame the simulation selection problem as a “stoppable” version of a Bayesian bandit problem that treats the ability to simulate as a real option prior to project implementation. For a single proposed system, we solve a free boundary problem for a heat equation that approximates the solution to a dynamic program that finds optimal simulation project stopping times and that answers the managerial questions. For multiple proposed systems, we extend previous Bayesian selection procedures to account for discounting and simulation-tool development costs.

Call-Routing Schemes for Call-Center Outsourcing

Companies may choose to outsource parts, but not all, of their call-center operations. In some cases, they classify customers as high or low value, serving the former with their in-house operations and routing the latter to an outsourcer. Typically, they impose service-level constraints on the time each type of customer waits on hold. This paper considers four schemes for routing low-value calls between the client company and the outsourcer. These schemes vary in the complexity of their routing algorithms, as well as the sophistication of the telephone and information technology infrastructure they require of the two operations. For three of these schemes, this paper provides a direct characterization of system performance. For the fourth, most complex, scheme the paper provides performance bounds for the important special case in which the service requirements of high- and low-value callers are the same. These results allow the systematic comparison of the performance of the various routing schemes. The results suggest that, for clients with large outsourcing requirements, the simpler schemes that require little client-outsourcer coordination can perform very well.

Securing the Containerized Supply Chain: Analysis of Government Incentives for Private Investment

To mitigate the threat that terrorists smuggle weapons of mass destruction into the United States through maritime containers, the U.S. Bureau of Customs and Border Protection (CBP) inspects containers upon entry to domestic ports. Inspection-driven congestion is costly, and CBP provides incentives to firms to improve security upstream in the supply chain, thereby reducing the inspection burden at U.S. ports. We perform an economic analysis of this incentive program, called Customs-Trade Partnership Against Terrorism (C-TPAT), modeling in a game-theoretic framework the strategic interaction between CBP, trading firms, and terrorists. Our equilibrium results highlight the possibility that a properly run program can efficiently shift some of CBPs security burden to private industry. These results also suggest that CBP may have the opportunity to use strategic delay as an incentive for firms to join. Analysis of comparative statics shows that, with increasing capacity, membership in C-TPAT systematically declines.

Pricing and Capacity Rationing for Rentals with Uncertain Durations

We consider a rental firm with two types of customers. Contract customers pay fixed, prenegotiated rental fees and expect a high quality of service. Walk-in customers have no contractual relations with the firm and are “shopping for price.” Given multiple contract and walk-in classes, the rental firm has to decide when to offer service to contract customers and what fees to charge walk-in customers for service. We formulate this rental management problem as a problem in stochastic control and characterize optimal policies for managing contract and walk-in customers. We also consider static, myopic controls that are simpler to implement, and we analytically establish conditions under which these policies perform optimally. Complementary numerical tests provide a sense of the range of systems for which myopic policies are effective.

Estimating the Operational Impact of Container Inspections at International Ports

A U.S. law mandating nonintrusive imaging and radiation detection for 100% of U.S.-bound containers at international ports has provoked widespread concern that the resulting congestion would hinder trade significantly. Using detailed data on container movements, gathered from two large international terminals, we simulate the impact of the two most important inspection policies that are being considered. We find that the current inspection regime being advanced by the U.S. Department of Homeland Security can only handle a small percentage of the total load. An alternate inspection protocol that emphasizes screening---a rapid primary scan of all containers, followed by a more careful secondary scan of only a few containers that fail the primary test---holds promise as a feasible solution for meeting the 100% scanning requirement. This paper was accepted by Yossi Aviv, operations management.

Optimal Control of a Multiclass, Flexible Queueing System

We consider a general class of queueing systems with multiple job types and a flexible service facility. The arrival times and sizes of incoming jobs are random, and correlations among the sizes of arriving job types are allowed. By choosing among a finite set of configurations, the facility can dynamically control the rates at which it serves the various job types. We define system work at any given time as the minimum time required to process all jobs currently in the backlog. This quantity is determined by solving a linear program defined by the set of processing configurations. The problem we study is how to dynamically choose configurations to minimize the time average system work. Using bounds and heuristics, we analyze a class of service policies that is provably asymptotically optimal as system utilization approaches one, as well as a policy that in numerical studies performs near-optimally in moderate traffic. Our analysis also yields a closed-form expression for the optimal, average work in heav...

Dynamic Vehicle Dispatching: Optimal Heavy Traffic Performance and Practical Insights

We analyze a general model of dynamic vehicle dispatching systems in which congestion is the primary measure of performance. In the model, a finite collection of tours are dynamically dispatched to deliver loads that arrive randomly over time. A load waits in queue until it is assigned to a tour. This representation, which is analogous to classical set-covering models, can be used to study a variety of dynamic routing and load consolidation problems. We characterize the optimal work in the system in heavy traffic using a lower bound from our earlier work (Gans and van Ryzin 1997) and an upper bound which is based on a simple batching policy. These results give considerable insight into how various parameters of the problem affect system congestion. In addition, our analysis suggests a practical heuristic which, in simulation experiments, significantly outperforms more conventional dispatching policies. The heuristic uses a few simple principles to control congestion, principles which can be easily incorporated within classical, static routing algorithms.

Optimal Hiring and Retention Policies for Heterogeneous Workers Who Learn

We study the hiring and retention of heterogeneous workers who learn over time. We show that the problem can be analyzed as an infinite-armed bandit with switching costs, and we apply results from Bergemann and Valimaki [Bergemann D, Valimaki J 2001 Stationary multi-choice bandit problems. J. Econom. Dynam. Control 2510:1585--1594] to characterize the optimal hiring and retention policy. For problems with Gaussian data, we develop approximations that allow the efficient implementation of the optimal policy and the evaluation of its performance. Our numerical examples demonstrate that the value of active monitoring and screening of employees can be substantial. This paper was accepted by Yossi Aviv, operations management.