PRODUCTION AND OPERATIONS MANAGEMENT …

1 Coping with Time-Varying Demand WhenSetting Staffing Requirements for aService SystemLinda V. Green Peter J. Kolesar Ward WhittGraduate School of Business, Columbia UniversityGraduate School of Business, Columbia UniversityIEOR Department, Columbia review queueing-theory methods for setting staffing requirements in service systems wherecustomer demand varies in a predictable pattern over the day. Analyzing these systems is notstraightforward, because standard queueing theory focuses on the long-run steady-state behavior ofstationary models.

2 We show how to adapt stationary queueing models for use in nonstationaryenvironments so that time-dependent performance is captured and staffing requirements can be little modification of straightforward stationary analysis applies in systems where servicetimes are short and the targeted quality of service is high. When service times are moderate and thetargeted quality of service is still high, time-lag refinements can improve traditional stationary inde-pendent period-by-period and peak-hour approximations. Time-varying infinite-server models helpdevelop refinements, because closed-form expressions exist for their time-dependent behavior.

3 Moredifficult cases with very long service times and other complicated features, such as end-of-day effects,can often be treated by a modified-offered-load approximation, which is based on an associatedinfinite-server model. Numerical algorithms and deterministic fluid models are useful when the systemis overloaded for an extensive period of time. Our discussion focuses on telephone call centers, butapplications to police patrol, banking, and hospital emergency rooms are also words: staffing; call centers; time-varying demand; queues with time-varying arrival rate; policepatrol; banking; hospital emergency roomsSubmissions and Acceptance: Received April 2005, Accepted October 20051.

4 IntroductionA common feature of many service systems rangingfrom telephone call centers to police patrol and hos-pital emergency rooms is that the demand for ser-vice often varies greatly by time of day. This is illus-trated by the plot of hourly arrival rates from afinancial-services call center in Figure 1, taken fromSection 4 of Green, Kolesar, and Soares (2001). In thispaper we discuss ways to cope with that time-varyingdemand when setting staffing it helps to have a definite context in mind,we primarily focus on telephone call centers, wherethere already is a relatively high level of managerialcontrol and sophistication, and extensiveinformation-and-communication-t echnology(ICT) equipment, includ-ingautomatic call distributors, personal computers, andassorted databases.

5 In many call centers, staffing isperformed byworkforce managementsoftware, whichprocesses data and performs simple queueing analy-ses. For background on call centers, see the survey byGans, Koole, and Mandelbaum (2003).Many of our suggestions for call centers applyrather directly to other service systems, such as banktellers, airlines ticket counters, and tollbooths; , seethe classic toll-booth paper by Edie (1954). Moreover,the ideas apply in principle to other service systems,such as air-terminal queues ( , runways; Koopman1972), police patrol (Green and Kolesar 1984a, b, 1989,2004), and hospital emergency rooms (Green, Wyer,and Giglio 2002; Green et al.)

6 2006) but the complexitiesPOMSPRODUCTION AND OPERATIONS MANAGEMENTVol. 16, No. 1, January-February 2007, pp. 13 39issn1059-1478 07 1601 013$ 2007 PRODUCTION and OPERATIONS MANAGEMENT Society13of these systems invite more research. At the end ofthe paper we discuss the implications of our proposalsfor staffing in other service systems, including policepatrol and hospital emergency of the Section 2 we definethe staffing problem and place it in context. In Section3 we explain how stationary models can be used in anonstationary manner to solve the staffing problem inthe easiest cases those systems with short servicetimes and a high quality-of-service standard.

7 In Sec-tion 4 we discuss refinements for harder cases withmedium to long service times, but still with a highquality-of-service standard. We show how an associ-ated infinite-server model can be used to develop andunderstand these refinements. In Section 5 we discussthe most difficult case, in which the system may beoverloaded for an extensive period of time. In Section6 we discuss staffing in other systems. In Section 7 wemake concluding remarks, discussing extensions andother complications not addressed in the main paper,such as service systems with networks of facilities andsystems with customer The Staffing ProblemOne Decision in a Hierarchy of requirements is one in a hierarchy of decisionsthat must be made in the design and MANAGEMENT ofa service system.

8 In a long-term planning horizon,managers set the systemcapacity. That usually in-volves hardware choices; , in a call center manag-ers determine the number of possible agent positionsand the amount and capacity of supporting ICT equip-ment. In an intermediate-time planning horizon, man-agers set the overall size of the workforce, makingimportant hiring and training decisions. The (daily)staffing decision specifies the number of customer ser-vice representatives (agents) needed to work duringeach staffing interval over the the staffing requirements are set, managersmake agentschedulingdecisions, specifying the num-ber of agents to work on specific tours of duty, periodby period, in conformance with the previously deter-mined staffing levels, work rules, and legal con-straints.

9 The scheduling decision is often determinedby solving aninteger linear program(Dantzig 1954;Segal 1974; Kolesar et al. 1975). It is important torecognize that the staffing requirements could in prin-ciple be set by a larger algorithm that also addressesactual employee real time, managers often make further adjust-ments flexingdecisions which move agents in andout of the line of duty (to and from offline work).This is accomplished by having extra agents on sitedoing alternative work or being trained or by beingable to use remote agents on short notice.

10 If flexibilitycan be achieved, then it is often possible to efficientlyprovide a very high quality of service (Whitt 1999a).In call centers, where the actual services required bycustomers are diverse and agent skills can be matchedto them, we must be concerned with the numbers ofagents with different combinations of skills, not justthe total number of agents. Telephone callers mayspeak different languages or may require special ser-vice. For example, we may need to ensure that enoughagents are present to provide technical support inFrench and respond to billing inquiries in 1 Arrivals Per Hour to a Medium-Sized Financial-Services Call Center1 2 3 4 5 6 7 8 9 1011121314151617181920212223240 500 1000150020002500hour of daycalls per hourGreen, Kolesar, and Whitt:Time-Varying Demand and Staffing Requirements14 PRODUCTION and OPERATIONS MANAGEMENT 16(1), pp.