Measuring the Effect of Queues on Customer Purchases

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Measuring the Effect of Queues on Customer PurchasesYina Lu, Andrés Musalem, Marcelo Olivares, Ariel Schilkrut,

To cite this article:Yina Lu, Andrés Musalem, Marcelo Olivares, Ariel Schilkrut, (2013) Measuring the Effect of Queues on Customer Purchases.Management Science 59(8):1743-1763. http://dx.doi.org/10.1287/mnsc.1120.1686

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MANAGEMENT SCIENCEVol. 59, No. 8, August 2013, pp. 1743–1763ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.1120.1686

© 2013 INFORMS

Measuring the Effect of Queues onCustomer Purchases

Yina LuDecision, Risk, and Operations Division, Columbia Business School, Columbia University, New York, New York 10027,

[email protected]

Andrés MusalemFuqua School of Business, Duke University, Durham, North Carolina 27708, [email protected]

Marcelo OlivaresDecision, Risk, and Operations Division, Columbia Business School, Columbia University, New York, New York 10027;

and Department of Industrial Engineering, University of Chile, Santiago, Chile 8370439, [email protected]

Ariel SchilkrutSCOPIX, Burlingame, California 94010, [email protected]

We conduct an empirical study to analyze how waiting in queue in the context of a retail store affectscustomers’ purchasing behavior. Our methodology combines a novel data set with periodic information

about the queuing system (collected via video recognition technology) with point-of-sales data. We find thatwaiting in queue has a nonlinear impact on purchase incidence and that customers appear to focus mostly onthe length of the queue, without adjusting enough for the speed at which the line moves. An implication ofthis finding is that pooling multiple queues into a single queue may increase the length of the queue observedby customers and thereby lead to lower revenues. We also find that customers’ sensitivity to waiting is het-erogeneous and negatively correlated with price sensitivity, which has important implications for pricing in amultiproduct category subject to congestion effects.

Key words : queuing; service operations; retail; choice modeling; empirical operations management;operations/marketing interface; Bayesian estimation; service quality

History : Received May 24, 2011; accepted August 10, 2012, by Martin Lariviere, operations management.Published online in Articles in Advance April 22, 2013.

1. IntroductionCapacity management is an important aspect in thedesign of service operations. These decisions involvea trade-off between the costs of sustaining a servicelevel standard and the value that customers attachto it. Most work in the operations management liter-ature has focused on the first issue developing mod-els that are useful to quantify the costs of attaining agiven level of service. Because these operating costsare more salient, it is frequent in practice to observeservice operations rules designed to attain a quantifi-able target service level. For example, a common rulein retail stores is to open additional checkouts whenthe length of the queue surpasses a given thresh-old. However, there isn’t much research focusing onhow to choose an appropriate target service level.This requires measuring the value that customersassign to objective service level measures and howthis translates into revenue. The focus of this paperis to measure the effect of service levels—in particu-lar, customers waiting in queue—on actual customerpurchases, which can be used to attach an economicvalue to customer service.

Lack of objective data is an important limitation tostudy empirically the effect of waiting on customerbehavior. A notable exception is call centers, wheresome recent studies have focused on measuring cus-tomer impatience while waiting on the phone line(Gans et al. 2003). Instead, our focus is to study phys-ical queues in services, where customers are physi-cally present at the service facility during the wait.This type of queue is common, for example, in retailstores, banks, amusement parks, and healthcare deliv-ery. Because objective data on customer service aretypically not available in these service facilities, mostprevious research relies on surveys to study howcustomers’ perceptions of waiting affect their intendedbehavior. However, previous work has also shownthat customer perceptions of service do not neces-sarily match with the actual service level received,and purchase intentions do not always translate intoactual revenue (e.g., Chandon et al. 2005). In contrast,our work uses objective measures of actual servicecollected through a novel technology—digital imag-ing with image recognition—that tracks operationalmetrics such as the number of customers waiting in

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Lu et al.: Measuring the Effect of Queues on Customer Purchases1744 Management Science 59(8), pp. 1743–1763, © 2013 INFORMS

line. We develop an econometric framework that usesthese data together with point-of-sales (POS) infor-mation to estimate the impact of customer servicelevels on purchase incidence and choice decisions.We apply our methodology using field data collectedin a pilot study conducted at the deli section of abig-box supermarket. An important advantage of ourapproach over survey data is that the regular and fre-quent collection of the store operational data allowsus to construct a large panel data set that is essentialto identifying each customer’s sensitivity to waiting.

There are two important challenges in our estima-tion. A first issue is that congestion is highly depen-dent on store traffic, and therefore periods of highsales are typically concurrent with long waiting lines.Consequently, we face a reverse causality problem:whereas we are interested in measuring the causaleffect of waiting on sales, there is also a reverseeffect whereby spikes in sales generate congestionand longer waits. The correlation between waitingtimes and aggregate sales is a combination of thesetwo competing effects and therefore cannot be useddirectly to estimate the causal effect of waiting onsales. The detailed panel data with purchase historiesof individual customers is used to address this issue.

Using customer transaction data produces a secondestimation challenge. The imaging technology cap-tures snapshots that describe the queue length andstaffing level at specific time epochs but does not pro-vide an exact measure of what is observed by eachcustomer (technological limitations and consumer pri-vacy issues preclude us from tracking the identityof customers in the queue). A rigorous approachis developed to infer these missing data from peri-odic snapshot information by analyzing the tran-sient behavior of the underlying stochastic processof the queue. We believe this is a valuable contribu-tion that will facilitate the use of periodic operationaldata in other studies involving customer transactionsobtained from POS information.

Our model also provides several metrics that areuseful for the management of service facilities. First,it provides estimates on how service levels affectthe effective arrivals to a queuing system when cus-tomers may balk. This is a necessary input to setservice and staffing levels optimally balancing oper-ating costs against lost revenue. In this regard, ourwork contributes to the stream of empirical researchrelated to retail staffing decisions (e.g., Fisher et al.2009, Perdikaki et al. 2012). Second, it can be used toidentify the relevant visible factors in a physical queu-ing system that drive customer behavior, which canbe useful for the design of a service facility. Third, ourmodels provide estimates of how the performance ofa queuing system may affect how customers substi-tute among alternative products or services account-ing for heterogeneous customer preferences. Finally,

our methodology can be used to attach a dollar valueto the cost of waiting experienced by customers andto segment customers based on their sensitivity towaiting.

In terms of our results, our empirical analysis sug-gests that the number of customers in the queuehas a significant impact on the purchase incidenceof products sold in the deli, and this effect appearsto be nonlinear and economically significant. Moder-ate increases in the number of customers in queuecan generate sales reduction equivalent to a 5% priceincrease. Interestingly, the service capacity—whichdetermines the speed at which the line moves—seemsto have a much smaller impact relative to the num-ber of customers in line. This is consistent with cus-tomers using the number of people waiting in lineas the primary visible cue to assess the expectedwaiting time. This empirical finding has importantimplications for the design of the service facility. Forexample, we show that pooling multiple queues intoa single queue with multiple servers may lead tomore customers walking away without purchasingand therefore lower revenues (relative to a systemwith multiple queues). We also find significant het-erogeneity in customer sensitivity to waiting, and thatthe degree of waiting sensitivity is negatively corre-lated with customers’ sensitivity to price. We showthat this result has important implications for pricingdecisions in the presence of congestion and, conse-quently, should be an important element to considerin the formulation of analytical models of waitingsystems.

2. Related WorkIn this section, we provide a brief review of the lit-erature studying the effect of waiting on customerbehavior and its implications for the managementof queues. Extensive empirical research using exper-imental and observational data has been done inthe fields of operations management, marketing, andeconomics. We focus this review on a selection of theliterature that helps us to identify relevant behavioralpatterns that are useful in developing our economet-ric model (described in §3). At the same time, we alsoreference survey articles that provide a more exhaus-tive review of different literature streams.

Recent studies in the service engineering litera-ture have analyzed customer transaction data in thecontext of call centers. See Gans et al. (2003) fora survey on this stream of work. Customers arriv-ing to a call center are modeled as a Poisson pro-cess where each arriving customer has a “patiencethreshold”: one abandons the queue after waitingmore than his patience threshold. This is typicallyreferred to as the Erlang-A model or the M/M/c+G,

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Lu et al.: Measuring the Effect of Queues on Customer PurchasesManagement Science 59(8), pp. 1743–1763, © 2013 INFORMS 1745

where G denotes the generic distribution of the cus-tomer patience threshold. Brown et al. (2005) estimatethe distribution of the patience threshold based oncall-center transactional data and use it to measurethe effect of waiting time on the number of lost (aban-doned) customers.

Customers arriving to a call center typically donot directly observe the number of customers aheadin the line, so the estimated waiting time may bebased on delay estimates announced by the serviceprovider or their prior experience with the service(Ibrahim and Whitt 2011). In contrast, for physicalcustomer queues at a retail store, the length of theline is observed and may become a visible cue affect-ing their perceived waiting time. Hence, queue lengthbecomes an important factor in customers’ decision tojoin the queue, which is not captured in the Erlang-Amodel. In these settings, arrivals to the system canbe modeled as a Poisson process where a fraction ofthe arriving customers may balk—that is, not join thequeue—depending on the number of people alreadyin queue (see Gross et al. 2008, Chap. 2.10). Ourwork focuses on estimating how visible aspects ofphysical queues, such as queue length and capacity,affect choices of arriving customers, which providesan important input to normative models.

Png and Reitman (1994) empirically study the effectof waiting time on the demand for gas stations andidentify service time as an important differentiatingfactor in this retail industry. Their estimation is basedon aggregate data on gas station sales and uses mea-sures of a station’s capacity as a proxy for waitingtime. Allon et al. (2011) study how service time affectsdemand across outlets in the fast food industry, usinga structural estimation approach that captures pricecompetition across outlets. Both studies use aggregatedata from a cross-section of outlets in local markets.The data for our study are more detailed because theyuse individual customer panel information and peri-odic measurements of the queue, but it is limited toa single service facility. None of the aforementionedpapers examine heterogeneity in waiting sensitivity atthe individual level as we do in our work.

Several empirical studies suggest that customerresponses to waiting time are not necessarily linear.Larson (1987) provides anecdotal evidence of non-linear customer disutility under different service sce-narios. Laboratory and field experiments have shownthat customer’s perceptions of waiting are importantdrivers of dissatisfaction and that these perceptionsmay be different from the actual (objective) waitingtime, sometimes in a nonlinear pattern (e.g., Davisand Vollmann 1993, Berry et al. 2002, Antonides et al.2002). Mandelbaum and Zeltyn (2004) use analyticalqueuing models with customer impatience to explainnonlinear relationships between waiting time and

customer abandonment. Indeed, in the context of call-center outsourcing, the common use of service levelagreements based on delay thresholds at the upper tailof the distribution (e.g., 95% of the customers wait lessthan two minutes) is consistent with nonlinear effectsof waiting on customer behavior (Hasija et al. 2008).

Larson (1987) provides several examples of factorsthat affect customers’ perceptions of waiting, suchas (1) whether the waiting is perceived as sociallyfair, (2) whether the wait occurs before or after theactual service begins, and (3) feedback provided tothe customer on waiting estimates and the root causesgenerating the wait, among other examples. Berryet al. (2002) provide a survey of empirical work test-ing some of these effects. Part of this research hasused controlled laboratory experiments to analyzefactors that affect customers perceptions of waiting.For example, the experiments by Hui and Tse (1996)suggest that queue length has no significant impact onservice evaluation in short-wait conditions, althoughit has a significant impact on service evaluation inlong-wait conditions. Janakiraman et al. (2011) useexperiments to analyze customer abandonments andpropose two competing effects that explain why aban-donments tend to peak at the midpoint of waits. Huiet al. (1997) and Katz et al. (1991) explore several fac-tors, including music and other distractions, that mayaffect customers’ perception of waiting time.

In contrast, our study relies on field data to analyzethe effect of queues on customer purchases. Muchof the existing field research relies on surveys tomeasure objective and subjective waiting times, link-ing these to customer satisfaction and intentions ofbehavior. For example, Taylor (1994) studies a sur-vey of delayed airline passengers and finds that delaydecreases service evaluations by invoking uncertaintyand anger affective reactions. Deacon and Sonstelie(1985) evaluate customers’ time value of waitingbased on a survey on gasoline purchases. Althoughsurveys are useful to uncover the behavioral processby which waiting affects customer behavior and thefactors that mediate this effect, they also suffer fromsome disadvantages. In particular, there is a poten-tial sample selection because nonrespondents tend tohave a higher opportunity cost for their time. In addi-tion, several papers report that customer purchaseintentions do not always match actual purchasingbehavior (e.g., Chandon et al. 2005). Moreover, rely-ing on surveys to construct a customer panel data setwith the required operational data is difficult (all thereferenced articles use a cross-section of customers).Our work uses measures of not only actual customerpurchases, but also operational drivers of waitingtime (e.g., queue length and capacity at the time ofeach customer visit) to construct a panel with objec-tive metrics of purchasing behavior and waiting. Our

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approach, however, is somewhat limited for studyingsome of the underlying behavioral process driving theeffect of waiting time.

Several other studies use primary and secondaryobservational data to measure the effect of servicetime on customer behavior. Forbes (2008) analyzesthe impact of airline delays on customer complaints,showing that customer expectations play an impor-tant role mediating this effect. Campbell and Frei(2011) study multiple branches of a bank, providingempirical evidence that teller waiting times affect cus-tomer satisfaction and retention. Their empirical studyreveals significant heterogeneity in customer sensitiv-ity to waiting time, some of which can be explainedthrough demographics and the intensity of competi-tion faced by the branch. Aksin et al. (2013) modelcallers’ abandonment decisions as an optimal stop-ping problem in a call-center context and find hetero-geneity in callers’ waiting behavior. Our study alsolooks at customer heterogeneity in waiting sensitivity,but in addition we relate this sensitivity to customers’price sensitivity. This association between price andwaiting sensitivity has important managerial impli-cations; for example, Afèche and Mendelson (2004)and Afanasyev and Mendelson (2010) show that itplays an important role for setting priorities in queueand it affects the level of competition among serviceproviders. Section 5 discusses other managerial impli-cations of this price/waiting sensitivity relationship inthe context of category pricing.

Our study uses discrete choice models based onrandom utility maximization to measure substitutioneffects driven by waiting. The same approach wasused by Allon et al. (2011), who incorporated waitingtime factors into customers’ utility using a multino-mial logit (MNL) model. We instead use a randomcoefficient MNL, which incorporates heterogeneityand allows for more flexible substitution patterns(Train 2003). The random coefficient MNL model hasalso been used in the transportation literature toincorporate the value of time in consumer choice (e.g.,Hess et al. 2005).

Finally, all of the studies mentioned so far focuson settings where waiting time and congestion gener-ate disutility to customers. However, there is theorysuggesting that longer queues could create value toa customer. For example, if a customers’ utility for agood depends on the number of customers that con-sume it (as with positive network externalities), thenlonger queues could attract more customers. Anotherexample is given by herding effects, which may arisewhen customers have asymmetric information aboutthe quality of a product. In such a setting, longerqueues provide a signal of higher value to unin-formed customers, making them more likely to jointhe queue (see Debo and Veeraraghavan 2009 for sev-eral examples).

3. EstimationThis section describes the data and models used inour estimation. The literature review of §2 providesseveral possible behavioral patterns that are includedin our econometric specification: (1) the effect ofwaiting time on customer purchasing behavior maybe nonlinear, such that customers’ sensitivity to amarginal increase in waiting time may vary at dif-ferent levels of waiting time; (2) the effect may notbe monotone (for example, although more antici-pated waiting is likely to negatively affect customers’purchase intentions, herding effects could potentiallymake longer queues attractive to customers); (3) cus-tomer purchasing behavior is affected by percep-tions of waiting time, which may be formed basedon the observed queue length and the correspond-ing staffing level; (4) customers’ sensitivity to wait-ing time may be heterogeneous and possibly relatedto demographic factors, such as income or pricesensitivity.

Subsection 3.1 describes the data used in ourempirical study, which motivates the econometricframework developed in the rest of the section.Subsection 3.2 describes an econometric model tomeasure the effect of queues on purchase incidence.It uses a flexible functional form to measure theeffect of the queue on purchasing behavior that per-mits potential nonlinear and nonmonotone effects.Different specifications are estimated to test for fac-tors that may affect customers’ perceptions of wait-ing. Subsection 3.3 describes how to incorporate theperiodic queue information contained in the snap-shot data into the estimation of this model. Sub-section 3.4 conducts a simulation study to validatethis estimation methodology. Subsection 3.5 devel-ops a discrete choice model that captures additionalfactors not incorporated into the purchase incidencemodel, including substitution among products, prices,promotions, and state-dependent variables that affectpurchases (e.g., household inventory). This choicemodel is also used to measure heterogeneity in cus-tomer sensitivity to waiting.

3.1. DataWe conducted a pilot study at the deli section ofa supercenter located in a major metropolitan areain Latin America. The store belongs to a leadingsupermarket chain in this country and is located ina working-class neighborhood. The deli section sellsabout eight product categories, most of which arefresh cold-cuts sold by the pound.

During a pilot study running from October 2008to May 2009 (approximately seven months), we useddigital snapshots analyzed by image recognition tech-nology to periodically track the number of peoplewaiting at the deli and the number of sales associates

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Figure 1 Example of a Deli Snapshot Showing the Number of Customers Waiting (Left) and the Number of Employees Attending (Right)

Source. Courtesy of SCOPIX.

serving it. Snapshots were taken periodically every30 minutes during the open hours of the deli, from9 a.m. to 9 p.m. on a daily basis. Figure 1 shows asample snapshot that counts the number of customerswaiting (left panel) and the number of employeesattending customers behind the deli counter (rightpanel). Throughout this paper, we denote the lengthof the deli queue at snapshot t by Qt and the numberof employees serving the deli by Et .

During peak hours, the deli uses numbered tick-ets to implement a first-come, first-served priorityin the queue. The counter displays a visible panelintended to show the ticket number of the last cus-tomer attended by a sales associate. This informa-tion would be relevant for the purpose of our studyto complement the data collected through the snap-shots; for example, Campbell and Frei (2011) useticket-queue data to estimate customer waiting time.However, in our case the ticket information was notstored in the POS database of the retailer, and welearned from other supermarkets that this informa-tion is rarely recorded. Nevertheless, the methodsproposed in this paper could also be used with peri-odic data collected via a ticket queue, human inspec-tion, or other data collection procedures.

In addition to the queue and staffing information,we also collected POS data for all transactions involv-ing grocery purchases from January 1, 2008, until theend of the study period. In the market area of ourstudy, grocery purchases typically include bread, andabout 78% of the transactions that include deli prod-ucts also include bread. For this reason, we selectedbasket transactions that included bread to obtain asample of grocery-related shopping visits. Each trans-action contains checkout data, including a time stampof the checkout and the stock-keeping units (SKUs)bought along with unit quantities and prices (afterpromotions). We use the POS data prior to the pilotstudy period—from January to September of 2008—to

calculate metrics employed in the estimation of someour models (we refer to this subset of the data as thecalibration data).

Using detailed information on the list of productsoffered at this supermarket, each cold-cut SKU wasassigned to a product category (e.g., ham, turkey,bologna, salami, etc.). Some of these cold-cut SKUsinclude prepackaged products that are not sold bythe pound and therefore are located in a differentsection of the store.1 For each SKU, we defined anattribute indicating whether it was sold in the deli orprepackaged section. About 29.5% of the transactionsin our sample include deli products, suggesting thatdeli products are quite popular in this supermarket.

An examination on the hourly variation of thenumber of transactions, queue length, and numberof employees reveals the following interesting pat-terns. In weekdays, peak traffic hours are observedaround midday, between 11 a.m. and 2 p.m., and inthe evenings, between 6 p.m. and 8 p.m. Althoughthere is some adjustment in the number of employ-ees attending, this adjustment is insufficient, andtherefore queue lengths exhibit an hour-of-day pat-tern similar to the one for traffic. A similar effect isobserved for weekends, although the peak hours aredifferent. In other words, congestion generates a pos-itive correlation between aggregate sales and queuelengths, making it difficult to study the causal effectof queues on traffic using aggregate POS data. In ourempirical study, detailed customer transaction data areused instead to address this problem. More specifi-cally, the supermarket chain in our study operates apopular loyalty program such that more than 60% ofthe transactions are matched with a loyalty card iden-tification number, allowing us to construct a panel of

1 This prepackaged section can be seen to the right of customernumbered 1 in the left panel of Figure 1 (top-right corner).

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Table 1 Summary Statistics of the Snapshot Data, Point-of-SalesData, and Loyalty Card Data

No. of Std.obs. Mean dev. Min Max

Periodic snapshot dataLength of the queue (Q )

Weekday 31671 3076 3081 0 26Weekend 11465 6042 4090 0 27

Number of employees (E)Weekday 31671 2011 1026 0 7Weekend 11465 2084 1046 0 9

Point-of-sales dataPurchase incidence of deli products 2841709 2205%

Loyalty card dataNumber of visits per customer 131103 2107 1806 7 162

individual customer purchases. Although this sampleselection limits the generalizability of our findings,we believe this limitation is not too critical becauseloyalty card customers are perceived as the most prof-itable customers by the store. To better control for cus-tomer heterogeneity, we focus on grocery purchasesof loyalty card customers who visit the store one ormore times per month on average. This accounts fora total of 284,709 transactions from 13,103 customers.Table 1 provides some summary statistics describ-ing the queue snapshots and the POS and loyaltycard data.

3.2. Purchase Incidence ModelRecall that the POS and loyalty card data are usedto construct a panel of observations for each indi-vidual customer. Each customer is indexed by i, andeach store visit by v. Let yiv = 1 if the customer pur-chased a deli product in that visit, and zero otherwise.Denote Qiv and Eiv as the number of people in queueand the number of employees, respectively, that wereobserved by the customer during visit v. Through-out this paper we refer to Qiv and Eiv altogether asthe state of the queue. The objective of the purchaseincidence model is to estimate how the state of thequeue affects the probability of purchase of productssold in the deli. Note that we (the researchers) do notobserve the state of the queue directly in the data,which complicates the estimation. Our approach is toinfer the distribution of the state of queue using snap-shot and transaction data and then plug estimates ofQiv and Eiv into a purchase incidence model. Thismethodology is summarized in Figure 2. In this sub-section, we describe the purchase incidence modelassuming the state of the queue estimates are given(Step 1 in Figure 2); later, §3.3 describes how to handlethe unobserved state of the queue.

In the purchase incidence model, the probabilityof a deli purchase, defined as p4Qiv1 Eiv5 ≡ Pr6yiv =

1 � Qiv1 Eiv7, is modeled as

h4p4Qiv1 Eiv55= f 4Qiv1 Eiv1�q5+�xXiv1 (1)

Figure 2 Outline of the Estimation Procedure

• Step 0.(a) Calculate the average store traffic åt using all cashier

transactions (including those without deli purchases) for differenthours of the day and days of the week (e.g., Mondays between9 a.m. and 11 a.m.).

(b) Initialize the state of the queue 4Qiv1 Eiv5 observed by cus-tomer i in visit v as the second previous snapshot before checkouttime.

(c) Group the snapshot data into time buckets with observationsfor the same time of the day, day of the week and the same num-ber of employees. For example, one bucket could contain snapshotstaken on Mondays between 9 a.m. and 11 a.m. with two employeesattending. For each time bucket, compute the empirical distributionof the queue length based on the snapshot data.

• Step 1. Estimate purchase incidence model (1) via ML assum-ing state of queue 4Qiv1 Eiv5 is observed.

• Step 2. Estimate the queue intensity �t on each time bucket.(a) Based on the estimated store traffic åt and purchase

incidence probability p4Q1E5, calculate the effective arrival rate�t4Q1E5=åtp4Q1E) for each possible state of the queue in timebucket t.

(b) Compute the stationary distribution of the queue lengthon each time bucket t as a function of the queue intensity �t and�t4Q1E5: for each time bucket, choose the queue intensity �t thatbest matches the predicted stationary distribution to the observedempirical distribution of the length of the queue (computed inStep 0(c)).

• Step 3. Update the distribution of the observed queuelength Qiv.

(a) Compute the transition probability matrix Pt4s5.(b) For a given deli visit time � , calculate the distribution of

Q� using Pt4s5.(c) Integrate over all possible deli visit times � to find the

distribution of Qiv. Update Qiv by its expectation based on thisdistribution.

(d) Repeat from Step 1 until the estimated length of the queue,Qiv, converges.

where h4 · 5 is a link function, f 4Qiv1 Eiv1�q5 is a para-metric function that captures the impact of the stateof the queue, �q is a parameter vector to be estimated,and Xiv is a set of covariates that capture other fac-tors that affect purchase incidence (including an inter-cept). We use a logit link function, h4x5= ln6x/41−x57,which leads to a logistic regression model that can beestimated via maximum likelihood (ML) methods. Wetested alternative link functions and found the resultsto be similar.

Now we turn to the specification of the effect ofthe state of the queue, f 4Qiv1 Eiv1�q5. Previous workhas documented that customer behavior is affectedby perceptions of waiting that may not be equal tothe expected waiting time. Upon observing the stateof the queue 4Qiv1 Eiv5, the measure Wiv = Qiv/Eiv

(number of customers in line divided by the numberof servers) is proportional to the expected time to waitin line, and hence is an objective measure of waiting.Throughout this paper, we use the term expected wait-ing time to refer to the objective average waiting timefaced by customers for a given state of the queue,

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which can be different from the perceived waiting timethey form based on the observed state of the queue.Our first specification uses Wiv to measure the effectof this objective waiting factor on customer behavior.

Note that the function f 4Wiv1�q5 captures the over-all effect of expected waiting time on customer behav-ior, which includes the disutility of waiting but alsopotential herding effects. The disutility of waiting hasa negative effect, whereas the herding effect has a pos-itive effect. Because both effects occur simultaneously,the estimated overall effect is the sum of both. Hence,the sign of the estimated effect can be used to testwhich effect dominates. Moreover, as suggested byLarson (1987), the perceived disutility from waitingmay be nonlinear. This implies that f 4Wiv1�q5 maynot be monotone—herding effects could dominate insome regions whereas waiting disutility could domi-nate in other regions. To account for this, we specifyf 4Wiv1�q5 in a flexible manner using piecewise linearand quadratic functions.

We also estimate other specifications to test foralternative effects. As shown in some of the experi-mental results reported in Carmon (1991), customersmay use the length of the line, Qiv, as a visible cue toassess their waiting time, ignoring the speed at whichthe queue moves. In the setting of our pilot study, thelength of the queue is highly visible, whereas deter-mining the number of employees attending is notalways straightforward. Hence, it is possible for a cus-tomer to balk from the queue based on the observedlength of the line, without fully accounting for thespeed at which the line moves. To test for this, weconsider specifications where the effect of the stateof the queue is only a function of the queue length,f 4Qiv1�q5. As before, we use a flexible specificationthat allows for nonlinear and nonmonotone effects.

The two aforementioned models look at extremecases where the state of the queue is fully capturedeither by the objective expected time to wait (Wiv)or by the length of the queue (ignoring the speedof service). These two extreme cases are interestingbecause there is prior work suggesting each of themas the relevant driver of customer behavior. In addi-tion, f 4Qiv1 Eiv1�q5 could also be specified by placingseparate weights on the length of the queue (Qiv) andthe capacity (Eiv); we also consider these additionalspecifications in §4.

There are two important challenges to estimate themodel in Equation (1). The first is that we are seekingto estimate a causal effect—the impact of 4Qiv1 Eiv5 onpurchase incidence—using observational data ratherthan a controlled experiment. In an ideal experimenta customer would be exposed to multiple 4Qiv1 Eiv5conditions holding all other factors (e.g., prices, timeof the day, seasonality) constant. For each of theseconditions, her purchasing behavior would then be

recorded. In the context of our pilot study, however,there is only one 4Qiv1 Eiv5 observation for each cus-tomer visit. This could be problematic if, for exam-ple, customers with a high purchase intention visitthe store around the same time. These visits wouldthen exhibit long queues and high purchase probabil-ity, generating a bias in the estimation of the causaleffect. In fact, the data do suggest such an effect:the average purchase probability is 34.2% on week-ends at 8 p.m., when the average queue length is10.3, and it drops to 28.3% on weekdays at 4 p.m.,when the average queue length is only 2.2. Anotherexample of this potential bias is when the deli runspromotions: price discounts attract more customers,which increases purchase incidence and also gener-ates higher congestion levels.

To partially overcome this challenge, we includecovariates in X that control for customer heterogene-ity. A flexible way to control for this heterogeneity isto include customer fixed effects to account for eachcustomer’s average purchase incidence. Purchase inci-dence could also exhibit seasonality—for example,consumption of fresh deli products could be higherduring a Sunday morning in preparation for a familygathering during Sunday lunch. To control for sea-sonality, the model includes a set of time-of-day dum-mies interacted with weekend/weekday indicators.This set of dummies also helps to control for a poten-tial endogeneity in the staffing of the deli, because itcontrols for planned changes in the staffing schedule.Finally, we also include a set of dummies for each dayin the sample, which controls for seasonality, trends,and promotional activities (because promotions typi-cally last at least a full day).

Although customer fixed effects account for pur-chase incidence heterogeneity across customers, theydon’t control for heterogeneity in purchase incidenceacross visits of the same customer. Furthermore, someof this heterogeneity across visits may be customerspecific, so that they are not fully controlled by theseasonal dummies in the model. State-dependent fac-tors, which are frequently used in the marketing lit-erature (Neslin and van Heerde 2008), could help topartially control for this heterogeneity. Another lim-itation of the purchase incidence model is that (1)cannot be used to characterize substitution effectswith products sold in the prepackaged section, whichcould be important to measure the overall effectof queue-related factors on total store revenue andprofit. To address these limitations, we develop thechoice model described in §3.5. Nevertheless, theseadditions require focusing on a single product cate-gory, whereas the purchase incidence model capturesall product categories sold in the deli. For this reasonand because of its relative simplicity, the estimationof the purchase incidence model (1) provides valuable

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Figure 3 Sequence of Events Related to a Customer PurchaseTransaction

t – 2 t – 1

B (τ) A (τ)τ (deli)

t t + 1 t + 2

ts (checkout)

insights about how consumers react to different levelsof service.

A second challenge in the estimation of (1) is that4Qiv1 Eiv5 are not directly observable in our data set.The next subsection provides a methodology to infer4Qiv1 Eiv5 based on the periodic data captured by thesnapshots 4Qt1Et5 and describes how to incorporatethese inferences into the estimation procedure.

3.3. Inferring Queues from Periodic DataWe start by defining some notation regarding eventtimes, as summarized in Figure 3. Time ts denotes theobserved checkout time stamp of the customer trans-action. Time � < ts is the time at which the customerobserved the deli queue and made her decision onwhether to join the line (whereas in reality customerscould revisit the deli during the same visit hoping tosee a shorter line, we assume a single deli visit to keepthe econometric model tractable; see Footnote 8 forfurther discussion). The snapshot data of the queuewere collected periodically, generating time intervals6t − 11 t5, 6t1 t + 15, etc. For example, if the checkouttime ts falls in the interval 6t1 t+15, � could fall in theintervals 6t − 11 t5, 6t1 t + 15, or in any other intervalbefore ts (but not after). Let B4�5 and A4�5 denote theindex of the snapshots just before and after time � . Inour application, � is not observed, and we model it asa random variable and denote F 4� � ts5 its conditionaldistribution given the checkout time ts.2

In addition, the state of the queue is only observedat prespecified time epochs, so even if the deli visittime � is known, the state of the queue is still notknown exactly. It is then necessary to estimate 4Q�1E�5for any given � based on the observed snapshot data4Qt1Et5. The snapshot data reveal that the numberof employees in the system, Et , is more stable: forabout 60% of the snapshots, consecutive observationsof Et are identical. When they change, it is typicallyby one unit (81% of the samples).3 When Et−1 = Et = c,it seems reasonable to assume that the number ofemployees remained to be c in the interval 6t − 11 t5.

2 Note that in applications where the time of joining the queue isobserved—for example, as provided by a ticket time stamp in aticket queue—it may still be unobserved for customers that decidednot to join the queue. In those cases, � may also be modeled as arandom variable for customers that did not join the queue.3 However, there is still sufficient variance of Et to estimate theeffect of this variable with precision; a regression of Et on dummiesfor day and hour of the day has an R2 equal to 0.44.

When changes between two consecutive snapshotsEt−1 and Et are observed, we assume (for simplic-ity) that the number of employees is equal to Et−1throughout the interval 6t − 11 t).

Assumption 1. In any interval 6t − 11 t5, the numberof servers in the queuing system is equal to Et−1.

A natural approach to estimate Q� would be to takea weighted average of the snapshots around time � ,for example, an average of QB4�5 and QA4�5. However,this naive approach may generate biased estimates, aswe will show in §3.4. In what follows, we show a for-mal approach to using the snapshot data in the vicinityof � to get a point estimate of Q� . Our methodologyrequires the following additional assumption aboutthe evolution of the queuing system:

Assumption 2. In any snapshot interval 6t1 t + 15,arrivals follow a Poisson process with an effective arrivalrate �t4Q1E5 (after accounting for balking) that maydepend on the number of customers in queue and the num-ber of servers. The service times of each server follow anexponential distribution with similar rate but independentacross servers.

Assumptions (1) and (2) together imply that inany interval between two snapshots the queuingsystem behaves like an Erlang queue model (alsoknown as M/M/c) with balking rate that depends onthe state of queue. The Markovian property impliesthat the conditional distribution of Q� given the snap-shot data only depends on the most recent queueobservation before time � , QB4�5, which simplifies theestimation. We now provide some empirical evidenceto validate these assumptions.

Given that the snapshot intervals are relativelyshort (30 minutes), stationary Poisson arrivals withineach time interval seem a reasonable assumption.To corroborate this, we analyzed the number ofcashier transactions on every half-hour interval bycomparing the fit of a Poisson regression model witha negative binomial (NB) regression. The NB modelis a mixture model that nests the Poisson model butis more flexible, allowing for overdispersion—that is,a variance larger than the mean. This analysis sug-gests that there is a small overdispersion in the arrivalcounts, so that the Poisson model provides a reason-able fit to the data.4

The effective arrival rate during each time period�t4Q1E5 is modeled as �t4Q1E5 = åt · p4Q1E5, whereåt is the overall store traffic that captures seasonality

4 The NB model assumes Poisson arrivals with a rate � that isdrawn from a gamma distribution. The variance of � is a parameterestimated from the data; when this variance is close to zero, the NBmodel is equivalent to a Poisson process. The estimates of the NBmodel imply a coefficient of variation for � equal to 17%, which isrelatively low.

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and variations across times of the day; p4Q1E5 isthe purchase incidence probability defined in (1).To estimate åt , we first group the time intervals intodifferent days of the week and hours of the day andcalculate the average number of total transactions ineach group, including those without deli purchases(see Step 0(a) in Figure 2). For example, we calcu-late the average number of customer arrivals acrossall time periods corresponding to “Mondays between9 a.m. and 11 a.m.” and use this as an estimate ofåt for those periods. The purchase probability func-tion p4Q1E5 is also unknown; in fact, it is exactlywhat the purchase incidence model (1) seeks to esti-mate. To make the estimation feasible, we use an ini-tial rough estimate of p4Q1E5 by estimating model (1)replacing E� by EB4ts5−1 and Q� by QB4ts5−1 (Step 0(b) inFigure 2). We later show how this estimate is refinediteratively.

Provided an estimate of �t4Q1E5 (Step 2(a) in Fig-ure 2), the only unknown primitive of the Erlangmodel is the service rate �t , or alternatively, the queueintensity level �t = 4maxQ6�t4Q1E575/4Et ·�t5. Neither�t nor �t are observed, and have to be estimated fromthe data. To estimate �t and also to further validateAssumption 2, we compared the distribution of theobserved samples of Qt in the snapshot data with thestationary distribution predicted by the Erlang model.To do this, we first group the time intervals into buck-ets 8Ck9

Kk=1, such that intervals in the same bucket k

have the same number of servers Ek (see Step 0(c)in Figure 2). For example, one of these buckets cor-responds to “Mondays between 9 a.m. and 11 a.m.,with two servers.” Using the snapshots on each timebucket, we can compute the observed empirical dis-tribution of the queue. The idea is then to estimatea utilization level �k for each bucket so that the pre-dicted stationary distribution implied by the Erlangmodel best matches the empirical queue distribution(Step 2(b) in Figure 2). In our analysis, we estimated�k by minimizing the L2 distance between the empiri-cal distribution of the queue length and the predictedErlang distribution.

Overall, the Erlang model provides a good fit formost of the buckets: a chi-square goodness of fit testrejects the Erlang model only in 4 out of 61 buckets(at a 5% confidence level). By adjusting the utiliza-tion parameter �, the Erlang model is able to captureshifts and changes in the shape of the empirical distri-bution across different buckets. The implied estimatesof the service rate suggest an average service timeof 1.31 minutes, and the variation across hours anddays of the week is relatively small (the coefficientof variation of the average service time is approxi-mately 0.18).5

5 We find that this service rate has a negative correlation (−0046)with the average queue length, suggesting that servers speed up

Figure 4 Estimates of the Distribution of the Queue Length Observedby a Customer for Different Deli Visit Times 4� 5

20

18

16

14

12

10

Que

ue le

ngth

8

6

4

2

0

� = 5 � = 15 � = 25

0 5 10 15

t (min)20 25 30

Note. The previous snapshot is at t = 0 and shows two customers in queue.

Now we discuss how the estimate of Qiv is refined(Step 3 in Figure 2). The Markovian property (givenby Assumptions 1 and 2) implies that the distributionof Q� conditional on a prior snapshot taken at timet < � is independent of all other snapshots taken priorto t. Given the primitives of the Erlang model, we canuse the transient behavior of the queue to estimatethe distribution of Q� . The length of the queue can bemodeled as a birth–death process in continuous time,with transition rates determined by the primitives Et ,�t4Q1E5, and �t . Note that we already showed how toestimate these primitives. The transition rate matrixduring time interval 6t1 t+ 1), denoted Rt, is given by6Rt7i1 i+1 = �t4i1 Et5, 6Rt7i1 i−1 = min8i1 Et9 · �t , 6Rt7i1 i =

−èj 6=i6Rt7i1 j , and zero for the rest of the entries.The transition rate matrix Rt can be used to calcu-

late the transition probability matrix for any elapsedtime s, denoted Pt4s5.6 For any deli visit time � , thedistribution of Q� conditional on any previous snap-shot Qt(t < �) can be calculated as Pr4Q� = k � Qt5 =

6Pt4� − t57Qtkfor all k ≥ 0.7

Figure 4 illustrates some estimates of the distribu-tion of Q� for different values of � . (For display pur-poses, the figure shows a continuous distribution butin practice it is a discrete distribution.) In this exam-ple, the snapshot information indicates that Qt = 2,the arrival rate is åt = 102 arrivals/minute and theutilization rate is � = 80%. For � = 5 minutes afterthe first snapshot, the distribution is concentratedaround Qt = 2, whereas for � = 25 minutes after, the

when the queue is longer (Kc and Terwiesch 2009 found a similareffect in the context of a healthcare delivery service).6 Using the Kolmogorov forward equations, one can show thatPt4s5 = eRt s . See Kulkarni (1995) for further details on obtaining atransition matrix from a transition rate matrix.7 It is tempting to also use the snapshot after � , A4�5, to estimate thedistribution of Q� . Note, however, that QA4�5 depends on whetherthe customer joined the queue or not, and is therefore endogenous.Simulation studies in §3.4 show that using QA4�5 in the estimationof Q� can lead to biased estimates.

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distribution is flatter and is closer to the steady statequeue distribution. The proposed methodology pro-vides a rigorous approach, based on queuing theoryand the periodic snapshot information, to estimate thedistribution of the unobserved data Q� at any point intime.

In our application where � is not observed, it isnecessary to integrate over all possible values of �to obtain the posterior distribution of Qiv, so thatPr4Qiv = k � tsiv5 =

∫

�Pr4Q� = k5dF 4� � tsiv5, where tsiv

is the observed checkout time of the customer trans-action. Therefore, given a distribution for � , F 4� � tsiv5,we can compute the distribution of Qiv, which canthen be used in Equation (1) for model estima-tion. In particular, the unobserved value Qiv can bereplaced by the point estimate that minimizes themean square prediction error, i.e., its expected valueE6Qiv7 (Step 3(b) in Figure 2).8

In our application, we discretize the support of � sothat each 30-minute snapshot interval is divided intoa grid of 1-minute increments, and calculate the queuedistribution accordingly. However, because we do nothave precise data to determine the distribution of theelapsed time between a deli visit and the cashier timestamp, an indirect method (described in the appendix)is used to estimate this distribution based on esti-mates of the duration of store visits and the locationof the deli within the supermarket. Based on this anal-ysis, we determined that a uniform in range 601307minutes prior to checkout time is a reasonable distri-bution for � .

Assumption 3. Customers visit the deli once, and thisvisiting time is uniformly distributed with range 601307minutes before checkout time.

To avoid problems of endogeneity, we determinethe distribution of Qiv conditioning on a snapshot thatis at least 30 minutes before checkout time (that is, thesecond snapshot before checkout time) to ensure thatwe are using a snapshot that occurs before the delivisit time.

Finally, Steps 1–3 in Figure 2 are run iteratively torefine the estimates of effective arrival rate �t4Q1E5,the system intensity �k, and the queue length Qiv.In our application, we find that the estimates con-verge quickly after three iterations.9

3.4. Simulation TestOur estimation procedure has several sources of miss-ing data that need to be inferred: time at which

8 Although formally the model assumes a single visit to the deli, theestimation is actually using a weighted average of many possiblevisit times to the deli. This makes the estimation more robust if inreality customers revisit the queue more than once in the hope offacing a shorter queue.9 As a convergence criteria, we used a relative difference of 0.1% orless between two successive steps.

the customer arrives at the deli is inferred from hercheckout time, and the state of the queue observedby a customer is estimated from the snapshot data.This subsection describes experiments using simu-lated data to test whether the proposed methodologycan indeed recover the underlying model parametersunder Assumptions 1 and 2.

The simulated data are generated as follows. First,we simulate a Markov queuing process with a singleserver: customers arrive following a Poisson processand join the queue with probability logit4f 4Q55, wheref 4Q5 is quadratic in Q and has the same shape aswe obtained from the empirical purchase incidencemodel. (We also considered piecewise linear specifi-cations, and the effectiveness of the method was sim-ilar.) After visiting the queue, the customer spendssome additional random time in the store (whichfollows a uniform 601307 minutes) and checks out.Snapshots are taken to record the queue length every30 minutes. The arrival rate and traffic intensity areset to be equal to the empirical average value.

Figure 5 shows a comparison of different estima-tion approaches. The black line, labeled True response,represents the customer’s purchase probabilities thatwere used to simulate the data. A consistent esti-mation should generate estimates that are close tothis line. Three estimation approaches, shown withdashed lines in the figure, were compared:

(i) Using the true state of the queue, Q� . Although thisinformation is unknown in our data, we use it as abenchmark to compare with the other methods. Asexpected, the purchase probability is estimated accu-rately with this method, as shown in the black dashedline.

(ii) Using the average of the neighboring snapshots12 4QB4�5 + QA4�55 and integrating over all possible valuesof � . Although the average of neighboring snapshotsprovides an intuitive estimate of Q� , this method gives

Figure 5 Estimation Results of the Purchase Incidence Model UsingSimulated Data

0 2 4 6 8 10 12 14 16 18 20

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

0.31

Queue

Pur

chas

e pr

obab

ility

True responseUsing actual queue length Q�

Using average of neighboringsnapshotsUsing conditional mean basedon previous snapshot

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biased estimates of the effect of the state of the queueon purchase incidence (the blue dotted-dashed line).This is because QA4�5, the queue length in the snapshotfollowing � , depends on whether the customer pur-chased or not, and therefore is endogenous (if the cus-tomer joins the queue, then the queue following herpurchase is likely to be longer). The bias appears to bemore pronounced when the queue is short, producinga (biased) positive slope for small values of Q� .

(iii) Using the inference method described in §3.3 toestimate Q� , depicted by the red dotted line. This givesan accurate estimate of the true curve. We conductedmore tests using different specifications for the effectof the state of the queue and the effectiveness of theestimation method was similar.

3.5. Choice ModelThere are three important limitations of using the pur-chase incidence model (1). The first is that it does notaccount for changes in a customer’s purchase proba-bility over time, other than through seasonality vari-ables. This could be troublesome if customers plantheir purchases ahead of time, as we illustrate withthe following example. A customer who does weeklyshopping on Saturdays and is planning to buy hamat the deli section visits the store early in the morn-ing when the deli is less crowded. This customervisits the store again on Sunday to make a few “fill-in” purchases at a busy time for the deli and doesnot buy any ham products at the deli because shepurchased ham products the day before. In the pur-chase incidence model, controls are indeed includedto capture the average purchase probability at the delifor this customer. However, these controls don’t cap-ture the changes to this purchase probability betweenthe Saturday and Sunday visits. Therefore, the modelwould mistakenly attribute the lower purchase inci-dence on the Sunday visit to the higher congestion atthe deli, whereas in reality the customer would nothave purchased regardless of the level of congestionat the deli on that visit.

A second limitation of the purchase incidencemodel (1) is that it cannot be used to attach an eco-nomic value to the disutility of waiting by customers.One possible approach would be to calculate anequivalent price reduction that would compensate thedisutility generated by a marginal increase in waiting.Model (1) cannot be used for this purpose becauseit does not provide a measure of price sensitivity.A third limitation is that model (1) does not explicitlycapture substitution with products that do not requirewaiting (e.g., the prepackaged section), which can beuseful to quantify the overall impact of waiting onstore revenues and profit.

To overcome these limitations, we use a randomutility model (RUM) to explain customer choice.

Table 2 Statistics for the 10 Most Popular Ham Products, asMeasured by the Percentage of Transactions in theCategory Accounted by the Product (Share)

Product Avg. price Std. dev. price Share (%)

1 0.67 0.10 21.232 0.40 0.04 9.373 0.53 0.06 7.124 0.59 0.06 6.135 0.64 0.07 5.666 0.24 0.01 5.497 0.52 0.07 3.978 0.54 0.07 3.109 0.56 0.07 2.8510 0.54 0.08 2.20

Note. Prices are measured in local currency per kilogram (one unit oflocal currency equals approximately US$21).

Because it is common in this type of model, the util-ity of a customer i for product j during a visit v,denoted Uijv, is modeled as a function of prod-uct attributes and parameters that we seek to esti-mate. Researchers in marketing and economics haveestimated RUM specifications using scanner datafrom a single product category (e.g., Guadagni andLittle 1983 model choices of ground coffee products;Bucklin and Lattin 1991 model saltine crackers pur-chases; Fader and Hardie 1996 model fabric softenerchoices; Rossi et al. 1996 model choices among tunaproducts). Note that although deli purchases includemultiple product categories, using a RUM to modelcustomer choice requires us to select a single productcategory for which purchase decisions are indepen-dent from choices in other categories and where cus-tomers typically choose to purchase at most one SKUin the category. The ham category appears to meetthese criteria. The correlations between purchases ofham and other cold-cut categories are relatively small(all less than 8% in magnitude). About 93% of thetransactions with ham purchases included only oneham SKU. In addition, it is the most popular cate-gory among cold-cuts, accounting for more than 33%of the total sales. The ham category has 75 SKUs, 38 ofwhich are sold in the deli and the rest in the prepack-aged section, and about 85% of ham sales are gener-ated in the deli section. In what follows, we describean RUM framework to model choices among prod-ucts in the ham category. Table 2 shows statistics fora selection of products in the ham category.

One advantage of using a RUM to characterizechoices among SKUs in a category is that it allowsus to include product specific factors that affectsubstitution patterns. Although many of the prod-uct characteristics do not change over time and canbe controlled by a SKU-specific dummy, our datareveal that prices do fluctuate over time and could bean important driver of substitution patterns. Accord-ingly, we incorporate product-specific dummies, �j ,

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and product prices for each customer visit (Pricevj )as factors influencing customers’ utility for product j .Including prices in the model also allows us to esti-mate customer price sensitivity, which we use to puta dollar tag on the cost of waiting.

As in the purchase incidence model (1), it is impor-tant to control for customer heterogeneity. Because ofthe size of the data set, it is computationally chal-lenging to estimate a choice model including fixedeffects for each customer. Instead, we control for eachcustomer’s average buying propensity by includinga covariate measuring the average consumption rateof that customer, denoted CRi. This consumption ratewas estimated using calibration data as done by Belland Lattin (1998). We also use the methods developedby these authors to estimate customers’ inventory ofham products at the time of purchase, based on a cus-tomers’ prior purchases and their consumption rateof ham products. This measure is constructed at thecategory level and is denoted by Inviv.

We use the following notation to specify the RUM.Let J be the set of products in the product categoryof interest (i.e., ham); JW is the set of products thatare sold at the deli section and, therefore, potentiallyrequire the customer to wait, JNW = J\JW is the setof products sold in the prepackaged section, whichrequire no waiting. Let Tv be a vector of covariatesthat capture seasonal sales patterns, such as holidaysand time trends. Define Freshj as a binary variableindicating whether product j is sold in the deli (i.e.,j ∈ JW ). Using these definitions, customer i’s utility forpurchasing product j during store visit v is specifiedas follows:

Uijv = �j +�qi f 4Qiv1 Eiv5Freshj +�Fresh

i Freshj

+�Pricei Pricejv +�crCRi +� invInviv

+�T Tv + �ijv1 (2)

where �ijv is an error term capturing idiosyncraticpreferences of the customer, and f 4Qiv1 Eiv5 capturesthe effect of the state of the queue on customers’preference. Note that the indicator function 16j ∈ JW 7adds the effect of the queue only to the utility ofthose products that are sold at the deli section (i.e.,j ∈ JW ) and not to products that do not require wait-ing. As in the purchase incidence model (1), the stateof the queue 4Qiv1 Eiv5 is not perfectly observed, butthe method developed in §3.3 can be used to replacethese by point estimates.10 An outside good, denoted

10 In our empirical analysis, we also performed a robustness checkwhere instead of replacing the unobserved queue length Qiv bypoint estimates, we sampled different queue lengths from theestimated distribution of Qiv. The results obtained with the twoapproaches are similar.

by j = 0, accounts for the option of not purchasingham, with utility normalized to Ui0v = �i0v. The inclu-sion of an outside good in the model enables us toestimate how changes in waiting time affect the totalsales of products in this category (i.e., category sales).

Assuming a standard extreme value distributionfor �ijv, the RUM described by Equation (2) becomesa random coefficients multinomial logit. Specifically,the model includes consumer-specific coefficients forPrice (�Price

i ); the dummy variable for products soldin the deli (�Fresh

i ), as opposed to products sold inthe prepackage section; and some of the coefficientsassociated with the effect of the queue (�q

i ). These ran-dom coefficients are assumed to follow a multivariatenormal distribution with mean È = 4�Price1 �Fresh1 �q5′

and covariance matrix ì, which we seek to estimatefrom the data. Including random coefficients for Priceand Fresh is useful to accommodate more flexiblesubstitution patterns based on these characteristics,overcoming some of the limitations imposed by theindependence of irrelevant alternatives of standardmultinomial logit models. For example, if customersare more likely to switch between products with simi-lar prices or between products that are sold in the deli(or alternatively, in the prepackaged section), thenthe inclusion of these random coefficients will enableus to model that behavior. In addition, allowing forcovariation between �Price

i , �Freshi , and �

qi provides use-

ful information on how customers’ sensitivity to thestate of the queue relates to the sensitivity to the othertwo characteristics.

The estimation of the model parameters is imple-mented using standard Bayesian methods (see Rossiand Allenby 2003). The goal is to estimate (i) theSKU dummies �j ; (ii) the effects of the consumptionrate (�cr ), inventory (� inv), and seasonality controls(�T ) on consumer utility; and (iii) the distributionof the price and queue sensitivity parameters, whichis governed by È and ì. To implement this estima-tion, we define prior distributions on each of theseparameters of interest: �j ∼ N4�1��5, � ∼ N4�1��5,� ∼ N4�1��5, and ì ∼ Inverse Wishart(df, Scale). Forestimation, we specify the following parameter val-ues for these prior distributions: � = � = � = 0, �� =

�� = �� = 100, df = 3, and Scale equal to the identitymatrix. These choices produce weak priors for param-eter estimation. Finally, the estimation is carried outusing Markov chain Monte Carlo (MCMC) methods.In particular, each parameter is sampled from its pos-terior distribution conditioning on the data and allother parameter values (Gibbs sampling). When thereis no closed-form expression for these full-conditionaldistributions, we employ Metropolis–Hastings meth-ods (see Rossi and Allenby 2003). The outcome of thisestimation process is a sample of values from the pos-terior distribution of each parameter. Using these val-ues, a researcher can estimate any relevant statistic of

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the posterior distribution, such as the posterior mean,variance, and quantiles of each parameter.

4. Empirical ResultsThis section reports the estimates of the purchase inci-dence model (1) and the choice model (2) using themethodology described in §3.3 to impute the unob-served state of the queue.

4.1. Purchase Incidence Model ResultsTable 3 reports a summary of alternative specifica-tions of the purchase incidence model (1). All of thespecifications include customer fixed effects (11,487 ofthem), daily dummies (192 of them), and hour-of-daydummies interacted with weekend/holiday dummies(30 of them). A likelihood ratio test indicates thatthe daily dummies and hour of the day interactedwith weekend/holiday dummies are jointly signifi-cant (p-value < 000001), and so are the customer fixedeffects (p-value < 000001).

Different specifications of the state of the queueeffect are compared, which differ in terms of (1) thefunctional form for the queuing effect f 4Q1 E1�q5,including linear, piecewise linear, and quadratic poly-nomial; and (2) the measure capturing the effect of thestate of the queue, including (i) expected time towait, W = Q/E, and (ii) the queue length, Q (weomit the tilde in the table). In particular, Models I–IIIare linear, quadratic, and piecewise linear (with seg-ments at 40151101155) functions of W ; Model IV–VIare the corresponding models of Q. We discuss othermodels later in this section. The table reports thenumber of parameters associated with the queu-ing effects (dim(�q)), the log-likelihood achieved inthe maximum likelihood estimation (MLE), and twoadditional measures of goodness of fit, the Akaikeinformation criterion (AIC) and Bayesian informationcriterion (BIC), that are used for model selection.

Using AIC and BIC to rank the models, the specifica-tions with Q as explanatory variables (Models IV–VI)all fit significantly better than the corresponding oneswith W (Models I–III), suggesting that purchase inci-dence appears to be affected more by the length ofthe queue rather than the speed of the service. A com-parison of the estimates of the models based on Q

Table 3 Goodness-of-Fit Results on Alternative Specifications of the Purchase Incidence Model (Equation (1))

Model Function form Metric Dim(�q ) Log-likelihood AIC Rank BIC Rank

I Linear W 1 −118119503 259180806 5 382102304 3II Quadratic W 2 −118119301 259180602 4 382103105 4III Piecewise W 4 −118119208 259180907 6 382105508 6IV Linear Q 1 −118118905 259179700 3 382101108 1V Quadratic Q 2 −118118504 259179008 1 382101600 2VI Piecewise Q 4 −118118409 259179307 2 382103908 5

is shown in Table 4 and Figure 6 (which plots theresults of Model IV–VI). Considering Models V andVI, which allow for a nonlinear effect of Q, the pat-tern obtained in both models is similar: customersappear to be insensitive to the queue length when itis short, but they balk when experiencing long lines.This impact on purchase incidence can become quitelarge for queue lengths of 10 customers and more.In fact, our estimation indicates that increasing thequeue length from 10 to 15 customers would reducepurchase incidence from 30% to 27%, correspondingto a 10% drop in sales.

The AIC scores in Table 3 also suggest that themore flexible models, V and VI, tend to provide a bet-ter fit than the less flexible linear model, IV. The BICscore, which puts a higher penalization for the addi-tional parameters, tends to favor the more parsimo-nious quadratic models, V, and the linear model, IV.Considering both the AIC and BIC score, we con-clude that the quadratic specification on queue length(Model V) provides a good balance of flexibility andparsimony, and hence we use this specification as abase for further study.

To further compare the models including queuelength versus expected time to wait, we estimated aspecification that includes quadratic polynomials ofboth measures, Q and W . Note that this specificationnests Models II and V (but it is not shown in thetable). We conducted a likelihood ratio test by com-paring log-likelihoods of this unrestricted model withthe restricted models II and V. The test shows thatthe coefficients associated with W are not statisticallysignificant, whereas the coefficients associated withQ are. This provides further support that customersput more weight on the length of the line rather thanon the expected waiting time when making purchaseincidence decisions.

In addition, we consider the possibility that themeasure W = Q/E may not be a good proxy forexpected time to wait if the service rate of the attend-ing employees varies over time and customers cananticipate these changes in the service rate. Recall,however, that our analysis in §3.3 estimates separateservice rates for different days and hours, and showsthat there is small variation across time. Neverthe-less, we constructed an alternative proxy of expected

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Table 4 MLE Results for Purchase Incidence Model (Equation (1))

Variable Coef. Std. err. z

Model IV (linear) Q −000133 000024 −5046

Model V (quadratic) Q− 507 −0000646 0000340 −10904Q− 50752 −0000166 0000066 −2050

Model VI (piecewise) Q0−5 000056 000079 0071Q5−10 −000106 000042 −3054Q10−15 −000199 000068 −2092Q15+ −000303 000210 −1044

Note. In the quadratic model (Model V), the length of the queue is centeredat the mean of 5.7.

time to wait that accounts for changes in the ser-vice rate: W ′ = W /�, where � is the estimated servicerate for the corresponding time period. Replacing Wby W ′ leads to estimates that are similar to modelsreported in Table 3.

Although the expected waiting time does not seemto affect customer purchase incidence as much asthe queue length, it is possible that customers dotake into account the capacity at which the systemis operating—i.e., the number of employees—in addi-tion to the length of the line. To test this, we esti-mated a specification that includes both the queuelength Q (as a quadratic polynomial) and the num-ber of servers E as separate covariates.11 The resultssuggest that the number of servers, E, has a posi-tive impact that is statistically significant, but smallin magnitude (the coefficient is 0.0201 with standarderror 0.0072). Increasing staff from 1 to 2 at the aver-age queue length only increases the purchase prob-ability by 0.9%. To compare, shortening the queuelength from 12 to 6 customers, which is the aver-age length, would increase the purchase probabilityby 5%. Because both scenarios halve the waiting time,this provides further evidence that customers focusmore on the queue length than the objective expectedwaiting time when making purchase decisions. Wealso found that the effect of the queue length inthis model is almost identical to the one estimatedin Model V (which omits the number of servers).We therefore conclude that although the capacity doesseem to play a role in customer behavior, its effect isminor relative to the effect of the length of the queue.

Finally, we emphasize that the estimates provide anoverall effect of the state of the queue on customerpurchases. The estimates suggest that, for queuelengths above the mean (about five customers in line)the effect is significantly negative, which implies thatthe disutility of waiting seems to dominate any poten-tial herding effects of the queue, whereas for queue

11 We also estimated models with a quadratic term for E, but thisadditional coefficient was not significant.

Figure 6 Results from Three Different Specifications of the PurchaseIncidence Model

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lengths below the mean, neither effect is dominant.In our context, herding effects could still be observed,for example, if customers passing by the deli sectioninfer from a long line that the retailer must be offer-ing an attractive deal, or if long lines make the delisection more salient. While the absence of a domi-nant herding effect seems robust for the average cus-tomer, we further tested Model V on subsamples offrequent customers (i.e., customers that made 30 ormore visits during the study period) and infrequentcustomers (i.e., customers that made less than 30 vis-its), with the idea that infrequent customers would beless informed and might potentially learn more fromthe length of the line. However, we found no signif-icant differences between the estimates. We also par-titioned customers into new customers and existingcustomers (customers are considered to be new withinthe first two months of their first visit), with the ideathat new customers should be less informed.12 Again,we found no significant differences in the estimatedresults for the two groups. In summary, the statisti-cal evidence in our results are not conclusive on thepresence of dominant herding effects.

4.2. Choice Model ResultsIn this subsection, we present and discuss the resultsobtained for the choice model described in §3.5. Thespecification for the queuing effect f 4Q1 E5 is basedon the results of the purchase incidence model. In par-ticular, we used a quadratic function of Q, which bal-anced goodness of fit and parsimony in the purchaseincidence model. The utility specification includesproduct-specific intercepts, prices, consumption rate,

12 We used one year of transaction data prior to the study period toverify the first customer visit date. We also tried other definitionsof new customers (within three months of the first visit), and theresults were similar.

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Table 5 Estimation Results for the Choice Model (Equation (2))

Average effect Variance/covariance (ì)

Estimate Std. err. Estimate Std. err.

Inv −00091 00026 ì4Price,Price) 310516 10671CR 10975 00150 ì4Fresh, Fresh) 70719 00436Fresh 00403 00112 ì4Q1 Q) 00403 00083Price −90692 00203 ì4Fresh, Q) 00020 00144Q −00058 00061 ì4Price, Fresh) −140782 00821Q2 −00193 00122 ì4Price, Q) −00508 00267

Note. The estimate and standard error of each parameter correspond to themean and standard deviation of its posterior distribution, respectively.

household inventory, and controls for seasonality asexplanatory variables. The model incorporates het-erogeneity through random coefficients for Price, theFresh dummy, and the linear term of the length of thequeue (Q). We use 2,000 randomly selected customersin our estimation. After running 20,000 MCMC iter-ations and discarding the first 10,000 iterations, weobtained the results presented in Table 5 (the tableomits the estimates of the product-specific interceptand seasonality). The left part of the table shows theestimates of the average effects, with the estimatedstandard error (measured by the standard devia-tion of the posterior distribution of each parameter).The right part of the table shows the estimates ofthe variance–covariance matrix (ì) characterizing theheterogeneity of the random coefficients �Price

i , �Freshi ,

and �qi .

Price, inventory, and consumption rate all have thepredicted signs and are estimated precisely. The aver-age of the implied price elasticities of demand is −3.The average effects of the queue coefficients implyqualitatively similar effects as those obtained in thepurchase incidence model: consumers are relativelyinsensitive to changes in the queue length in the Q = 0to Q = 5 range, and then the purchase probabilitystarts exhibiting a sharper decrease for queue lengthvalues at or above Q = 6.

These results can also be used to assign a monetaryvalue to customers’ cost of waiting. For example, foran average customer in the sample, an increase from5 to 10 customers in queue is equivalent to a 1.7%increase in price. Instead, an increase from 10 to 15customers is equivalent to a 5.5% increase in price,illustrating the strong nonlinear effect of waiting oncustomer purchasing behavior.

The estimates also suggest substantial heterogene-ity in customers’ price sensitivities (estimates on theright side of Table 5). The estimated standard devia-tion of the random price coefficients is 5.614, whichimplies a coefficient of variation of 57.9%. There isalso significant heterogeneity in customer sensitivityto waiting, as measured by the standard deviation ofthe linear queue effect, which is estimated to be 0.635.

The results also show a negative relationship betweenprice and waiting sensitivity and between price andthe fresh indicator variable.

To illustrate the implications of the model esti-mates in terms of customer heterogeneity, we mea-sured the effect of the length of the queue on threecustomer segments with different levels of price sen-sitivity: price coefficients equal to the mean, onestandard deviation below the mean (labeled highprice sensitivity), and one standard deviation abovethe mean (labeled low price sensitivity), respectively.To compute these choice probabilities, we consideredcustomer visits with average levels of prices, con-sumption rate, and consumer inventory. Given thenegative correlation between the price random coef-ficient and the two other random coefficients, cus-tomers with a weaker price sensitivity will in turnhave stronger preferences for fresh products and ahigher sensitivity to the length of the queue and,hence, be more willing to wait to buy fresh products.Figure 7 illustrates this pattern, showing a strongereffect of the length of the queue in the purchase prob-ability of the low price sensitivity segment. Interest-ingly, the low price sensitivity segment is also themost profitable, with a purchase incidence that morethan doubles that of the high price sensitivity segment(for small values of the queue length). This has impor-tant implications for pricing product categories undercongestion effects, as we discuss in the next section.

Finally, because our choice model also considersproducts that do not require waiting, we measurethe extent by which lost sales of fresh products dueto a higher queue length are substituted by sales ofthe prepackaged products. In this regard, our resultsshow that when the length of the line increases, forexample, from 5 to 10 customers, only 7% of thedeli lost sales are replaced by nondeli purchases.This small substitution effect can be explained by thelarge heterogeneity of the Fresh random coefficient

Figure 7 Purchase Probability for Ham Products in the Deli Sectionvs. Queue Length for Three Customer Segments withDifferent Price Sensitivity

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together with the relatively small share of purchasesof prepackaged products that we observe in the data.

5. Managerial ImplicationsThe results of the previous section suggest that(1) purchase incidence appears to be affected moreby the length of the line rather than the speed of theservice, and (2) there is heterogeneity in customers’sensitivity to the queue length, which is negativelycorrelated with their price sensitivity. We discussthree important managerial insights implied by thesefindings. The first shows that pooling multiple identi-cal queues into a single multiserver queue may lead toan increase in lost sales. The second considers the ben-efit of adding servers when making staffing decisions.The third discusses the implications of the externali-ties generated by congestion for pricing and promo-tion management in a product category.

5.1. Queuing DesignThe result from the purchase incidence model thatcustomers react more to the length of the queue thanthe speed of service has implications on queuing man-agement policies. In particular, we are interested incomparing policies between splitting versus mergingqueues.

It is well known that an M/M/c pooled queuingsystem achieves a much lower waiting time than asystem with separate M/M/1 queues at the same uti-lization levels. Therefore, if waiting time is the onlymeasure of customer service, then pooling queues isbeneficial. However, Rothkopf and Rech (1987) pro-vide several reasons for why pooling queues could beless desirable. For example, there could be gains fromserver specialization that can be achieved in the sep-arate queue setting. Cachon and Zhang (2007) look atthis issue in a setting where two separate queues com-pete against each other for the allocation of (exoge-nous) demand and show that using a system withseparate queues is more effective (relative to a pooledsystem) at providing the servers with incentives toincrease the service rate. The results in our paper pro-vide another argument for why splitting queues maybe beneficial: although the waiting time in the pooledsystem is shorter, the queue is longer, and this caninfluence demand. If customers make their decision ofjoining a queue based on its queue length, as we findin our empirical study, then a pooled system can leadto fewer customers joining the system and thereforeincrease lost sales. We illustrate this in more detailwith the following example.

Consider the following queuing systems: a pooledsystem given by an M/M/2 queue with constantarrival rate � and a split join-shortest-queue ( JSQ) sys-tem with two parallel single-server queues with thesame overall arrival given by a Poisson process with

rate � and where customers join the shortest queueupon arrival, and assuming that after joining a linecustomers don’t switch to a different line (i.e., no jock-eying). If there is no balking—that is, all customersjoin the queue—it can be shown that the pooledsystem dominates the split JSQ system in terms ofwaiting time. However, the queues are longer in thepooled system, so if customers may walk away uponarrival and this balking rate increases with the queuelength, then the pooled system may lead to fewersales.

To evaluate the differences between the two sys-tems, we numerically compute the average wait-ing time and revenue for both. For the split JSQsystem, the approximate model proposed by Raoand Posner (1987) is used to numerically evaluatethe system performance. When the queue length isequal to n and the number of servers is c, thearrival rate is �Pr4join � Q = n1E = c5, where Pr4join �

Q = n1E = c5 is customers’ purchase probability.In this numerical example, we set the purchase prob-abilities based on the estimates of the purchase inci-dence model that includes the quadratic specificationof Q. Traffic intensity is defined as �= maxn �Pr4join �

Q = n1E = c5/�, and revenue is defined as the num-ber of customers that join the queue. Figure 8 showsthe long-run steady-state average waiting time andaverage revenue of the two systems. As expected,the pooled M/M/2 system always achieves a shorterwaiting time. However the M/M/2 system generatesless revenue because it suffers more traffic loss due tolong queues, and the difference increases as the traf-fic intensity approaches one. In our particular case,the split JSQ system gains 2.7% more revenue whileincreasing the average waiting time by more than 70%at the highest level of utilization compared to thepooled system. These results imply that when movingtoward a pooled system, it may be critical to provide

Figure 8 Comparison Between the Split JSQ and Pooled Systems

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information about the expected waiting time so thatcustomers do not anchor their decisions primarily onthe length of the line, which tends to increase whenthe system is pooled.

5.2. Implications for Staffing DecisionThe model used in 5.1 also provides insights formaking staffing decisions. For example, consider atypical weekday 11:00–12:00 time window versus aweekend 11:00–12:00 window. Given the average cus-tomer arrival rates observed at the deli, the minimumcapacity needed to meet the demand is one server forthe weekday and two for the weekend. The impliedutilizations are 75% and 97% for weekdays and week-ends, respectively. We use our empirical results toevaluate whether it pays off to add one server in eachof these time windows.

In our sample, the average amount that a cus-tomer spends at the deli is US$3.3. The estimates fromthe purchase incidence model suggest that adding aserver leads to an increase on purchases of 2% and 7%for the weekday and weekend windows, respectively.This translates into a US$2.3 increase of hourly rev-enue for the weekday, and a US$20.7 increase for theweekend. In the supermarket of our study, an addi-tional server costs approximately US$3.75 per hour(for full-time staff). The contribution margin is typi-cally in the 10%–25% range for this product category.Hence, it may be profitable to add a server duringthe weekend 11:00–12:00 period (when the margin is18% or higher), but not profitable during the week-day 11:00–12:00 period. Interestingly, the supermarketstaffing policy seems to be aligned with this result:the snapshot data reveal that between 30%–40% ofthe time, the deli had a single server staffed dur-ing that hour on weekdays, whereas for the weekendmore than 75% of the snapshots showed three or moreservers.13

5.3. Implications for Category PricingThe empirical results suggest that customers whoare more sensitive to prices are less likely to changetheir probability of purchasing fresh deli productswhen the length of the queue increases. This can haveimportant implications for the pricing of productsunder congestion effects, as we show in the followingillustrative example.

Consider two vertically differentiated products,H and L, of high and low quality respectively, withrespective prices pH > pL. Customers arrive accord-ing to a Poisson process to join an M/M/1 queue to

13 The revenue increase was estimated using specification V fromTable 4. We repeated the analysis using a model where customersalso account for the number of employees staffed, and the resultswere similar.

buy at most one of these two products. Followingmodel (2), customer preferences are described by anMNL model, where the utility for customer i if buyingproduct j ∈ 8L1H9 is given by Uij = �j −�

pi pj −�

qiQ+

�i + �ij . Customer may also choose not to join thequeue and get a utility equal to Ui0 = �i0. In this RUM,�j denotes the quality of the product, and Q is a ran-dom variable representing the queue length observedby the customer upon arrival. Customers have hetero-geneous price and waiting sensitivity characterizedby the parameters �

pi and �

qi . In particular, hetero-

geneity is modeled through two discrete segments,s = 81129, with low and high price sensitivity, respec-tively, and each segment accounts for 50% of thecustomer population. (Later in this section we willalso consider a continuous heterogeneity distributionbased on our empirical results.) Let �p

1 and �p2 be the

price coefficients for these segments, with 0 <�p1 <�

p2.

In addition, the waiting sensitivity, �qi , is a random

coefficient that can take two values: �h with probabil-ity rs , and �l with probability 1 − rs , where s denotesthe customer segment and �l < �h. This characteri-zation allows for price and waiting sensitivity to becorrelated: if r1 > r2, then a customer with low pricesensitivity is more likely to be more waiting-sensitive;if r1 = r2, then there is no correlation.

Consider first a setting with no congestion so that Qis always zero (for example, if there is ample ser-vice capacity). For illustration purposes, we fixedthe parameters as follows: �H = 151 pH = 51 �L = 51pL = 105, �

p1 = 11 �1 = 01 �p

2 = 10, and �2 = 12. In thisexample, the difference in quality and prices betweenthe two products is sufficiently large so that most ofthe price sensitive customers (s = 2) buy the low qual-ity product L. Moreover, define the cross-price elastic-ity of demand EHL as the percentage increase in salesof H product from increasing the price of L by 1%,and vice versa for ELH . In this numerical example,we allow for significant heterogeneity with respectto price sensitivity such that, in the absence of con-gestion, the cross-elasticities between the two prod-ucts are close to zero (to be exact, EHL = 00002 andELH = 00008).

Now consider the case where customers observequeues. This generates an externality: increasing thedemand of one product generates longer queues,which decreases the utility of some customers whomay in turn decide not to purchase. Hence, lower-ing the price of one product increases congestion andthereby has an indirect effect on the demand of theother product, which we refer to as the indirect cross-elasticity effect.

We now show how customer heterogeneity andnegative correlation between price and waiting sensi-tivity can increase the magnitude of the indirect cross-elasticity between the two products. We parametrized

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the waiting sensitivity of each segment as �l = 1025 −

005ã and �h = 1025 + 005ã, where ã is a measure ofheterogeneity in waiting sensitivity. We also variedthe conditional probabilities r1 and r2 to vary the cor-relation between waiting and price sensitivity whilekeeping the marginal distribution of waiting sensi-tivity constant (50% �l and 50% �h). Fixing all theparameters of the model (including prices pH and pL),it is possible to calculate the stationary probabilities ofthe queue length Q. Using the RUM together with thisstationary distribution, it is then possible to calculatethe share of each product (defined as the fraction ofarriving customers that buy each product). Applyingfinite differences with respect to prices, one can thencalculate cross-elasticities that account for the indirecteffect through congestion.

Based on this approach, we evaluated the cross-elasticity of the demand for the H product whenchanging the price of the L product (EHL) for dif-ferent degrees of heterogeneity in customer sensitiv-ity to wait (ã) and several correlation patterns. Theresults of this numerical experiment are presented inTable 6. Note that in the absence of heterogeneity—that is, ã= 0—the cross-price elasticity is low: thetwo products H and L appeal to different customersegments, and there is little substitution betweenthem. However, adding heterogeneity and correla-tion can lead to a different effect. In the presenceof heterogeneity, a negative correlation between priceand waiting sensitivity increases EHL, showing thatthe indirect cross-elasticity increases when the wait-ing sensitive customers are also the least sensitiveto price. The changes in cross-elasticity due to corre-lation can become quite large for higher degrees ofcustomer heterogeneity. In the example, when ã = 2,the cross-elasticity changes from 0.011 to 0.735 whenmoving from positive to negative correlation patterns.

We now discuss the intuition behind the patternsobserved in the example of Table 6. When there is het-erogeneity in price sensitivity, lowering the price ofthe L product attracts customers who were not pur-chasing before the price reduction (as opposed to can-nibalizing the sales of the H product). Because of thisincrease in traffic, congestion in the queue increases,generating longer waiting times for all customers.

Table 6 Cross-Price Elasticities Describing Changes in theProbability of Purchase of the High-Price Product 4H5 fromChanges in the Price of the Low-Price Product 4L5

Correlation between price and waiting sensitivity

Heterogeneity −009 −005 0 0.5 0.9

ã= 000 — — 00042 — —ã= 100 00342 00228 00120 00047 00010ã= 200 00735 00447 00209 00070 00011

But when price and waiting sensitivity are negativelycorrelated, the disutility generated by the congestionwill be higher for the less price sensitive customers,and they will be more likely to walk away after theprice reduction in L. Because a larger portion of thedemand for the H product comes from the less pricesensitive buyers, the indirect cross-price elasticity willincrease as the correlation between price and waitingsensitivity becomes more negative.

Although the above example uses discrete cus-tomer segments, similar effects occur when consid-ering heterogeneity described through a continuousdistribution, as in our empirical model. Similar to theprevious discrete case example, we assume the util-ity for customer i to purchase j is given by Uij =

�j − �pi pj + f 4�

qi 1Q5+ �i. But now the queue effect is

specified by the quadratic form with random coeffi-cients for 4�p1�q1 �5, which are normally distributedwith the same covariance matrix as the one estimatedin Table 5. Prices pL and pH are picked to reflect thetrue price of high end and low end products, and �to reflect the empirical average arrival rate in the delisession. In this case, our calculation shows a cross-price elasticity equal to EHL = 0081. In a counterfactualthat forces the waiting sensitivity �q to be indepen-dent of the other random coefficients 4�p1 �5, the priceelasticity EHL drops to 0.083, one order of magnitudesmaller, showing qualitatively similar results to thosefrom the discrete heterogeneity example.

In summary, the relationship between price andwaiting sensitivity is an important factor affecting theprices in a product category when congestion effectsare present. Congestion can induce price–demandinteractions among products that in the absenceof congestion would have a low direct cross-price-elasticity of demand. Our analysis illustrates howheterogeneity and negative correlation between priceand waiting sensitivity can exacerbate these interac-tions through stronger indirect cross-elasticity effects.This can have important implications on how to setprices in the presence of congestion.

6. ConclusionsIn this paper, we use a new data set that linksthe purchase history of customers in a super-market with objective service level measures tostudy how an important component of the serviceexperience—waiting in queue—affects customer pur-chasing behavior.

An important contribution of this paper is method-ological. An existing barrier to studying the impactof service levels on customer buying behavior inretail environments comes from the lack of objec-tive data on waiting time and other customer ser-vice metrics. This work uses a novel data collection

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technique to gather high frequency store operationalmetrics related to the actual level of service deliveredto customers. Because of the periodic nature of thesedata, an important challenge arises in linking the storeoperational data with actual customer transactions.We develop a new econometric approach that relieson queuing theory to infer the level of service asso-ciated with each customer transaction. In our view,this methodology could be extended to other contextswhere periodic service level metrics and customertransaction data are available. This methodology alsoenables us to estimate a comprehensive descriptivemodel of how waiting in queue affects customer pur-chase decisions. Based on this model, we provide use-ful prescriptions for the management of queues andother important aspects of service management inretail. In this regard, a contribution of our work is tomeasure the overall impact of the state of the queueon customer purchase incidence, thereby attaching aneconomic value to the level of service provided. Thisvalue of service together with an estimate of the rel-evant operating costs can be used to determine anoptimal target service level, a useful input for capac-ity and staffing decisions.

Second, our approach empirically determines themost important factors in a queuing system that influ-ence customer behavior. The results suggest that cus-tomers seem to focus primarily on the length of theline when deciding to join a queue, whereas thenumber of servers attending the queue, which deter-mines the speed at which the queue advances, hasa much smaller impact on customers’ decisions. Thishas implications for the design of a queuing sys-tem. For example, although there are several benefitsof pooling queues, the results in this paper suggestthat some precautions should be taken. In movingtoward a pooled system, it may be critical to pro-vide information about the expected waiting time sothat customers are not drawn away by longer queues.In addition, our empirical analysis provides strongevidence that the effect of waiting on customer pur-chases is nonlinear. Hence, measuring extremes in thewaiting distribution—for example, the fraction of thetime that 10 or more customers are waiting in queue—may be more appropriate than using average waitingtime to evaluate the system’s performance.

Third, our econometric model can be used to seg-ment customers based on their waiting and pricesensitivities. The results show that there is indeed asubstantial degree of heterogeneity in how customersreact to waiting and price, and moreover, the wait-ing and price sensitivity are negatively correlated.This has important implications for the pricing of aproduct category where congestion effects are present.Lowering prices for one product increases demand forthat alternative, but also raises congestion, generating

a negative externality for the demand of other prod-ucts from that category. Heterogeneity and negativecorrelation in price and waiting sensitivity exacerbatethis externality, and therefore should be accountedfor in category pricing decisions. We hope that thisempirical finding fosters future analytical work tostudy further implications of customer heterogeneityon pricing decisions under congestion.

Finally, our study has some limitations that couldbe explored in future research. For example, our anal-ysis focuses on studying the short-term implicationsof queues by looking at how customer purchases areaffected during a store visit. There could be long-termeffects whereby a negative service experience alsoinfluences future customer purchases, for example,the frequency of visits and retention. Another possi-ble extension would be to measure how observablecustomer characteristics—such as demographics—arerelated to their sensitivity to wait. This would be use-ful, for example, to prescribe target service levels fora new store based on the demographics of the mar-ket. Competition could also be an important aspect toconsider; this would probably require data from mul-tiple markets to study how market structure mediatesthe effect of queues on customer purchases.

On a final note, this study highlights the impor-tance of integrating advanced methodologies fromthe fields of operations management and marketing.We hope that this work stimulates further research onthe interface between these two academic disciplines.

AcknowledgmentsThe authors thank seminar participants at the University ofNorth Carolina at Chapel Hill, Duke University, HarvardUniversity, the Consortium for Operational Excellence inRetail, the 2011 Marketing Science Conference, the 2011INFORMS Conference, the 2011 London Business SchoolInnovation in Operations Conference, the 2011 M&SOMService Operations Special Interest Group, and the 2011Wharton Workshop on Empirical Research in OperationsManagement for their useful feedback. They also thankSCOPIX for providing the data used in this study. MarceloOlivares acknowledges partial support from FONDECYT[Grant 1120898].

Appendix. Determining the Distribution forDeli Visit TimeOur estimation method requires integrating over differentpossible values of deli visit time. This appendix describeshow to obtain an approximation of this distribution. Ourapproach follows two steps. First, we seek to estimate thedistribution of the duration of a supermarket visit. Second,based on the store layout and previous research on cus-tomer paths in supermarket stores, we determine (approxi-mately) in which portion of the store visit customers wouldcross the deli.

In terms of the first step, to get an assessment of the dura-tion of a customer visit to the store, we conducted some

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Table A.1 Regression Results

Estimate Std. err.

�0 00069∗ 4000215�1 00107∗ 4000265�2 00101∗ 4000275�3 00063∗ 4000275�4 00066∗ 4000275�5 00029 4000275�6 −00020 4000235N 879R2 00928

Note. Standard errors are in parentheses.∗p < 0005.

additional empirical analysis using store foot traffic data.Specifically, we collected data on the number of customersthat entered the store during 15-minute intervals (for themonth of February of 2009). With these data, our approachrequires discretizing the duration of a visit in 15-minutetime intervals. Accordingly, let T denote a random variablerepresenting the duration of visit, from entry until finish-ing the purchase transaction at the cashier, with support in801 11 21 31 41 51 69; T = 0 is a visit of 15 minutes or less,T = 1 corresponds to a visit between 15 and 30 minutes, andsimilarly for the other values. Let �t = Pr4T = t5 denote theprobability mass function of this random variable. Not allcustomers that enter the store go through the cashier: withprobability � a customer leaves the store without purchas-ing anything. Hence,

∑6t=0 �t + � = 1. Note that 8�t9t=0100016

and � completely characterize the distribution of the visitduration T .

Let Xt be the number of entries observed during period t,and let Yt be the total number of observed transactions inthe cashiers during that period. We have

E4Yt � 8Xr 9r≤t5=

6∑

s=0

Xt−s�s 0

Because the conditional expectation of Yt is linear in thecontemporaneous and lagged entries Xt1 0 0 0 1Xt−6, the distri-bution of the duration of the visit can be estimated throughthe linear regression

Yt =

6∑

s=0

Xt−s�s +ut 0

Note that the regression does not have an intercept.Table A.1 shows the ordinary least squares (OLS) estimatesof this regression.14

The parameters �0 through �4 are positive and statisti-cally significant. (The other parameters are close to zeroand insignificant, so we consider those being equal tozero.) Conditional on going through the cashier, approxi-mately 70% of the customers spend 45 minutes or less inthe store (calculated as

∑2t=0 �t/

∑4t=0 �t), and 85% of them

14 The parameters of the regression could be constrained to be pos-itive and to sum to less than one. However, in the unconstrainedOLS estimates, all the parameters that are statistically significantsatisfy these constraints.

less than an hour. The average duration of a visit is approx-imately 35 minutes. Moreover, the distribution of the dura-tion of the store visit could be approximated reasonablywell by a uniform distribution with range 601757 minutes.

To further understand the time at which a customer vis-its the deli, it is useful to understand the path that a cus-tomer follows during a store visit. In this regard, the studyby Larson et al. (2005) provides some information of typ-ical customer shopping paths in supermarket stores. Theyshow that most customers tend to follow a shopping paththrough the “racetrack”—the outer ring of the store that iscommon in most supermarket layouts. In fact, the super-market where we base our study has the deli section locatedin the middle of the racetrack. Moreover, Hui et al. (2009)show that customers tend to buy products in a sequencethat minimizes total travel distance. Hence, if customer bas-kets are evenly distributed through the racetrack, it is likelythat the visit to the deli is done during the middle of thestore visit. Given that the visit duration tends to follow auniform distribution between 601757 minutes, we approxi-mate the distribution of deli visit time by a uniform distri-bution with range 601307 minutes before checkout time.

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Documents

Measuring the Effect of Queues on Customer Purchases