Cross-Selling in a Call Center with a Heterogeneous ... · O PERATIONS R ESEARCH...

OPERATIONS RESEARCHVol. 57, No. 2, March–April 2009, pp. 299–313issn 0030-364X eissn 1526-5463 09 5702 0299

informs ®

doi 10.1287/opre.1080.0568©2009 INFORMS

Cross-Selling in a Call Center witha Heterogeneous Customer Population

Itay GurvichKellogg School of Management, Northwestern University, Evanston, Illinois 60208, [email protected]

Mor ArmonyStern School of Business, New York University, New York, New York 10012, [email protected]

Constantinos MaglarasColumbia Business School, New York, New York 10027, [email protected]

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

Subject classifications : call centers; cross-selling; queueing systems; revenue management; pricing.Area of review : Manufacturing, Service, and Supply Chain Operations.History : Received September 2006; revisions received May 2007, September 2007, December 2007; acceptedDecember 2007. Published online in Articles in Advance January 5, 2009.

1. IntroductionMany organizations consider their call centers to be oneof the most important channels of interaction with theircustomers, acting both as a service center and a point ofsales—an opportunity for the firm to generate extra revenueby offering new or existing products to their customers. Thesignificant revenue potential of this cross-selling strategyis underscored by the nature of the interaction that takesplace in a call center and the wealth of information thatis available through state-of-the-art customer relationshipmanagement (CRM) systems. Together, they enable firmsto segment their customer pools effectively and to tailortheir product offerings to each such segment to increase thelikelihood of purchase and the associated expected revenue.A familiar and successful example of cross-selling practiceis in the financial services industry, where customers whocall for service, such as for account balance inquiries, areoften offered new financial products.1

Alongside its potential benefits, cross-selling may sub-stantially increase the total workload that needs to be han-dled by the call center’s agents,2 which may degrade thesystem’s quality of service and, in turn, have an adverseeffect on the overall customer experience, as well as theeffectiveness of cross-selling itself. It is important to care-fully select which cross-selling opportunities to pursue andwhen to do so, and to account for the impact of thesedecisions in determining the staffing level of the call cen-ter. This paper considers a call center with cross-selling

capabilities that serves a heterogenous pool of customers,and studies the operational decisions of staffing, call rout-ing, and cross-selling under various forms of customer seg-mentation. It derives near-optimal controls in each of thesettings analyzed, and characterizes the impact of morerefined customer segmentation on the structure of thesepolicies and the center’s profitability.In more detail, we consider a call center with a single

pool of fully flexible agents that first handle inbound callservice requests, and subsequently decide whether or not toattempt to cross-sell to some of these customers a certainproduct or service whenever such an opportunity arises.Cross-selling attempts are handled by the same agent thathas served the customer’s original request, upon comple-tion of that task. Each cross-selling attempt is preceded byan instantaneous step that captures the customer’s decisionof whether or not to agree to listen to the cross-sellingoffer. The processing times for the original service requestand the cross-selling phase are exponentially distributedwith potentially different parameters. Finally, the heteroge-neous pool of potential customers comprises a discrete setof types or segments. (The terms “type” and “segment” areused in this paper interchangeably.) Types differ in termsof their delay sensitivity and revenue potential. These arecaptured through the probability that a customer will agreeto listen to a cross-selling offer as a function of the waitingtime that he encountered, and through a demand relationthat specifies the probability that a customer decides to buy

299

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Gurvich et al.: Cross-Selling in a Call Center with a Heterogeneous Customer Population300 Operations Research 57(2), pp. 299–313, © 2009 INFORMS

the offered product as a function of the quoted price andthe waiting time.The ability to segment the caller population allows the

call center to customize the product offered to each callersegment. In this paper, we assume that the degree of seg-mentation is exogenously specified, for example, as theoutput of an upstream marketing analysis. Depending onthe application setting at hand, product customization mayinvolve charging a different price to different segments forthe same product, or could involve changing the attributes,as well as the price of the product offered to each segment.In both settings, the goal is to better exploit the prefer-ences of each caller segment so as to increase the expectedprofitability from cross-selling. The output of this pricingand/or product attribute customization process is summa-rized by the segment-specific expected revenue per cross-selling attempt. As we show, the latter is crucial in decidingto whom to cross-sell and how to staff the call center. Forpurposes of the analysis in this paper, we consider the sim-pler of the two settings mentioned above, in which the callcenter only customizes the price of the product offered toeach segment, keeping all other characteristics of that prod-uct common across segments. We acknowledge this factby using the term price customization as opposed to prod-uct customization, again keeping in mind that the essentialconsequence of the customization capability is that it leadsto different expected revenues per cross-selling attempt foreach segment.As an example of price customization, one may con-

sider the pricing of CD (certificate of deposit) productsoffered by banks to different customers. It is natural tothink of the price of the CD as its associated interest rate,although two important product attributes are the minimumcapital contribution and the length of the time over whichthe promised interest rate is guaranteed. An increasinglyimportant application of quantitative pricing and revenuemanagement tools in the financial services industry is indeciding the terms, and more importantly the interest rate,of the CD product that is offered to existing customersto entice them to roll their expiring CD contribution fromone product to another. Although pricing to initially attractcustomers who may be “shopping around” for such a prod-uct is quite competitive, the subsequent repricing decisionstend to be less constrained and, indeed, an area of intenseactivity in that industry.We study three variants of this model with an increasing

availability of information regarding customer segmenta-tion and, as a result, increasing flexibility in terms of theaforementioned operational and pricing decisions. The sim-plest model is one where customers are not segmented, orequivalently, where their types are not observable. In thiscase, the manager is limited to making the cross-sellingdecisions based solely on the aggregate load in the sys-tem, and charging all customers the same price. The secondmodel is one where types are observed sometime duringtheir service, and this information can therefore be used

together with the actual waiting time experienced by thecustomer in deciding whether to cross-sell to a customer,and if so, what price to charge. The third model is onewhere customer types are observable upon arrival, in whichcase the manager can also decide how to route customersof different types to the available agents. For each of thesemodels, the call center manager’s problem is to select itsstaffing, routing, cross-selling, and pricing policies to maxi-mize the center’s expected profit rate, given by its revenuesminus the staffing cost minus a linear waiting time costthat is experienced by all customers and is incurred by thecenter.The controlled two-stage service sequence of each cus-

tomer and the dependence of the cross-selling phase ondynamic waiting time information makes an exact analy-sis of this model cumbersome and difficult, even if cus-tomers are treated as one segment. Our approach considersa deterministic relaxation of this problem, which is solvedin closed form. Its solution suggests different staffing andcross-selling policies for each of the model variants listedabove. In each case, we show that our proposed policy isasymptotically optimal in systems with increasing call vol-ume, and as such is appropriate for call centers with highdemand volumes.Our contribution is twofold: From a practical view-

point—we propose a concrete, simple, and provably near-optimal solution for the complex problem of cross-sellingin environments with multiple customer classes. Our solu-tion will allow firms to extract the revenue potential em-bedded in their CRM systems through smart operationalmanagement of their marketing interface. From a manage-rial viewpoint—our tractable deterministic analysis and theasymptotic performance guarantees of the proposed poli-cies lead to several insights. The first one is that themarketing decisions of customer segmentation and pricecustomization are effectively decoupled from the opera-tional decisions of staffing, routing, and cross-selling. Spe-cifically, once the set of customer segments has beenidentified through an appropriate marketing and statisti-cal analysis, and their respective characteristics have beenidentified using observed data,3 the firm can precomputeits price customization strategy ahead of time, instead ofdynamically choosing the price charged to each customer.In particular, the prices are static and are identical acrosscustomers of the same type. These prices are then fed intothe operational control problem that involves staffing, rout-ing, and cross-selling decisions.The availability of information on customer segmenta-

tion has many important consequences, which can alsobe easily seen from our deterministic relaxation. To startwith, roughly speaking, the center will only cross-sell tocustomers that generate an expected revenue that exceedsthe capacity cost involved in pursuing this attempt; theexpected revenue is equal to the quoted price times theprobability that this customer will buy the offered prod-uct, provided that his waiting time was zero. If the cen-ter can segment its customers, then it will only cross-sell

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Gurvich et al.: Cross-Selling in a Call Center with a Heterogeneous Customer PopulationOperations Research 57(2), pp. 299–313, © 2009 INFORMS 301

to its profitable types; if no segmentation capability is inplace, then it will either cross-sell to all customers or tonone, depending again on the expected profitability of thesecross-selling attempts. In each case, the center will staff soas to handle all regular service requests plus the additionalnominal workload generated by its expected cross-sellingactivities. Because the cross-selling is controllable, it canprovide enough flexibility in the use of the center’s capac-ity, which eliminates the need to add “safety staffing” asis typically done according to the “square-root” rule to sta-bilize the system and guarantee moderate congestion. It ispossible that even though it is profitable to cross-sell ina system that segments its customers, this is not the casewithout segmentation. Our analysis outlines such cases.Overall, customer segmentation increases the center’s prof-itability in two ways: first, through a more efficient use ofcapacity achieved by reducing the volume of cross-sellingattempts that are unlikely to be profitable, and second, bycustomizing the price for each customer type so as to max-imize the resulting expected revenue. Finally, we note thatthe effect of observing the customer type upon arrival, asopposed to after service has commenced, is small. Thisis explained by the fact that even when the system doesnot differentiate between types in its routing decisions andhandles all external calls through a common first-come-first-served (FCFS) queue for all these types, the resultingwaiting times are small; these are moderated through thedynamic cross-selling decisions of the call center and arereinforced by the customers’ delay averseness.The remainder of this paper is organized as follows. This

section concludes with a brief literature survey. Section 2describes the two models with observable types, empha-sizing mostly the model where customer type is revealedonce his service starts. These two models are analyzedin §3. Section 4 shows how the pricing problem can betreated separately from all other decisions, which is thenused in §5 to analyze a model with no customer segmenta-tion. Section 6 provides results from our numerical exper-iments. Section 7 contains concluding remarks. The elec-tronic companion contains all of our proofs and is avail-able as part of the online version that can be found athttp://or.journal.informs.org/.

Literature Review. The literature on the operationalaspects of call centers is extensive and has grown rapidlyover the past decade. A survey of this literature and a tuto-rial on the subject can be found in Gans et al. (2003).Of particular relevance to our work is the literature onstaffing of call centers. The most commonly used staffingrule in the literature is the so-called square-root safetystaffing rule, according to which the number of serversrequired to handle an offered load of size R is R+

√R

for some constant . The square-root safety staffing ruledates back to Erlang in his 1923 paper (that appearedin Erlang 1948). This rule was formalized by Halfin andWhitt (1981), who showed that this square-root safetystaffing rule guarantees very short delays in an appropriate

asymptotic regime, and was shown to be nearly optimalfor a pure service center that handles a homogeneouscustomer population in Borst et al. (2004). Square-rootsafety staffing has been observed to be fairly robust withrespect to changes in model assumptions to include fea-tures such as customer abandonment (Garnett et al. 2002,Mandelbaum and Zeltyn 2009), multiple customer classes(Armony and Maglaras 2004a, b; Gurvich et al. 2008), mul-tiple server pools (Armony 2005), and nonstationary arrivalrates (Feldman et al. 2008). In contrast to the above set ofpapers, our work shows that the issue of safety staffing isof lesser importance in call centers with significant cross-selling activity because by adjusting the latter the managercan also control congestion.There is a small but growing portion of the recent lit-

erature on call centers that in broad terms studies howto best manage the cross-selling capability of such sys-tems. In more detail, the cross-selling control problem, i.e.,the question of when and to whom the center should tryto cross-sell, has been studied by several authors, includ-ing Aksin in a series of papers with Aksin and Harker(1999), Günes and Aksin (2004), and Örmeci and Aksin(2007), and by Byers and So in two papers (Byers andSo 2007a, b). These papers consider various aspects of theabove dynamic control problem under three assumptions:(a) the staffing levels are exogenously fixed; (b) the prod-ucts and prices offered to the various customers are homo-geneous even though the center may be able to segmentits customer pool according to their preferences; and (c) asimplified model of the service system that treats customersthat go through the cross-selling phase as a separate classof service requests with longer service times, as opposedto as a two-phase service. This latter restriction impliesthat cross-selling decisions have to be made in the begin-ning of the interaction with the customer, and it cannot useupdated state information that may be available at the com-pletion of a customer’s nominal service request. The servicefacility is either modeled as a single-server queue, a multi-server queue, or a multiserver loss system (i.e., customersthat do not find an idle server upon their arrival are lost).For the single-server model, Byers and So (2007a) showedthat the optimal cross-selling policy is of a threshold type;the center cross-sells as long as the number of customersin the system is below a certain threshold. The optimal-ity of the threshold policy in the multiserver case has notbeen established. Despite the restrictive assumptions listedabove, these papers made significant contributions to theliterature by being the first to address the important moti-vating questions mentioned earlier, and by deriving insightsthat seem to be fairly intuitive and, to some extent, robust.They also raised interesting questions: Are these insightsrobust to more representative models of the service deliveryprocess? What is their impact on staffing decisions? In whatway would the staffing decision affect the structure of thecross-selling policy and the profitability of cross-selling?And, finally, what is the impact of customer segmentationon all of the above?

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Recently, Armony and Gurvich (2006) proposed a morerealistic stochastic model for the cross-selling process,whereby the service time of each customer comprises twodistinct phases—the first captures the handling of the cus-tomer’s nominal service request, and the second, which isoptional, captures the duration of the cross-selling attempt.The main analytical contribution of Armony and Gurvich(2006) is to rigorously show that a threshold-type cross-selling policy is asymptotically optimal for this more com-plex service model as the nominal demand and the sizeof the call center grow large. Armony and Gurvich (2006)also conducted a preliminary analysis of the joint staffingand cross-selling control problem for the case where theentire pool of customers is either homogeneous in termsof its preferences, or is treated as such by the system; thelatter would correspond to settings where the customers areheterogeneous, but the system does not have segmentationcapability.Our paper applies the service model proposed in Armony

and Gurvich (2006) to a setting with heterogeneous anddelay-sensitive customers to address the joint price cus-tomization, staffing, and cross-selling control problem. Oureconomic model is more general than those used in earlierpapers, and the consideration of customer delay sensitivityis new. Our model allows for an insightful analysis of thejoint pricing, staffing, and cross-selling problems, whichemphasizes the trade-offs among customer segmentation,price customization, staffing costs, and the system prof-itability. Our work reinforces the insights derived in the var-ious papers listed thus far. It also highlights that the abilityto segment the customer pool and customize the respec-tive prices leads to significantly different staffing and cross-selling policy recommendation from those derived in thepapers mentioned above.An important ingredient of our solution methodology

hinges on the use of a deterministic relaxation for the orig-inal joint pricing, staffing, and cross-selling dynamic opti-mization problem, which is motivated from the work ofMaglaras and Zeevi (2005). Finally, the economic modelthat we adopt and the notion of price discrimination thatunderlies our work are related to a vast literature in eco-nomics, marketing, and revenue management. We refer thereader to the book by Talluri and Van Ryzin (2004) for anintroduction to these subjects.

2. Model FormulationWe consider a call center with a single pool of N fullyflexible agents that serves a heterogeneous customer popu-lation, comprising K distinct segments, or types, or classes.We study three model variants depending on the extentto which the customer types are observable by the sys-tem. These are graphically depicted in Figure 1. Model (a)assumes that types are unobservable, or that the call cen-ter does not segment its customers. In model (b), the typeof a customer is observed when s/he is being served, and

Figure 1. Three cross-selling models.

(a) (b) (c)

this information is subsequently used in the center’s cross-selling decisions. Finally, model (c) is one where the cus-tomer type is immediately observed upon arrival, e.g., byrequiring customers to enter an account number, and cantherefore be used in routing as well as in cross-selling deci-sions. We will focus on model (b), and treat model (c) asan extension and model (a) as a one-segment special caseof this multisegment model.

Basic Service. Type-i customers call the center accord-ing to a Poisson process, Ait t 0, with rate i. LetAt =∑K

i=1 Ait, and define =∑Ki=1 i to be the total

arrival rate into the system. All customers require the sametype of service and the processing requirement is exponen-tially distributed with rate s , independent of the customertype. Under the assumption that types are unobservablebefore service begins (model (b)), all customers join a sin-gle queue and get processed in an FCFS manner.

Cross-Selling. Once regular service is completed, acustomer either leaves the system or enters a cross-sellingphase that is handled by the same agent. A cross-sellingattempt is preceded by an instantaneous step in which thecustomer is asked to listen to the actual offer. The lengthof time required for the cross-selling attempt may dependon the customer segment and is assumed to be exponen-tially distributed with rate cs

i for type-i customers. Allprocessing times (regular service and cross-selling) andinterarrival times are assumed to be independent.The probability that a type-i customer will agree to lis-

ten to the cross-selling offer after experiencing waitingtime w is given by an arbitrary nonincreasing continuousfunction qiw + → 01, with limw→ qiw = 0. Weset qi = qi0 and note that it is possible to have qi < 1.This allows us to model cases where some customers mayalways decline to listen to the cross-selling offer.If a customer of class-i agrees to listen to a cross-

selling offer, he will be offered the product at a certainprice that might depend on both his class and his actualwaiting time. Class-i customers have i.i.d. valuations for

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this product, denoted by vi, drawn from a continuous dis-tribution function Fi·. The perceived “cost” of the offeredproduct may also depend on the waiting time s/he hasexperienced. This dependence may arise in some practi-cal settings, such as when signing up for help desk ser-vices where the waiting acts as a proxy for the futurequality of service. In other applications, the cost of theoffered product should not depend on the waiting time,and this is also allowed by our model. Specifically, weassume that class-i customers have a delay-sensitivity con-stant ci 0. Then, conditional on agreeing to listen toa cross-selling offer, a class-i customer who has waitedfor w time units before starting her service will buythe product with probability Fipiw = Fipi + ciw =Pvi > pi+ciw. Applications where the cost of the offeredproduct is independent of the waiting time are captured bysetting ci = 0. The resulting conditional expected revenuefrom a customer of class-i who waited w time units is givenby ripw = pi

Fipi+ciw. For simplicity of notation, welet rip = rip0= limw↓0 ripw. We will also assumethat the functions ripi are unimodal in the pis for each i;this is satisfied by many commonly used demand functions(see Talluri and van Ryzin 2004). For the first few sections,we will assume a fixed vector of prices p = p1 pK.Hence, we will use the simplified notation riw instead ofripiw and the notation ri for ri0. We will return tothe more general notation in §4, in which we consider thepricing problem.

Control Decisions. The call center manager selects thenumber of agents N for the system and has discretion withrespect to the cross-selling and pricing decisions. We willconsider policies, , that decide whether to cross-sell tothe jth type-i customer and which price to charge him as afunction of all the information available up to the decisionpoint. In particular, the cross-selling and pricing decisionsare dynamic and may depend on the customer’s type, thewaiting time encountered by this customer prior to his ser-vice, which we denote by w

i j , the number of customersin the queue, and the number of customers of each type-ithat are currently in service, denoted by Q

i t and Zi t,

respectively. We let Qt=∑Ki=1 Q

i t be the total queue

length at time t under . To guarantee the existence ofsteady state or at least the existence of long-run averagesfor various quantities of interest, we will restrict the set ofadmissible controls as follows.

Definition 1 (Admissible Controls). Given a staffinglevel N , and parameters 1 Kscs

1 csK , we

say that is an admissible policy if it is nonpreemptive,nonanticipative, and limt→ EQt/t → 0. We denotethe family of admissible policies by 1 Kscs1 cs

K N .

Loosely speaking, 1 Kscs1 cs

K N isthe set of stabilizing policies under the given parameters.Definition 1 takes into account the fact that the set of

admissible policies depends on the parameters of the modelthrough the stability conditions of the system. To simplifynotation, we will omit the parameters 1 Ks andcs

i , i = 1 K, whenever these are exogenously fixed,and write N or simply whenever the staffing levelis clear from the context. Note that the above definitionimplies that our system must be able to handle all of thenominal demand, at least when no cross-selling is exer-cised; that is, the staffing choice must satisfy the constraintN > R =/s .

Performance Criterion. We first define two systemquantities that will play an important role in the call cen-ter’s cost and revenue terms, respectively. Observe that asteady state need not exist for any ∈N . With that inmind, for some ∈N and i= 1 K, we define

EWi =E

[lim sup

t→

∑Aitj=1 w

i j

Ait

]

xi =E

[lim inf

t→

∑Aitj=1 x

i j

Ait

]

and

rixi =E

[lim inf

t→

∑Aitj=1 riw

i j x

i j

Ait

]

where xi j is an indicator that is set to one whenever the

jth class-i customer goes through a cross-selling phase,and x

i j equals zero otherwise. The performance measurerixi should be interpreted as the long-run average rev-enue per class-i customer under the policy . When asteady state exists, EW

i and xi coincide with theexpected steady-state waiting time experienced by type-icustomers, and the steady-state fraction of class-i customersthat are asked and agree to listen to a cross-selling offerunder , respectively. rixi will then coincide with thesteady-state revenue from class-i customers. Because cus-tomers are processed FCFS, it must be that EW

i =EW

k for all i, k, which will also be denoted byEW.The call center incurs linear staffing and waiting time

costs per unit time, given by c ·N and hEW, respec-tively. The latter assumes that the waiting time cost is typeindependent. The waiting time cost can be thought of asa penalty that the system incurs in terms of lost goodwillfrom the customers. The type independence of the waitingcost can be relaxed with no effect on any of our results.Under an FCFS discipline it seems reasonable, however,to assign a common cost to all customers. The call centermanager’s optimization problem is the following:

supN∈+∈N

K∑i=1

i · rixi− cN −hEW (1)

Note that although it is not guaranteed that there existsa control that actually achieves the optimal profit rate,

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it is easy to establish the existence of an optimal N ∗

because N is discrete, the profit rate is bounded above by∑i iri − c ·R, and it decreases to − as N grows large.An alternate formulation to (1) would replace the wait-

ing time cost by an upper-bound constraint on the expectedwaiting time, typically in the order of 30 seconds, and con-sider the following problem:

supN∈+∈N

K∑i=1

i · rixi− cN EW W (2)

Indeed, one can view (2) as a more natural starting point,and (1) as a “dualized” version of the problem that is per-haps simpler to address. We will refer to (1) and (2) asthe waiting cost and constrained formulations, respectively.We will also make the following assumption:

Assumption 1. Types are labeled so that

r1 − c/cs1 · · · rK − c/cs

K

and r1 − c/cs1 > 0.

The labeling assumption is innocuous. The conditionr1 − c/cs

1 > 0 means that it is profitable to cross-sell toat least type-1 customers. As will be shown later, r1−c/cs

1

is roughly the expected revenue from cross-selling to aclass-1 customer minus the marginal staffing costs associ-ated with it. In the absence of this assumption, it makessense not to invest in extra capacity for cross-selling andto only attempt to cross-sell to a negligible fraction of thecustomers.

3. Observable Types: Analysis Basedon a Deterministic Relaxation

A direct analysis of the problems formulated above is verydifficult due to their multiclass nature and the dependenceof the cross-selling success probability on state-dependentinformation. Our approach looks at relaxations of the aboveproblems, where in addition to the staffing and cross-sellingdecisions, the manager can also select the waiting timesexperienced by its callers, which in reality are random vari-ables that depend on the system dynamics. These relax-ations are tractable, deterministic optimization problemsthat have insightful solutions and give rise to near-optimalheuristics. Focusing on model (b) (cf. Figure 1) first, §3.1studies the waiting cost formulation of (1). These resultsare extended to the constrained formulation of (2) in §3.2,whereas §3.3 extends our work to model (c), where thecustomer types are observable upon arrival. All proofs arerelegated to the online appendix.

3.1. The Waiting Cost Formulation

Throughout this section, we focus on model (b) and thewaiting cost formulation (1).

Deterministic Relaxation. Starting with (1), we for-mulate the following linear program:

maximizeK∑

i=1 iriwixi − c ·R1+ z−h

K∑i=1

iwi

s.t. xi qiwiK∑

i=1

ixi

csi

Rz

z 0 xi 0 wi 0

for all i= 1 K

(3)

where xi is interpreted as the fraction of class-i customersthat are being asked and agree to listen to a cross-sellingoffer; wi is the “fictitious” waiting time experienced byclass-i customers in this formulation; and z is the excess(normalized) staffing level beyond the nominal requirementof the offered load R (=/s) as a fraction of R. Thecondition z 0 implies that the staffing level is sufficientlylarge to handle all basic service requests (i.e., N R). Thename “deterministic relaxation” comes with a slight abuseof terminology. As to whether or not this is indeed a relax-ation for (1)—the answer to this question depends on theactual form of the function qi· and, more specifically,on its concavity or lack thereof. It is a matter of a sim-ple observation, however, that any optimal solution to (3)will have wi = 0 for all i and, consequently, that an opti-mal solution to (3) is necessarily an upper bound for anyoptimal solutions to (1) if such solutions exist. Hence, wechoose to refer to (3) as the deterministic relaxation.Recall the labeling convention in Assumption 1. Denot-

ing the optimal solution to the knapsack problem in (3)with an overbar, we have the following: set wi = 0 for alli= 1 K,

xi =

qi i k

0 otherwiseand z=

k∑i=1

iqi

Rcsi

(4)

where k = maxi ri − c/csi 0 qi0 > 0. In fact, we

will assume throughout that r k−c/csk

> 0, which is equiv-alent to assuming that the deterministic relaxation has aunique solution. In the presence of multiple solutions to thedeterministic relaxation, our approach might lead to mul-tiple asymptotically optimal solutions. By Assumption 1,z is guaranteed to be strictly positive. The resulting staffinglevel is R +∑k

i=1 iqi/csi . Note that the structure of the

deterministic relaxation is such that as long as i/ isknown and is kept constant (which we will assume hence-forth), the normalized quantities x, z do not change with .Therefore, the relevant profit depends on the entire vector 1 K through their sum only. Specifically, the profitrate associated with solution (4) is

*=−cR+k∑

i=1 iqiri − c/cs

i

=−cR+K∑

i=1 iqiri − c/cs

i ∨ 0 (5)

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which is an upper bound for the optimal profit in (1). (Hereand elsewhere x∨ y =maxx y.)

A Staffing and Cross-Selling Proposal. The nestedstructure of (4) is intuitive: we cross-sell to all types i forwhich their marginal revenue contribution, iriqi, exceedsthe increase in staffing cost, c iqi/

csi , resulting from the

additional cross-selling workload; this reduces to the condi-tion ri − c/cs

i > 0. The solution to the deterministic relax-ation suggests the following pair of policies for the originalstochastic system:(S) Staffing: Staff with N =R1+ z.(C) Cross-selling: Given a sequence of thresholds ,k

,k−1 · · · ,1: cross-sell to a customer of type i k thatcompletes service at time t if and only if Qt < ,i.The cross-selling policy (C) follows the solution of the

deterministic relaxation when the queue length is modest,and then starts to reduce the amount of cross-selling activ-ity as the system gets increasingly congested. The asymp-totic performance analysis that will follow does not usethe precise values of the above thresholds, and in fact onlymakes use of the smallest threshold ,k. Consequently, onemay prefer to use a simpler policy that uses only thissmallest threshold ,k. This single-threshold policy alwayscross-sells to classes 1 k−1 and stops cross-selling toclass k when the queue length exceeds the threshold. In oursetting, in which the arrivals rates, i, are known and sta-tionary, this single-threshold policy will be asymptoticallyequivalent to (C) in terms of the profits it generates. Still,we choose to present the results for the more elaborate con-trol (C). We motivate the use of multiple thresholds in anonstationary environment in §7.

Asymptotic Optimality of (S)-(C). Despite its sim-ple structure, (S)-(C) performs very well in the stochasticsystem under consideration, and is, in fact, asymptoticallyoptimal in large-scale systems, i.e., where is large. As astarting point, we will establish that the system is alwaysstable under (S)-(C) and that it admits a unique stationarydistribution. We do that by showing the stronger result thatthe system will be stable under (C) as long as N > R, evenif N < R1+ z.

Proposition 1 (Stability). Fix and assume that C isused for some set of thresholds ,k ,k−1 · · · ,1 .Then, N > R is a sufficient condition for stability. More-over, for any N > R, the underlying Markov process admitsa unique stationary distribution that is also its limitingdistribution.

This proposition illustrates the self-stabilizing nature ofthe cross-selling system. Note that the use of thresholds,i is not necessary for this result to hold. Indeed, they mayall be set equal to ; the stabilizing force stems from thedelay sensitivity of the customers. Intuitively, when the sys-tem is heavily loaded, the queue and the resulting waitingtime will grow large. In turn, fewer customers will agreeto listen to cross-selling offers, thus reducing the load.

The remainder of this subsection will characterize theasymptotic performance of the original stochastic call cen-ter system under (S)-(C) in settings with large call volumes,as measured by . One naturally expects that with a thresh-old policy, the best threshold values will be a function ofthe system size and in particular of , the overall arrivalrate. Let ,

k ,

1 be the threshold values correspond-ing to a system with arrival rate . Then, we will show inour subsequent results that, indeed, there is a dependenceof the threshold values on the system size and, moreover,that asymptotically optimal performance implies that thesethreshold values scale according to

,i = ,i

√ for i= 1 k (6)

and appropriate constants ,k · · · ,1. Let N∗, x∗

i ,and *∗ denote the (unknown) optimal staffing level,realized long-run average cross-selling rates, and the corre-sponding profit rate for (1), respectively, when the aggre-gate demand is . Also, let * be the profit obtainedwhen using (S)-(C) in the stochastic system.In the sequel, we will make use of the following nota-

tion: for two positive sequences we say that x is oy ifx/y → 0 as →.

Theorem 1 (Asymptotic Optimality). Let grow large,keeping i/ constant for all i. Then, with thresholds sat-isfying 6, S-C is asymptotically optimal in the sensethat

*=*∗− o (7)

Alternatively, one could write (7) in the form */*∗→ 1 as →. The proof of the above result fol-lows by showing the stronger result that * approaches*, which itself is an upper bound for *∗. Because*∗ is sandwiched between * and *, it mustalso be close to *. This leads to a partial characteriza-tion of the unknown optimal policy in large-scale systems.

Theorem 2 (Estimates of the Optimal Solution). Let grow large, keeping i/ constant for all i. Then,(a) *∗= *− o, (b) N ∗=R1+ z± o,and (c) x∗

i = xi + o1.

Theorems 1 and 2 together demonstrate how the solu-tion of the deterministic relaxation captures the first-orderbehavior of the optimal policy for (1), both in terms ofits staffing and cross-selling decisions as well as its result-ing profits. A key component of the asymptotic optimalityproof is the next lemma that shows that if the thresholds, are of order

√ (as in (6)), then the steady-state wait-

ing times that characterize the system are of order 1/√

and in particular of order o1; this is the nominal timeit takes an order servers to clear a queue length oforder

√. Thresholds of smaller magnitudes would result

in even smaller waiting times.

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Lemma 1. Let grow large, keeping i/ constant forall i. Denote by EW the steady-state expected wait-ing time under policy S-C. Then, with thresholdssatisfying 6, EW = O1/

√, or equivalently,

lim sup→√

EW <. In particular, EW→ 0 as→.

The next lemma then shows that, actually, it would bealways optimal to staff and cross-sell so that the wait-ing times are very small. We denote by EW∗ theexpected steady-state waiting time under the optimal con-trol N ∗x∗.

Lemma 2. Let grow large, keeping i/ constant forall i. If an optimal policy N ∗x∗ exists for all

large enough, then lim sup→ EW∗= 0.

Remark 1 (Strengthening the Notion of AsymptoticOptimality). The main technical problem in proving The-orems 1 and 2 lies in the so-called limit interchange prob-lem. Specifically, although it might be relatively simple toget performance guarantees on finite time intervals, it ismuch harder to characterize the asymptotic performance,as → , of the system’s steady state. The technicalarguments in that respect are quite complex, as the onlineappendix illustrates. The interested reader is referred topart B of the online appendix for a further discussion of theunderlying complexities. Consequently, refining the perfor-mance bounds by showing, for example, an O

√ devia-

tion from optimality, is complicated even in much simplersettings than the system we consider—especially when onewants to establish convergence of moments.

Remark 2 (Choosing the Threshold Values). For thecost formulation, the values of the thresholds ,i can beselected via simulation. In most call centers, however, theconstrained formulation (considered in the next section) ismore natural. Fortunately, for the constrained formulationwe have a very simple rule to determine the threshold value.

3.2. The Constrained Formulation

Lemmas 1 and 2 illustrate that the waiting times expe-rienced in an optimally controlled call center will be oforder o1. With that in mind, a waiting time constraint ofthe form EW W will become irrelevant as growslarge because the actual waiting times will be much smallerthan the desired target W . A more appropriate formulationthat is meaningful as grows large replaces the upper-bound constraint by a quantity that itself changes with

such as W = W/√

for an appropriate choice of W .4

This would result in the following problem:

supN∈+∈N

K∑i=1

i · rixi− cN EW W

(8)

where W = W/√

for an appropriate choice of W . Alongthe lines of (3), the following is a deterministic relaxationof (8):

maximizeK∑

i=1 irixi − c ·R1+ z

s.t.K∑

i=1

i

wi

W

xi qiwiK∑

i=1

ixi

csi

Rz

xi 0 wi 0 for all i= 1 K

(9)

The linear program described above has the same optimalsolution as (3), making our solution insensitive to the pre-cise articulation of the effect of customer waiting times.The resulting staffing and cross-selling heuristics are againthe ones described by (S)-(C) in the previous subsection.In the case of the constrained formulation, one can alsoget a crude estimate for the threshold ,k to be ,k =W,which is consistent with (6). Intuitively, if the queue lengthis maintained below that threshold, then by a heuristicapplication of Little’s law, one would expect the waitingtimes to be below W. The next theorem establishes thisresult in an asymptotic sense as grows large. With aslight abuse of notation, we use * and *∗ to denotethe profit rate for the constrained formulation under (S)-(C)and the optimal policy, respectively.

Theorem 3 (Asymptotic Optimality). Let grow large,keeping i/ constant for all i. Then, with thresholds satis-fying 6, and such that ,

k=W, (a) *=*∗+

o and (b) EW W + oW.

Theorem 3 shows that the waiting time constraint willbe violated only by a negligible amount if one sets ,k =W. Of course, if one is interested in strict satisfaction ofthe threshold, one may start with the recommended thresh-old and fine-tune it in real time with small perturbationsaround the recommended value.

3.3. The Value of Customer Type IdentificationUpon Arrival

We complete the analysis of the model with observabletypes by comparing the model analyzed thus far (model (b)in Figure 1) with the one where the type of each customer isobserved at the time of his arrival to the system (model (c)).The latter could be achieved by requiring callers to identifythemselves through a PIN or an account number.

Routing Capability. Once the call center observes thetype of each arriving customer, it can maintain different(virtual) queues for customers of each type, and use thatadded flexibility in routing calls to available agents. Thiswill eventually trade off the delay sensitivity and wait-ing time cost of each type against its potential revenue

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contribution. It is clear that this added element of controlcan only improve the call center’s profitability. The ques-tion is by how much. The main result of this sectionshows that the performance difference between FCFS rout-ing (used when types are unobservable upon arrival) andany other routing policy that makes use of the type informa-tion, including the optimal one, is small and asymptoticallynegligible. The crude asymptotic analysis of this subsec-tion uses a sandwich argument, similar to the one appliedin Theorem 2, and does not need a detailed articulation ofthe set of admissible routing policies. We refer the readerto Bassamboo et al. (2006) for one possible definition ofthese controls.We henceforth drop the distinction between the wait-

ing cost and constrained formulations. The results in theremainder of this section as well as those in §§4 and 5 holdfor both formulations. Let *∗∗ be the optimal achiev-able profit for the system where customer types are observ-able upon their arrival, and note that *∗∗*∗. Thekey to our analysis is that the deterministic relaxations formodels (b) and (c) are identical. The routing capability ofmodel (c) can only serve to improve the vector of expectedwaiting times EWi. Because the relaxation treats these asfree optimization variables, denoted by wi, and sets themequal to zero, its solution will coincide with that of (3).It follows that **∗∗*∗. From Theorem 2we have that *∗= *− o, which leads to thefollowing conclusion:

Proposition 2. Let grow large, keeping i/ constantfor all i. Then, *∗∗−*∗= o.

Therefore, although routing control capability mayimprove the quality of service enjoyed by some types andpotentially simultaneously increase the revenue extractedfrom them, it will not lead to a significant overall profitgain. Moreover, the asymptotically optimal staffing andcross-selling recommendations that emerge from our anal-ysis are insensitive (up to first order) to the use of thisinformation.The question that arises is whether segmentation at the

cross-selling stage leads to significantly different results incomparison to no segmentation at all. To address this ques-tion, we first study the issue of type-dependent price cus-tomization in §4, and then assess the value of customersegmentation in §5.

4. The Price Customization ProblemCustomer segmentation in a call center setting allows firmsto customize their products to better match the character-istics of each customer type and extract higher revenues.In our model, the product offered to all customers isassumed to be the same, but the firm can customize theprice quoted to each customer type. In this section, weshow that the optimal prices can be computed separatelyfrom the operational decisions of staffing and cross-selling.

Towards this end, note that due to the dependence of thewillingness to pay on the waiting times of customers,one expects the true optimal pricing mechanism to be adynamic one that takes into account these realized waitingtimes. Hence, the pricing mechanism should be regardedas a mapping from waiting times to prices. Specifically, weassume that prices may assume values in the space =1⊗2⊗· · ·⊗K , where for i= 1 K, i is assumedto be a compact interval in +. The pricing mechanism isthen a function p·= p1· pK· + →; we let be the space of these functions. Accordingly, we expandthe notation used earlier to let *∗/p· and N ∗/p·be the optimal profit rate and staffing level, respectively,for (1) for a given and pricing function pw. Wethen redefine *∗ = supp·∈ *∗p to be the opti-mal achievable profit rate when the call center is allowedto optimize over its price function over the set . Letp∗ = p∗· be the optimal price function, whichis assumed to exist, and N ∗ the corresponding staffinglevel. We also let *p be the profit rate achieved in thedeterministic relaxation of (3) for a given constant value ofp, * =maxp∈ *p be the profit rate when opti-mizing over the price, and let p denote the correspondingoptimizer, which will most likely be different than the func-tion p∗. Whereas identifying p∗ is hard, the deter-ministic price vector p is easy to characterize by rewritingthe objective function as

*p=−cR+K∑

i=1 iqiripi− c/cs

i ∨ 0 (10)

where ripi = piFipi0; this expression reflects the fact

that the center only cross-sells to and receives revenue fromtypes for which ripi c/cs

i , and that it staffs accord-ingly. It follows that the corresponding optimal price in (10)is static (waiting time independent) and satisfies

pi = argmaxpi∈i

piFipi0 (11)

and * = −cR + ∑Ki=1 iqiripi − c/cs

i ∨ 0 =* p. The corresponding staffing level is R1+ zp,where

zp=kp∑i=1

iqi

Rcsi

and

kp=maxi ripi c/csi /

(12)

the above expressions assume w.l.o.g that types are rela-belled so that r1p1 · · · rKpK. We also assume thatr1p1 > c/cs

1 and that rkppkp > c/cskp

, which guar-antee, respectively, that Assumption 1 holds and that thesolution of the deterministic relaxation given p is unique.It is straightforward to show that p, zp, and kp jointlycharacterize the optimal solution of the deterministic relax-ation, and that this solution does not change if one were

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to scale large, while keeping i/ constant (this is theasymptotic setup adopted thus far). Note that although pmay be different than p∗, * p is still an upperbound for *∗p∗. Using this observation and apply-ing Theorem 2 (with the fixed-price vector p), we find thefollowing:

Proposition 3. Define p, zp through 11 and 12, re-spectively. Let grow large, keeping i/ constant for all i.Then: (a) *∗p∗ = * p − o, (b) N ∗p∗ =R1+ zp± o, and (c) p∗0= p+ o1.

Consequently, we recommend adding the static pricevector p to the staffing and cross-selling rules proposedin §3. By Theorem 1 and Proposition 3 above, the resultingjoint pricing, staffing, and cross-selling solution is asymp-totically optimal for the original stochastic system.

Decoupling of Pricing and Staffing. An importantconsequence of the above result is that the pricing deci-sions can be made independently of the operational ones ofstaffing and cross-selling. This insight is valid in the systemwhere types are observed upon arrival (model (c)), as wellas in settings where products are customized along othernonprice attributes that do not involve capacity and quality-of-service specifications. This decoupling trivially followsin settings where the perceived cost of a product is indepen-dent of the waiting time encountered by the customer, butneed not be true in the more general model considered inour paper. Moreover, because the waiting time of the cus-tomer is known to the agent, the center may want to invokea dynamic pricing policy to optimize the expected revenueper customer. The fact that a static pricing policy is shownto perform very close to optimal is an appealing charac-teristic of our solution that allows the system manager tomake the pricing and operational decision in a hierarchicalsequence.

5. The Effect of Customer SegmentationThis section compares the profitability and behavior of thesystem studied in §§3 and 4 against one that does not usea segmentation mechanism and instead treats its entire cus-tomer pool as one segment. The latter is offered a commonproduct, i.e., at the same price, and cross-selling decisionsare made without the customer type information; this ismodel (a) in Figure 1.

A System with No Customer Segmentation. Thecharacteristics of this combined segment are a single delaysensitivity function q· and a corresponding willingness-to-pay distribution F · that are appropriate mixtures of thecorresponding quantities for the various types. The delaysensitivity function, qw, is given by

qw =K∑

i=1

i

qiw

The mean cross-selling time for the combined segment isestimated by

1

cs=

K∑i=1

iqi∑Kj=1 jqj

· 1cs

i

This is a reasonably precise estimate assuming that thewaiting times are small. Moreover, the comparison resultin Proposition 4 below holds when one uses a more preciseestimate that takes into account the waiting times. The com-bined willingness-to-pay distribution F is computed indi-rectly as follows. Let F pw be equal to the probabilitythat the willingness-to-pay of a customer that agreed to lis-ten to the cross-selling offer after a waiting time of w timeunits is less than or equal to p. Then, qw and F pwsatisfy the following intuitive relation

qw F pw=K∑

i=1

i

qiw Fipw

from which we can solve for F pw.The deterministic relaxation for the combined segment

is now easy to solve by specializing the results of §3 toa single segment with characteristics qw and F pw.Specifically, it is again optimal to set w = 0, which togetherwith (10) gives the following objective:

*ap =−cR+qp F p0− c/cs∨ 0 (13)

where the superscript “a” is meant to associate this expres-sion to model (a), and

q =K∑

i=1

i

qi and F p0 =

K∑i=1

iqi∑Kj=1 jqj

Fip0 (14)

As shown in §4, one can study this deterministic formu-lation by separately optimizing over the price p, and thenconsidering the resulting staffing and cross-selling problemat that price.The pricing decision. The optimal price that the call cen-

ter should use in this deterministic relaxation is given bythe solution to the following problem:

maxp∈

p F p0 (15)

which we denote by pa, and let ra = pa F pa0 and =1 ∩ 2 ∩ · · · ∩ K . Note that despite our assumptionsregarding the unimodality of pi

Fipi0, p F p0 need notbe unimodal itself. However, one can always find its opti-mizer through a single-parameter search (assuming that theset is not empty).The staffing and cross-selling decisions. Plugging pa

into (13) and using the results of §3, the solution of thedeterministic relaxation can be divided into two cases:Case i. If ra c/cs: the call center cross-sells to all

customers and staffs with Rmax =R1+ za servers, whereza =q/Rcs.

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Case ii. If ra < c/cs: the call center will not cross-sellto any customer and staff with R servers. Using (13) and(14), the resulting profit rate in the deterministic relaxationis given by

*a =

−cR+

K∑i=1

iqipa Fip

a0− c/csi

if ra > c/cs

−cR otherwise

(16)

which is again an upper bound for the optimal profit rateof the stochastic call center system.As in §3, the natural implementation of the above poli-

cies in case i would be to cross-sell as long as the queue isbelow an appropriate threshold that serves to limit exces-sive delays. In case ii, the system may still elect to cross-sell, but only if either the queue is very small or there area sufficient number of agents that are idle. Moreover, inthat case the staffing level should be inflated to R+ x

√R

for an appropriate constant x to provide moderate delays.The asymptotic analysis of §3 does apply to the single-segment model when the solution of the deterministic relax-ation falls into case i, but it does not cover case ii, wherethe system exercises negligible cross-selling. That case wasstudied in detail in Armony and Gurvich (2006) and willnot be further reviewed here.

The Effect of Customer Segmentation. The key dif-ferences between the two systems, with and without seg-mentation, are best illustrated through their respectivedeterministic relaxations, which are simple and accurate,in the sense that they capture the structure of the under-lying optimal policies and their resulting performanceasymptotically.1. Cross-selling all-or-none versus selected types. For

both models, the call center will do significant cross-sellingonly if the expected revenue from doing so exceeds thecapacity cost involved in that activity. With no segmenta-tion capability in place, the system will either choose tocross-sell to all of its callers if ra c/cs , or to none.In the first case, this may involve cross-selling to cus-tomer segments to which it is strictly unprofitable to do so,whereas in the second case, it involves forgoing profitablecross-selling opportunities that cannot be singled out fromthe larger pool of callers (the latter follows from Assump-tion 1). Using customer segmentation, the system will onlycross-sell to types i = 1 k for which ripi c/cs

i ,i.e., for which cross-selling is profitable. Finally, we notethat although Assumption 1 guarantees that the call cen-ter will always choose to cross-sell to some subset of thecustomer types, if these can be segmented out, it does notguarantee that it is profitable to do so in a system with nosegmentation capability.2. Staffing. The model with no segmentation will either

staff with Rmax =R1+ za or R+x√

R servers, dependingon whether it will cross-sell or not. In contrast, the model

with segmentation will staff with R1+ z servers; z < za,unless it is profitable to cross-sell to all customer types.3. Uniform versus customized pricing and profitability.

Most structural differences between the two systems origi-nate from the pricing policies adopted by the call cen-ter in each case, and the corresponding expected revenuethat they will generate per customer that agrees to enterthe cross-selling phase. As explained earlier, the systemthat segments its customers will customize its prices, pi,for each type according to (11), whereas the system withno segmentation will use a uniform price, pa, definedthrough (15). An immediate consequence of the aboveis that

ra =K∑

i=1

iqi∑Kj=1 jqj

pa Fipa0

K∑i=1

iqi∑Kj=1 jqj

piFipi0

Premultiplying by∑K

j=1 jqj and subtracting out the corre-sponding capacity cost, we find that( K∑

j=1 jqj

)ra − c/cs

K∑i=1

iqiripi− c/csi

K∑i=1

iqiripi− c/csi +

The right-hand side (RHS) of the above expression is equalto the profit contribution due to cross-selling in the sys-tem with segmentation, which is clearly nonnegative. Thisallows us to strengthen this inequality to the following:( K∑

j=1 jqj

)ra − c/cs+

K∑i=1

iqiripi− c/csi + (17)

where, in turn, the left-hand side (LHS) of (17) is theprofit contribution due to cross-selling in the system withno segmentation. The above inequality is strict as long asthere exists a type i for which pa Fip

a0 < piFipi, which

by the definition of pi and the unimodality of p Fip0,reduces to

∃i ∈ 1 K for which pi = pa (18)

or equivalently, to

∃i j ∈ 1 K such that pi = pj (19)

Unless customer types have trivial differences with respectto their willingness to pay, conditions (18) or (19) are likelyto be satisfied, in which case the ability to segment thecustomer pool would lead to significant profit gains. Forexample, if the willingness-to-pay distributions for the var-ious types were exponential with parameters bi, then theabove conditions would require that at least two of thesetypes had different parameters bi = bj . If the distributionswere logistic with scale parameters bi (these are commonlyused in the literature in modelling different customer seg-ments), then again (18) would require that the parametersof at least two segments are different. A simple extensionof our previous results yields the following characteriza-tion of the potential value of customer segmentation in theunderlying stochastic call center systems.

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Proposition 4. Under Assumption 1, if 18 or equiva-lently 19 holds, then for all , * − *a = 2,where 2 is the difference of the RHS and LHS of 17 nor-malized by . Moreover, if we let grow large, keeping i/ constant for all i, then

*∗−*∗ a= 2+ o

where *∗*∗ a are the optimal expected profit ratesfor the underlying stochastic systems with and without seg-mentation, respectively.

The above proposition together with the results of The-orems 1 and 3 suggest that the staffing and cross-sellingpolicies proposed in this paper would realize most of theprofit differential that can be attributed to customer segmen-tation. Operationally, the latter also leads to more efficientcapacity utilization because call centers that do not segmenttheir callers, but try to cross-sell to them, end up pursu-ing too many customer prospects that are unlikely to leadto a sale. Our stylized yet insightful analysis can be usedto assess the magnitude of this potential benefit, which isuseful in deciding the value proposition of an investmentin technology and agent training that would be needed tosupport a sophisticated customer segmentation and cross-selling strategy.

6. Numerical ResultsOur results are organized in three categories. The first offersa representative numerical illustration of the accuracy ofour asymptotic analysis. The second examines the qualityof the proposed policies, and in particular shows the sensi-tivity of the system performance to changes in staffing andthreshold levels that are used in the cross-selling decisions.The last one gives some examples of the potential value ofusing customer segmentation in such a call center. For sim-plicity, we assume throughout this section that cs

i = cs

for all i.

The Accuracy of Large-Scale Asymptotics. We illus-trate the accuracy of the proposed (S)-(C) heuristic byexperimenting on a system with four customer classes. Theservice rates are s = 1 and cs = 2; one may regard allsubsequent parameters as normalized with respect to s .The arrival rates are 1 = 2 = 1

3 and 3 = 4 = 16,

whereas the aggregate arrival rate, , will be varied overa range of values in our experiment. The product pricesare exogenously given and result in expected revenues pertype-i customer who goes through cross-selling, given byr1 = 7, r2 = 5, and r3 = r4 = 04, regardless of the real-ized waiting time.5 For simplicity, we assume that the cus-tomers’ willingness-to-listen functions are common acrosstypes and given by the linear function qiw= 1−01w+.The staffing cost is normalized to c = 1, and for concrete-ness we consider the constrained formulation with an upperbound for the waiting time equal to 1/6; if the natural time

units are minutes, then this upper bound is 10 seconds.Under this choice of parameters, we have that z = 1

3 > 0and k = 2, i.e., the center will cross-sell to types 1 and 2only. These values of z and k and the above set of rev-enue and cost parameters give *= 267 as an upperbound on the system’s profit rate.We have simulated the system behavior under three vari-

ants of the policy (S)-(C) for ranging from 40 to 200.The first variant is a direct translation of the solution ofthe deterministic relaxation, with a threshold ,2 = 16chosen according to the recommendation in §3.2; recallthat type 2 is the least profitable type to which the systemcross-sells. (For simplicity, we set ,1 =, i.e., the systemwould always cross-sell to type 1 customers.) The othertwo policy variants had ,2 and the staffing level N furtheroptimized via exhaustive simulation, i.e, by performing asearch over all possible values of N . The simulation codewas written in c++. Each sample path contained 800,000customer arrivals from which we formed time averages ofthe queue length and of the fraction of customers of eachtype that were cross-sold to. The length of each simulatedpath ensured that our estimates were close to the actualsteady-state behavior.First, we note from Figure 2(a) that the absolute devi-

ation between the profits achieved through the three

Figure 2. Performance of (S)-(C).

(a) Realized profit = Π(Λ)

050

100150200250300350400450500550600650700750800

40 80 120 160 200

Arrival rate

Prof

it

(b) Scaled profit = Π(Λ)Λ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

40 80 120 160 200Arrival rate

Scal

ed p

rofi

t

Upper bound

+Threshold and staffing fine-tuning+Threshold fine-tuning

(S)-(C) profitO(√Λ)

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candidate policies as well as their difference against thedeterministic upper bound increases with the scale of thesystem, as measured by the aggregate call volume . How-ever, Figure 2(b) illustrates that if normalized by , whichis the multiplicative factor by which the above quantitiesare growing, then the respective difference decays to zero.In fact, this decay is of order 1/

√. The above findings

are representative of many examples that we tested. Sec-ond, we observe that as the size of the system increases,most of the profit gains from fine-tuning the cross-sellingthreshold parameter and staffing level can be attributed tothe former. This is practically appealing because it makesthe model more robust to forecasting errors, because adjust-ments can be made online. The next set of results that wepresent studies this issue in more detail, and also reviewsthe waiting time constraint qualification.

Performance Sensitivity with Respect to the Cross-Selling Threshold and the Staffing Level. Figure 3offers a more detailed look at the effect of these twoparameters to the center’s profitability and the steady-stateexpected waiting time experienced by its callers for the sys-tem examined above for = 120. The parameters extractedfrom the deterministic relaxation are z = 1/3 and k = 2,which would translate to a nominal staffing of N = 160servers, and a nominal threshold of ,2 = W = 20; i.e.,the center would stop cross-selling to type 2 customerswhen there are more than 20 customers in queue. Specif-ically, Figure 3(a) shows the distance between the real-ized profit and its upper bound for various values of ,2

and N . Figure 3(b) depicts the expected waiting time foreach of these parameter combinations; the respective con-straint requires that this falls below 0.167.It is worth noting that the center’s profitability is fairly

insensitive to the staffing level around its nominal value of160 servers because the effect of the latter can be com-pensated for by appropriately adjusting the cross-sellingthreshold. As expected, the waiting time is decreasing inthe staffing level and increasing in the value of the cross-selling threshold; i.e., more servers reduce the overall load,whereas higher thresholds imply that the system is will-ing to tolerate longer waiting times. In fact, as expectedfrom an informal application of Little’s law, the expectedwaiting time increases almost linearly as a function of thethreshold. The effect of the threshold on the profit is lesssignificant, which is consistent with our asymptotic resultsthat showed that (S)-(C) with practically any threshold levelperforms very close to the upper bound in large systems.Taken together, the above comments suggest that call cen-ters of reasonably large size can use the nominal staffinglevel extracted through the deterministic analysis, and sub-sequently select the cross-selling threshold to achieve con-straint qualification and improve profits.

The Value of Market Segmentation. We concludethis section through a set of numerical experiments that

Figure 3. Performance as function of staffing andthreshold levels.

0

16

32

48

64

80

140

144

148

152

156

160

164

168

172

176

180

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Dis

tanc

e fr

om u

pper

bou

nd

Threshold

Staffing

(a) Profit under (S)-(C)

1.6–1.81.4–1.61.2–1.41.0–1.20.8–1.00.6–0.80.4–0.60.2–0.40–0.2

0

16

32

48

64

80

140

144

148

152

156

160

164

168

172

176

180

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Wai

ting

time

Threshold

Staffing

(b) Waiting time under (S)-(C)

0.6–0.7

0.5–0.6

0.4–0.5

0.3–0.4

0.2–0.3

0.1–0.2

0–0.1

strive to illustrate the potential value of market segmenta-tion. The analysis here is crude in the sense that it is limitedto the deterministic relaxation. The asymptotic performanceguarantees and the numerical results presented above sug-gest that the profit gap between the respective deterministicrelaxations will persist in the stochastic systems as well.To facilitate the presentation of our results, we will mostlyfocus on a two-type system, for which s = 1, cs = 2,c = 1, = 100, and 1 = 2 = 05. The waiting cost dor the waiting time upper bound W do not play any role

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in the deterministic analysis, and hence there is no need tospecify them.It remains to specify the customer choice behavior. As in

the previous examples, we assume that the delay prefer-ences of both types are the same with qiw= 1−01w+.Type-i customers are assumed to have an exponentially dis-tributed willingness to pay with parameter bi for whichFipi = e−bipi , i = 12. We assume that prices can obtainvalues on the bounded interval 020 in each case. Forthe system that segments its customers, the optimal pricesare given by pi = 1/bi ∧ 20 (where x∧ y =minx y), forwhich ripi = 1/bi ∧ 20e−20bi∧1. Note that the optimalprice 1/bi in the absence of the price bound of $20 isequal to the average of the distribution Fi, and that ripiis linear in 1/bi as long as bi 005. The solution to thedeterministic relaxation will cross-sell to type-i providedthat ripi c/cs , which in this model translates to bi

074 =2/e, and that 1/bi 136. The optimal price forthe system that cannot segment the two customer typesdoes not admit a closed-form solution, and is computednumerically using (14) and (15).To test specific numerical system instances, we have

generated 250 independent realizations of the pair (b1 b2)by drawing each of the bis independently from a uni-form distribution on 02. For each realization of (b1 b2),we solved the deterministic relaxations with and with-out segmentation. This involved computing the optimalprices, deciding to which types to cross-sell, if any, cal-culating the corresponding staffing level, and finally theprofit rate. Figure 4 displays the relative increase in prof-its, *− *a/*a, versus the maximum of theaverage willingness to pay among the two types, given bymax1/b11/b2. The average profit increase through seg-mentation in this two-class experiment was around 24%.We have repeated this experiment several times, and inall of the experiments the average profit increase wasabove 20%. Figure 4 is rather intuitive. There will be noprofit gap between the two systems if b1 = b2 or if thebis are different, but are such that no system decides to

Figure 4. Profit comparison of systems with and with-out customer segmentation.

Value of segmentation

0

50

100

150

200

250

300

350

400

450

500

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Max average willingness to pay = max(1/b 1,1/b2)

Rel

ativ

e pr

ofit

incr

ease

(%

)

cross-sell to any customer. In settings where at least onetype has a very large average willingness to pay, both sys-tems will be very profitable in their cross-selling activi-ties, and the relative difference in profit will be small (theRHS of the figure). In settings where both parameters 1/bi

are small, then again the profit differential will be smallbecause cross-selling is barely compensating for the cost ofcapacity. The difference between the two systems is morepronounced when 1/b1 and 1/b2 are of moderate size, inwhich case the relative added value from (a) price cus-tomization and (b) selective cross-selling (i.e., the capabil-ity to cross-sell to only one of the two types) is significant.For example, 20% of the 250 instances that we generatedare such that the system with segmentation will choose toonly cross-sell to one type, whereas the system with nosegmentation capability will not cross-sell at all.Finally, as the number of customer types and the avail-

ability of information on potential segmentation increases,the overall profit contribution due to segmentation becomesmore substantial. In a set of experiments that we ran withfour customer types, the average relative profit increase was40% (up from 24% for the system with two types). Also,as the number of types was increased, we observed moreinstances where the cross-selling recommendations of thetwo systems would differ significantly.

7. Concluding RemarksTo summarize, this paper proposes a tractable determin-istic relaxation for studying the various control problemsthat arise in call center systems with cross-selling capa-bility, paying particular attention to the effect of customersegmentation on the structure of the staffing, cross-selling,and routing policies that the system may choose to adopt.The policies that are generated through this analysis aresimple to implement, intuitive, and achieve near-optimalperformance.Our analysis can be extended in several directions to bet-

ter model the operational complexity of modern call centersystems, as well as that of customer behavior. In the former,this may include systems that have multiple pools of agentswith different processing capabilities, as well as more com-plicated service requirements, that may need a sequence ofsteps to be handled by the same or different agents. Withrespect to the latter, one could allow the customer’s deci-sion of whether to listen to the cross-selling offer to includeinformation from the initial phase of service experienced bythe customer, such as his service time, whether his initialrequest was successfully resolved, etc. Another extensionwould be to allow for customers to abandon the queue iftheir waiting time is excessive. All of the above general-izations increase the complexity of the underlying systemsubstantially, but can be addressed using our approximateanalysis with little additional effort.Finally, an interesting extension would examine the

staffing and control decisions in the face of nonstation-ary arrival patterns or parameter estimation and forecasting

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errors. Our asymptotic optimality results in this paper applyonly to the stationary case with known arrival rates. For thissetting, our asymptotic analysis and experience with numer-ical examples show that only the smallest threshold, ,k, hasan important effect on the system performance. Still, thecontrol (C)—with its multiple thresholds—was designedwith more general settings in mind. Indeed, it seems plau-sible that in settings with nonstationarity and estimationerrors, these larger threshold will play an important role byproviding the system with a significant level of adaptabilityand robustness.

8. Electronic CompanionAn electronic companion to this paper is available as partof the online version that can be found at http://or.journal.informs.org/.

Endnotes1. A recent study by McKinsey & Co. (Eichfeld et al.2006) suggests that bank call centers can generate revenuesthat are equivalent to 10% of the revenue generated throughthe retail branch channels.2. In a recent study, a Purdue University research group(Anton 2005) has estimated that call centers may attemptto cross-sell to as many as 60% of all its callers.3. The first step involves the identification of appropriateattributes along which to segment the customer pool. Theaccuracy of the estimation of the customer-type characteris-tics will be greatly improved if the center can keep track ofdata on customers that refused to listen to the cross-sellingoffer, and on those that listened but did not buy. Finally,there is a trade-off between the number of customer seg-ments and the accuracy of this estimation procedure, whichmay result in coarse segmentation, as opposed to segment-ing down to the level of each customer.4. For example, if the problem of original interest has′ = 100 and W ′ = 20 seconds, then W is selected so thatW ′ = W/

√′, which in this case would give W = 200 sec-

onds. One should then study an asymptotic version of (2)as grows large and W is scaled according to 200/

√;

note that the original formulation is recovered for = 100.5. This is equivalent to assuming that in this case the will-ingness to pay is independent of the realized waiting time.

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Cross-Selling in a Call Center with

a Heterogeneous Customer Population: Online companion

Itay Gurvich† Mor Armony∗ Costis Maglaras ‡

Appendix A - Proofs of the main results

This appendix is dedicated to performance analysis of the (S)-(C) staffing and control rule as intro-

duced in the paper, “Cross-Selling in a Call Center with a Heterogeneous Customer Population,”

[5]. The analysis, which consists of several components, will eventually lead to the asymptotic

optimality results of Theorems 1, 2 and 3. Some of the results proved below are based partially on

auxiliary results whose proof is involved and very lengthy. Hence, while we do give here the proofs

for all the main results of the paper, the proofs for some of the auxiliary results are relegated to

Appendix B.

The following notational conventions will be used throughout this appendix. ZΛi,1(t) is the number

of type-i customers receiving service (not cross-selling), at time t. We set ZΛ1 (t) =

∑Ki=1 ZΛ

i,1(t).

Recall that QΛ(t) is the overall queue length at time t. Also, let ZΛi,2(t) be the number of class i

customers in the cross-selling phase at time t and ZΛ2 (t) =

∑Ki=1 ZΛ

i,2(t) be the overall number of

customers in the cross-selling phase at time t. We assume that all processes and random variables

are defined on a common probability space (Ω,F , P ) on which we make later additional probabilistic

assumptions. For any finite dimensional random variable X, |X| is the L1 norm. For any finite

dimensional process Y (t), t ≥ 0, Ey[|Y (t)|] stands for the expected value of the normed process

at time t given that Y (0) = y.

Proof of Proposition 1:

First note that the delay sensitivity of the customers dictates that we need to keep track of the

individual customers’ waiting times to generate the sample path of the system. Hence, the state

descriptor SΛ(t) = (ZΛ1 (t);ZΛ

i,2(t), i = 1, ..., K; QΛ(t)) is not large enough for a Markovian charac-

terization under (S)-(C). Instead, we use a larger state descriptor that contains additional workload

information. Specifically, let vΛj (t) be the residual handling time of the jth arriving customer (service

and cross-selling) at time t and let WΛ(t) be the virtual waiting time at time t. Then, we construct

†Columbia Business School, 4I Uris Hall, 3022 Broadway, NY, NY 10027. ([email protected])∗Stern School of Business, NYU, 44 West 4th Street, NY, NY 10012. ([email protected])‡Columbia Business School, 409 Uris Hall, 3022 Broadway, NY, NY 10027. ([email protected])

1

the vector vΛ(t) = vΛj (t)j≥1:vΛ

j (t)>0, and consider the process ΞΛ(t) = vΛ(t), QΛ(t),WΛ(t) so

that for all t ≥ 0, ΞΛ(t) ∈ Ξ := R∞+ × Z+ × R+.

The sample paths of the process ΞΛ(t) are generated as follows: we generate an infinite sequence

of I.I.D uniform [0, 1] random variables and let customer j agree to cross-selling if Uj ≤ q(W (τj))

where τj is the arrival time of customer j. We generate an infinite sequence s1j , s

2j∞j=1 of I.I.D

random variables, where for each j, s1j and s2

j are independent exponentially distributed random

variables with respective rates µs and µcs. We let sj(wj) be the handling time of customer j given

that he had to wait wj units of time. Then, if the agent decided to cross-sell to customer j, sj(wj)

will equal s1j + s2

j with probability q(wj) and it will equal s1j otherwise. Note, that under (S)-(C)

and with the given state descriptor one can calculate the waiting time (and in turn the actual

service time) of customer j immediately upon the customer’s arrival to the system. We maintain

the customers ordered in increasing order of their arrival times, so that if there are more than N

customers in the system (where N is the number of agents) the first N elements represent the

customers that are in service. Under these definitions ΞΛ(t) is a Markov process.

We are now ready to prove Proposition 1 which is more formally stated as follows: Fix Λ and

assume N > R. Then, the process ΞΛ(t) admits a unique stationary distribution, νΛ, which is also

the limit distribution, that is

ΞΛ(t) ⇒ ΞΛ(∞), as t →∞,

where ΞΛ(∞) has the distribution νΛ, and the convergence holds regardless of the distribution of

ΞΛ(0). We actually prove a stronger result than mere stability under (S)-(C). That is, we prove

that the there exists a unique limit distribution for any policy that does not idle agents when there

are customer waiting in queue. The proof can then be easily adapted to cover (C) with finite

thresholds. Since Λ is fixed we omit it from the notation throughout the proof. What we need

to show to establish the statement of the proposition is that there exists δ0 and K so that for all

ξ ∈ Ξ with |ξ| > K we have that

Eξ[|Ξ(|ξ|(1 + δ0))|] ≤ 1/2|ξ|. (A1)

Assuming (A1) holds one can follow the proof of Theorem 3.1 in Dai [3] to show positive Harris

recurrence of Ξ(·). Positive Harris recurrence implies existence and uniqueness of a stationary

distribution. The stationary and limit distributions agree by the obvious non-lattice structure of

2

the time between consecutive visits to the origin (the 0 state of R∞+ × Z+). The inequality (A1)

is established easily by adapting the fluid limits argument of [3] to our model. For the sake of

completeness we give the detailed proof in the Appendix B. ¥

We now turn to prove the results regarding the performance analysis and asymptotic optimality

of (S)-(C). Here, the superscript Λ is used to denote parameters or processes related to the Λth

system. We omit the superscript when dealing with parameters that do not scale with Λ.

In the rest of this appendix we use a different sample path construction than the one we used before

and, in particular, one that relies on the Poisson nature of most of the processes involved. A detailed

description of the construction is given in Appendix B. Throughout the rest of the section we assume

that the thresholds ηΛi , i ≤ k satisfy ηΛ

i = ηi

√Λ for some constants ∞ > η1 ≥ η2 . . . ≥ ηk.

Proof of Theorem 1: Since Π(Λ) ≥ Π∗(Λ) ≥ Π(Λ), it suffices to prove that Π(Λ) = Π(Λ) −o(Λ). Note that since (C) was proven to admit steady state we necessarily have that λixi(Λ) =

µcsi E[ZΛ

i,2(∞)]. We now write,

E[(ri(WΛ(∞))xi(Λ)] = E[ri(0)xi(Λ)] + E[(ri(WΛ(∞)− ri(0))xi(Λ)].

As the function ri(·) is a bounded function with limw↓0 ri(w) = 0, we have by Lemma 1 that

E[ri(WΛ(∞))] → 0 (Lemma 1 itself is proved in Appendix B; see Remark 3 there). Consequently,

λiE[(ri(WΛ(∞))xi(Λ)] = λiri(0)xi(Λ) + o(Λ). (A2)

Hence, the profit function is given by

Π(Λ) =k∑

i=1

riµcsi E[ZΛ

i,2(∞)]− cR(1 + z)− hΛE[WΛ(∞)] + o(1).

It follows that in order to show the result it suffices to prove that

E[ZΛi,2(∞)] =

λiqi

µcsi

− o(Λ), for all i ≤ k, and E[WΛ(∞)] = o(1). (A3)

Indeed, given that (A3) holds the proof is completed since

Π(Λ)− Π(Λ) =k∑

i=1

ri(λiqi − µcsi E[ZΛ

i,2(∞)]) + hΛE[WΛ(∞)] + o(Λ) = o(Λ). (A4)

3

Expression (A3) is rather intuitive. The left hand side says that in steady-state the expected

number of type i customers that are in the system and are being cross-sold to is equal to λiqi/µcsi ,

which is what the deterministic relaxation predicts, minus a small (second order) correction term.

Equivalently, the stochastic effects in the system result in small deviations around the deterministic

solution. Similarly, the right hand side, which is a consequence of Lemma 1 says that the expected

waiting times experienced by callers are of order o(1), which in turn are consistent with the second

order correction terms just mentioned. The proof of (A3) is rather technical. The complexity

emanates from the fact that, while one may use simpler methods to prove convergence of the related

processes (see, e.g., [1]), proving the convergence of the steady state variables is rather involved

even if one can compute the steady state distribution explicitly - which is not the case in our

setting. Following Gamarnik and Zeevi [4], we use an appropriate Lyapunov function constructed

via a fluid model analysis together with some probabilistic bounds obtained using tools form Strong

Approximations. The details of this argument is relegated to Appendix B.

Proof of Theorem 2: Item (a) of the proposition was already proved within the proof of

Theorem 1. The rest of the result is proved by contradiction. Assume first that (b) does not hold.

That is that

lim infΛ→∞

|N∗(Λ)−R(1 + z)|Λ

> 0. (A5)

Note that Π∗(Λ) ≤ Π(N∗(Λ),Λ), where the latter stands for the solution of the deterministic relax-

ation when the staffing level is fixed to N∗(Λ). Π(N∗(Λ),Λ) is obtained by solving the corresponding

fractional Knapsack problem. Assume now that

lim infΛ→∞

N∗(Λ)−R(1 + z)Λ

> 0. (A6)

That is, the staffing is higher than the one suggested by (S) (the proof for the other case is

essentially the same). Then, there exists a subsequence Λjj≥1, such that for all j, the solution

to the fractional knapsack problem is obtained by setting

xΛj

i = qi,∀i ≤ k.

4

Set i0 = maxi :∑i

k=1 λkqk/µcsk ≤ N∗(Λj) − R(1 + z). Since N∗(Λ) > R(1 + z) by (A6), we

necessarily have that i0 ≥ k. If i0 > k we also have that xΛj

i = qi for all k < i ≤ i0 and we set

xΛj

i0+1 =N∗(Λj)−R(1 + z)−∑i0

i=k+1 λiqi/µcsi

λji0

.

Finally, xΛj

i = 0 for all i > i0 + 1. Consequently, we have that

Π∗(Λj)− Π(Λj) ≤ Π(N∗(Λj), Λj)− Π(Λj) = −c(N∗(Λj)−R(1 + z)) +K∑

i=k+1

λji rix

Λj

i

≤i0∑

i=k+1

λiqi(ri − c/µcsi )

+ (µcsi0+1ri0+1 − c)

N∗(Λj)−R(1 + z)−

i0∑

i=k+1

λiqi/µcsi

. (A7)

Recalling that, by assumption, ri < c/µcsi for all i > k, we must have that

lim supΛ→∞

Π∗(Λ)− Π(Λ)Λ

< 0.

But we have already shown that Π∗(Λ) = Π(Λ)− o(1) leading to a contradiction.

The proof of (c) follows in essentially the same manner where we now assume, to reach a contra-

diction, that for some i, xi = xi + f(Λ), with lim infΛ→∞ |f(Λ)| > 0. More specifically, if i ≤ k we

assume that xΛi = qi + f(Λ) with f(Λ) negative for all Λ, and if i ≥ k we assume that xi = f(Λ)

with f(Λ) positive for all Λ. We then consider again the deterministic relaxation where, instead of

fixing the staffing level, we fix xΛi . The rest of the argument is now analogous to the proof of (b).

¥

Proof of Lemma 2: Let Π(N∗(Λ), x∗(Λ)) be the resulting profit in a system with arrival rate

Λ and equipped with the optimal policy. Then, it is trivial that,

Π(N∗(Λ), x∗(Λ)) ≤ −cR +K∑

i=1

λiqi(ri − c/µcsi )+ − ΛhE[WΛ,∗] = Π(Λ)− ΛhE[WΛ(N∗(Λ), x∗(Λ))].

(A8)

Assume, by contradiction, that

lim infΛ→∞

E[WΛ,∗] > 0.

5

Let (N ′(Λ), x′(Λ)) the staffing level and fraction of cross-selling attempts resulting from (S)-(C)

and recall that we have established in Lemma 1 that with thresholds satisfying (6), we have that

E[WΛ] = O

(1√Λ

), and Π(Λ)−ΠΛ(N ′(Λ), x′(Λ)) = o(1),

where WΛ is the steady state waiting time under (S)− (C). In particular,

lim supΛ→∞

Π(Λ)−Π(N∗(Λ), x∗(Λ))Π(Λ)−Π(N ′(Λ), x′(Λ))

= ∞, (A9)

contradicting the optimality of (N∗(Λ), x∗(Λ)). ¥

6

Appendix B - Proofs of supporting results

This is part B of the e-companion to the paper, “Cross-Selling in a Call Center with a Heterogeneous

Customer Population,” [5]. The organization of this appendix is as follows: we begin in §B.1 with

the completion of the proof of Proposition 1, whose sketch was given in Appendix A. We continue

in §B.2 with some preliminaries required for the performance analysis of (S)-(C). Specifically, we

provide a sample-path construction that uses a collection of independent rate-1 Poisson processes.

We also discuss some strong approximation tools. Finally, in §B.3 we prove the main performance-

analysis results which are used in the proofs in Appendix A. Some auxiliary results are proved in

§B.4.

B.1 A Detailed Proof of Proposition 1

This section is dedicated to the completion of the proof of Proposition 1 in Appendix A. Following

our proof sketch in Appendix A we show that there exists δ0 ≥ 0 such that for any sequence of

initial states ξn ⊆ Ξ with |ξn| → ∞:

lim supn→∞

1|ξn|Eξn [|Ξ(|ξn|(1 + δ0)|] = 0. (B1)

Whenever this holds we can always find a positive number K, such that for all |ξ| > K, (A1) holds.

Toward that end, let V (t) =∑

j≥1 vj(t) be the amount of residual work in the system. Also, let

Vs(t) and Vq(t) be, respectively, the residual work for the customers in service at time t, i.e. Vs(t) =∑

j≤N vj(t), and the residual work for the customers in queue at time t, i.e. Vq(t) = V (t)− Vs(t).

Set1µ

=K∑

i=1

λi

Λ

(1µs

+1

µcsi

),

so that 1/µ is the mean customer handling time in case cross-selling is performed. The following

Lemma is the analogue of Lemma 4.3 in Dai [3]. Its proof is the same and is hence omitted.

Lemma 4 Fix a sequence of initial conditions |ξn| with |ξn| → ∞. Then,

(a) Almost surely, uniformly on compact sets,

lim supn

Vs(|ξn|t)|ξn| ≤ N(1− t)+, (B2)

1

and

lim supn

Vq(|ξn|t)|ξn| ≤ 1 +

Λµ

t, (B3)

(b) For each t ≥ 0, the sequences

Vs(|ξn|t)|ξn| , n ≥ 1

and

Vq(|ξn|t)|ξn| , n ≥ 1

, are uniformly inte-

grable.

Finally, for any fixed t ≥ 0

lim supn

1µs

Q(|ξn|t)|ξn| ≤ lim sup

n

Vq(|ξn|t)|ξn| ≤ lim sup

n

1µ

Q(|ξn|t)|ξn| , (B4)

and

lim supn

1µs

Eξn [Q(|ξn|t)]|ξn| ≤ lim sup

n

Eξn [Vq(|ξn|t)]|ξn| ≤ lim sup

n

1µ

Eξn [Q(|ξn|t)]|ξn| . (B5)

For the following let D(t) be the cumulative number of customer departure from the system up

to time t. Also, let Wi(t) be the virtual waiting time for type-i customers at time t. Since

customers are assumed to be served FCFS we have that Wi(t) = Wj(t) for all i 6= j. Hence,

we can define W (t) to be the virtual waiting time for all the customers, regardless of their type.

Using the standard notation, we let Dd[0,∞) be the space of functions f(·) : [0,∞) 7→ Rd that

are Right Continuous with Left Limits (RCLL). A sequence of processes, Xn, in Dd is said to be

C-tight if, in addition to being tight (as random elements of Dd), every convergent subsequence

converges to a limit that is a.s. continuous. The following lemma establishes C-tightness of the

different processes in consideration and identifies important properties of the limit points. We let

µmax = maxµs, µcs1 , . . . , µcs

K.

Lemma 5 For any sequence of initial conditions ξn ⊆ Ξ with |ξn| → ∞ as n → ∞, and on

compact subsets of [1,∞), the sequence(

Q(|ξn|·)|ξn| , D(|ξn|·)

|ξn| ,Vq(|ξn|·)|ξn| , W (|ξn|·)

|ξn|)

is C-tight. Moreover,

any limit point (Q(t), D(t), Vq(t), W (t)) satisfies

D(t) ≤ Nµmaxt, (B6)

W (t) ≥ Q(t)Nµmax

, (B7)

1µ

Q(t) ≥ Vq(t), (B8)

2

for all t ≥ 0. Finally,

Vq(t) ≤[Vq(1) + (t− 1)

(Λµs−N

)]+

, (B9)

for all t ≥ 1.

Proof: The proof of C-tightness for the sequence(

Q(|ξn|t)|ξn| , D(|ξn|t)

|ξn|)

is reminiscent of the proof of

Theorem 4.1 in Dai [3] and we omit it. The inequality (B6) is trivial to establish. We proceed then

to prove (B7). The virtual waiting time satisfies the representation

W (t) = infs ≥ 0 : D(t + s)−D(t) ≥ Q(t), (B10)

and in particular,

W (|ξn|t)|ξn| = inf

s ≥ 0 :

D(|ξn|(t + s))−D(|ξn|t))|ξn| ≥ Q(|ξn|t)

|ξn|

.

Considering a convergent subsequence nj of (Q(|ξn|t)/|ξn|, D(|ξn|t)/|ξn|), we can now apply the

corollary in [10], to conclude that W (|ξnj |t)/|ξnj | also converges to a limit W (t). Consequently,

the sequence (Q(|ξn|t)/|ξn|, D(|ξn|t)/|ξn|,W (|ξnj |t)/|ξnj ) is also C-tight. Moreover, any limit point

satisfies W (t) ≥ Q(t)/Nµmax. The inequality (B8) follows from Lemma 4. Finally, to establish

(B9), fix ε > 0 and set τn := inft ≥ 1 : Vq(|ξn|t) ≤ ε|ξn|. Then, for 1 ≤ t ≤ τn and by work

conservation

Vq(|ξn|t) + Vs(|ξn|t) = Vq(|ξn|1) + Vs(|ξn|1)−N |ξn|(t− 1) +A(|ξn|t)∑

l=A(|ξn|)sl (W (τl)) . (B11)

Here τl is the time of the lth arrival. Consider a convergent subsequence

(Q(|ξnj |t)|ξnj | ,

D(|ξnj |t)|ξnj | ,

W (|ξnj |t)|ξnj |

).

Then, we claim that uniformly on compact subsets of [1, τnj ),

ynj (t) :=

∑A(|ξnj |t)l=A(|ξnj |) sl (W (τl))

|ξnj | → Λµs

(t− 1), in probability, as j →∞. (B12)

3

To prove this claim, fix T > 1 and consider the set

Ωnj :=

ω ∈ Ω : inf1≤t≤T

W (|ξnj |t) ≤ 12

µVq(|ξnj |t)Nµmax

.

By inequalities (B7) and (B8), P (Ωnj ) → 0 as j →∞. Then,

P

sup

1≤t≤T∧τnj

∣∣∣∣ynj (t)− Λµs

(t− 1)∣∣∣∣ > ε

≤ P

sup

1≤t≤T∧τnj


(t− 1)∣∣∣∣ > ε, (Ωnj )c

+ PΩnj → 0, as j →∞. (B13)

for any ε > 0, where the convergence

P

sup

1≤t≤T∧τnj


(t− 1)∣∣∣∣ > ε, (Ωnj )c

→ 0, as j →∞,

follows from the definition of Ωnj and a careful application of the strong law of large numbers using

the fact that on (Ωnj )c, for t ≤ τnj and for any ε > 0 there exists j large enough so that for all

k ≥ j, E[sl(W (τl))] ≤ 1µs + ε. Now, Let

y(t) :=[Vq(1) + (t− 1)

(Λµs−N

)]∨ 3

2ε.

Combining (B11) and (B13) we then have that for any ε > 0,

P

sup

0≤t≤τnj∧T

[Vq(|ξnj |t)|ξnj | − y(t)

]+

> ε

→ 0, as j →∞. (B14)

Define now two random times as follows:

τn = inft ≥ τn : Vq(|ξn|t) > 2ε|ξn| and τ′n = supt ≤ τn : Vq(|ξn|t) ≤ ε|ξn|.

Then, we can extend the arguments we used above to show that for any ε > 0,

P

sup

τ′nj∧T≤t≤τnj∧T

[Vq(|ξnj t)|ξnj | − y(t)

]+

> ε

→ 0, as j →∞.

4

Since for ε < ε/2,

τnj < T ⊂

supτ′nj∧T≤t≤τnj∧T


]+

> ε

,

we have that Pτnj < T → 0, as j →∞. In particular, since

sup

0≤t≤T


]+

> ε

⊆

sup

0≤t≤τnj∧T


]+

> ε

⋃τnj < T,

we conclude that for arbitrarily small ε,

P

sup

0≤t≤T


]+

> ε

→ 0, as j →∞. (B15)

The result of the Lemma now follows since ε was arbitrary. ¥

To complete the proof of Proposition 1, note that with δ0 = (1+Λ/µ)(N−R), and since by Lemma

4, Vq(1) ≤ 1 + Λ/µ, we have that

Vq(1) + (δ0)(

Λµs−N

)≤ 0.

To conclude the argument, we fix an arbitrary sequence of initial conditions ξn. Consider a

convergent subsequence njj≥1, of E[Vq(|ξn|(1+δ0))]|ξn| . By Lemma 5 each subsequence njk

⊂ njsuch that Vq(|ξnjk |(1+δ0))

|ξnjk | converges satisfies

limk→∞

E[Vq(|ξnjk |(1 + δ0))]|ξnjk | = 0.

In particular,

lim supn→∞

E[Vq(|ξn|(1 + δ0))]|ξn| = 0.

Since the sequence ξn was arbitrary the argument is complete. ¥

5

B.2 Sample Path Construction, Strong Approximations and Other

Preliminaries

For the rest of our results we replace the sample path construction from Proposition 1 with a differ-

ent one that takes advantage of the properties of the Poisson process. This alternative construction

follows an approach that is, by now, quite common; see Whitt [6] for an overview. Having con-

structed the sample paths using Poisson processes we can use strong approximations to bound the

distance of the underlying Poisson process from their respective rates. This section is divided then

to two subsections. First, in §B.2.1 we construct the sample path. Then, in §B.2.2 we introduce

the strong approximation tools that will be required for our proofs.

B.2.1 Sample-Path Construction

We base the sample-path construction on independent unit-rate Poisson processes A, S1 and Si,2,

i = 1, . . . , k, where k is the last customer class that is cross-sold under the control (C); see §3.1 in

[5]. We generate arrivals of customers using the Poisson process AΛ(t) := A(Λt). The arrival times

are registered in a dynamic array in which the customers are ordered in order of their arrival times.

Denote the value of this array at time t ≥ 0 by the vector T (t). Let T (t) be the process obtained by

setting Tk(t) = t−Tk(t). Whenever a customer leaves the queue to be served he is erased from this

array. We hold an NΛ dimensional vector w(t) = (w1(t), . . . , wNΛ(t)) with wj(t), j = 1, . . . , NΛ

standing for the registered waiting time of the customer currently in service with agent j. This

value is registered immediately when the customer is admitted to service.

We let S(t) be the set of agents giving service at time t, i.e., S(t) contains all the agents that are

not idle and not cross-selling at time t. We generate the process of phase 1 (service) completions

with type-i customers using a time change of a unit-rate Poisson process. Specifically, let DΛ1 (t) be

the number of phase 1 completion up to time t. Then, we write

DΛ1 (t) := S1

(µs

∫ t

0ZΛ

1 (s)ds

), (B16)

where ZΛ1 (t) is the number of agents giving service at time t. Whenever the process DΛ

1 (t) jumps we

generate a discrete random variable with values in S(t−) to identify the number of the agent that

completed service. Formally, for a subset J of 1, . . . , NΛ we let es(J, l), l ∈ N be a sequence

6

of i.i.d random variables distributed uniformly over J . We will use these to determine the actual

agents that completes services. Also, we let ec(l), l ∈ N be a sequence of i.i.d discrete random

variables on 1, . . . ,K with Pec(1) = i = λi/Λ. Finally, we let ecs(l), l ∈ N be i.i.d uniform

random variables on [0, 1].

The number of service completions followed by a cross-selling phase with a type-i customer, Di(t),

for i ≤ k is then given by

DΛi (t) =

DΛ1 (t)∑

l=1

∑

j∈S(t−)

1es(S(t−), l) = j, ec(l) = i, ecs(l) ≤ qi(wj(t−)), QΛ(t−) ≤ ηiΛ.

Here we used the fact that an agent that completes service will attempt cross-selling to a type-i

customer only if at the time of completion the queue is less than the threshold ηΛi . The process

DΛi (t) is then a non-homogenous Poisson process with an instantaneous rate that is bounded from

above byk∑

i=1

λi

Λµsqi(0)ZΛ

1 (t)1QΛ(t) ≤ ηi.

Finally, for i ≤ k, we generate cross-selling completions by a non-homogenous Poisson process with

an instantaneous rate µcsi ZΛ

i,2(t) at time t, i.e, we write

DΛi,2(t) = Si,2

(µcs

i

∫ t

0ZΛ

i,2(s)ds

),

where ZΛi,2(t) is the number of type-i customers undergoing cross-selling at time t.

The system state is then captured by the multi-dimensional Markov process

ΞΛ(t) := (ZΛ1 (t), ZΛ

i,2(t),SΛ(t), wΛ(t), QΛ(t), T Λ(t); i = 1, . . . ,K). (B17)

We let X be the domain of this process. The following Lemma is a corollary of Proposition 1 and

its proof is easily obtained by expanding the state-space of the Markov process constructed in the

proof of Proposition 1. Although the proof would use a different sample path construction, the

uniqueness of the stationary distribution is invariant to this construction as the probability law is

the same under both constructions.

Lemma 6 Fix Λ. Then the process ΞΛ(t) admits a unique stationary distribution πΛ which coin-

7

cides with the unique limit distribution.

We let ξ be a general element in X , and for given ξ we let q(ξ), z1(ξ), and zi,2(ξ), i = 1, k be

respectively the queue length, the number of agents giving service and the number of agents cross-

selling to a type-i customer in state ξ. The notation q(ξ) should not be confused with the integer

q that we will use as a general power nor with the delay sensitivity functions qi(·).

We conclude this section with some additional notation. For fixed T > 0, a positive integer d

and a function y ∈ Dd[0,∞), we define ‖y(·)‖T := sup0≤t≤T

∑di=1 |y(t)|. We let (Ω,F , P ) be the

probability space which will remain fixed throughout and use the notation ω to denote an element

in Ω. We let Pξ be the probability distribution under which Pξ

(ΞΛ(0) = ξ

)= 1, and put Eξ[·]

be the expectation operator with respect to the probability distribution Pξ. We let PπΛ be the

probability distribution under which ΞΛ(0) ∼ πΛ where πΛ is the unique stationary distribution

from Lemma 6 and we define EπΛ [·] accordingly. Finally for x, y ∈ R, we use the standard notations

x ∨ y = maxx, y and x ∧ y = minx, y as well as x+ = maxx, 0 and x− = max−x, 0.

B.2.2 Strong Approximations

Time changes of unit-rate Poisson processes, as the ones we have used above, can be approximated

by time-changed Brownian motion plus a logarithmic error term. We refer the reader to Mandel-

baum et. al. [9] and the references therein for a detailed discussion of strong approximations and

their application to Markovian queueing networks. For our purposes, it suffices to know that given

a unit-rate Poisson process N (·) and an instantaneous bounded rate function 0 ≤ λ(t) ≤ λ, for

some λ > 0, we have that

supt≥0

N(∫ t

0 λ(u)du)− ∫ t

0 λ(u)du−B(∫ t

0 λ(u)du)

log(λt ∨ 2)≤ C,

where C is a non-negative random variable with

PC > γ + βx ≤ c1e−c2x, (B18)

for some strictly positive constants γ, β, c1 and c2; see Lemma 9.4 in [9]. Since the rate of service

(or cross-selling completions) is bounded by NΛ(µs∨maxi=1,...,K µcsi ) and since the staffing rule (S)

dictates NΛ = R(1+ z) we can find m large enough so that all the instantaneous rates in the system

8

are bounded mΛ. We henceforth fix m to be such a value. We can then define a 3K-dimensional

Brownian motion, B(t), such that

∥∥AΛ(·)− Λ·∥∥

T+

∥∥∥∥DΛ1 (·)− µs

∫ ·

0ZΛ

1 (s)ds

∥∥∥∥T

+k∑

i=1

∥∥∥∥DΛi,2(·)− µcs

i

∫ ·

0ZΛ

i,2(s)ds

∥∥∥∥T

≤ ‖B(mΛ·)‖T + Clog(mΛT ∨ 2). (B19)

Finally, we will be using some basic facts about Brownian motion. Specifically, for a d-dimensional

Brownian motion and a constant m > 0, it can be easily shown that for all x ≥ 0, T > 0,

P‖B(mΛ·)‖T ≥ x

√Λ

≤ c3

√T

xe−c4

x2

T ,

for some strictly positive constants c3 and c4; see e.g. Problem 2.8.2 in [8]. Using (B18) we then

have that (B18) for all x > 0

P‖B(mΛ·)‖T + C log(mΛT ∨ 2) ≥ x

√Λ

≤ c5

(1 ∨

√T

x

)e−c6 minx2

T, x√

T, (B20)

for some strictly positive constants c5, c6. Denote by

Ω∗(Λ, T, x) = ω ∈ Ω : ‖B(mΛ·)‖T + C log(mΛT ∨ 2) ≤ x . (B21)

Then, by (B20)

P

Ω∗(Λ, T, x√

Λ)≥

(1− c5

(1 ∨

√T

x

)e−c6 minx2

T, x√

T)+

.

B.3 Performance Analysis for (S)-(C)

The aim of this section is two-fold. First, we want to establish that the queue length is, in some

sense, bounded by the smallest threshold ηΛk. Then, fixing ε > 0, we want to show that the

steady-state expected number of agents cross-selling type-i customers satisfies

∣∣∣∣E[ZΛi,2(∞)]− λiqi(0)

µcsi

∣∣∣∣ ≤ εΛ,

9

for all Λ large enough. Since ε is arbitrary, we will have that

E[ZΛi,2(∞)] =

λiqi(0)µcs

i

+ o(Λ),

which is what is used in the proof of Theorem 1 in Appendix A.

All the subsequent proofs share the same basic ideas. Using the strong approximation construction

we examine the behavior of the system on a subset of the sample paths (such as Ω∗(Λ, T, x))

where the stochastic fluctuations generated by the Brownian Motion are bounded. This allows

us to examine a deterministic version of the system dynamics. For the deterministic version, and

with arguments reminiscent of Lyapunov function tools used for stability proofs, we show that the

system is in some sense attracted back into a certain domain. Finally, we remove the conditioning

and apply some arguments from [4] to obtain the steady state bounds. Some of the arguments are

common to several proofs. We abbreviate the proofs whenever this is the case. Throughout we fix

T > 0 and ε > 0.

Most of the Propositions in this section share a common structure. The first part of each such

proposition states a steady-state bound. The second part essentially states that if the process is

initialized at time 0 close enough to its steady-state distribution (in a sense to be made precise), it

actually stays there.

Our first result towards our stated goals shows that the number of agents busy giving service (not

cross-selling) is close to the load R. It also shows that the queue length will not exceed, in some

sense, the largest threshold, ηΛ1 . We will use these results to refine the bound on the queue length

process in Proposition 7. For the following we define

X1 := ξ ∈ X : |z1(ξ)−R| ≤ εΛ, q(ξ) ≤ ηΛ1 + ε

√Λ (B22)

Proposition 5 Fix ε > 0 and q ≥ 2. Then,

P∣∣ZΛ

1 (∞)−R∣∣ > εΛ

≤ c7

(√

Λ)q−1, (B23)

and

lim supΛ→∞

E

[((QΛ(∞)− ηΛ

1 )+)q−1

]≤ CQ, (B24)

10

for all Λ large enough and some strictly positive constants c7 and CQ Moreover,

supξ∈X1

Pξ

∥∥ZΛ1 (·)−R

∥∥T≥ 2εΛ

≤ c9e−c10

√Λ, (B25)

and

supξ∈X1

Pξ

∥∥(QΛ(·)− ηΛ1 )+

∥∥T≥ 2ε

√Λ

≤ c9e

−c10ε√

Λ (B26)

for all Λ large enough and for some strictly positive constants c9 and c10.

Note that with the exception of the customer cross-selling probability being delay sensitive, when-

ever the queue-length process exceeds the greatest threshold, ηΛ1 , it behaves just as the queue length

process in the single class model of Armony and Gurvich [2] when it exceeds the unique threshold

there. Moreover, when the queue is above ηΛ1 the delay sensitivity does not play any role as by the

control mechanism (C), no cross-selling will be attempted until the queue goes below ηΛ1 . The proof

of this result will then follow some of the results in [2]. A detailed proof of Proposition 5 would

be obtained by expanding the proofs of Lemma B.1 and Proposition B.1 in [2] with the unique

threshold there replaced by ηΛ1 . We postpone the proof of this result to §B.4. We will also need the

following Proposition which establishes, the otherwise intuitive result, that the number of agents

busy cross-selling to class-i customers cannot exceed λiqi(0)/µcsi significantly. For the following we

define

X2 := ξ ∈ X1 : |zi,2(ξ)− λiqi(0)/µcsi | ≤ εΛ, i = 1, . . . , k, (B27)

where X1 was defined in equation (B22).

Proposition 6 Fix ε > 0. Then, for all Λ large enough and all x > 0,

P

(ZΛ

i,2(∞)− λiqi(0)µcs

i

)+

> εΛ

≤ c11e

−c12√

Λ, i = 1, . . . , K.

for some strictly positive constants c7 and c8. Moreover,

supξ∈X2

Pξ

∥∥∥∥∥(

ZΛi,2(·)−

λiqi(0)µcs

i

)+∥∥∥∥∥

T

≥ 2εΛ

≤ c11e

−c12√

Λ, i = 1, . . . , K, (B28)


The proof of Proposition 6 is very similar to the proofs of the estimate for ZΛ1 (t) that were stated

11

in Proposition 5. The detailed proof is omitted, and we turn now to the finer analysis of the queue

length and waiting time processes.

Towards than end, we let WΛi (t) be the virtual waiting time for type-i customers at time t. Since

customers are served in a FCFS fashion, we have that WΛi (t) = WΛ

j (t) for all i 6= j. Hence, we may

let WΛ(t) be the common virtual waiting time. Since, by Proposition 1, steady state exists, the

PASTA property guarantees that waiting time as seen by arriving customers is equal in distribution

to the steady-state virtual waiting time. The following proposition shows that using (S)-(C) the

queue length and the waiting time are small and in particular the queue exceeds the threshold ηk

at most by an amount that is o(√

Λ).

We now turn to a more refined analysis of the queue length process. Towards this end, define the

set

X3 := ξ ∈ X2 : q(ξ) ≤ ηΛk + ε

√Λ,

where X2 was defined in equation (B27). The following proposition is the most complicated one in

this appendix as it deals with a refined analysis of the queue length behavior and in particular one

that considers the behavior of the queue at an O(1) level. The other results are o(Λ) result and

are hence much simpler. The more refined analysis for the queue length is however necessary for

the other proofs as well as for the asymptotic feasibility result for the constrained case, as given in

Theorem 3 of the main paper [5].

Proposition 7 Fix an integer q ≥ 2. Then,

lim supΛ→∞

E

[((QΛ(∞)− ηΛ

k )+)q−1

]≤ Cq, and lim sup

Λ→∞

√ΛE[WΛ(∞)] < ∞, (B29)

for some constant Cq. In particular, for all x > 0 and all Λ large enough,

P(QΛ(∞)− ηΛk )+ > x

√Λ ≤ Cq

(x√

Λ)q−1. (B30)

Also,

supξ∈X3

Pξ

∥∥(QΛ(·)− ηΛk )+

∥∥T≥ 2ε

√Λ

≤ c11e

−c12ε√

Λ, (B31)

and

supξ∈X3

Pξ

‖WΛ(·)‖T ≥ M2 + ε√

Λ

≤ c11e

−c12ε√

Λ, (B32)

12

for all Λ large enough and some strictly positive constants c11, c12 and M2.

Remark 3 (Lemma 1 and Theorem 3 in [5]) Note that Lemma 1 in the main paper [5] is

covered as a special case of Proposition 7. Moreover, when the threshold ηΛk

is chosen according to

the recommendation in §3.2 in the main paper [5], Theorem 3 there is a consequence of Proposition

7 above.

Proof of Proposition 7: Fix a constant K > 1 and define a function Φ(x) : R+ 7→ R+ as

follows:

Φ(x) = (x− ηΛk )+ + K.

The proof now proceeds as follows: We first fix the integer q ≥ 2 and establish that there exist

t∗ > 0 and M > 0 so that

supξ∈X1:q(ξ)>ηΛ

k+M

Eξ

[Φ(QΛ(t∗/Λ))

q]− Φ(q(ξ))q ≤ −γΦ(q(ξ)q−1, (B33)

for some γ > 0. We will then use Lemma 8 from §B.4 and adapt the argument used in the proof

of Theorem 5.1 in [4] to obtain a bound for the steady-state queue length process. The bound for

the steady-state waiting time will then follow from an application of Little’s law. Finally, we will

establish the bound in (B31) and (B31).

We start, then, by establishing (B33). Towards this end, fix M > 0 and assume that QΛ(0) >

ηΛk

+ M . Fix 0 < η ≤ M/2 and define the random time τΛ = inft ≥ 0 : QΛ(t) ≤ QΛ(0)− η

.

Note that on [0, τΛ ∧ T ], the queue length process QΛ(t) satisfies

QΛ(t) > ηΛk , and QΛ(t) = QΛ(0) + AΛ(t)−

k∑

i=1

DΛi,2(t)− DΛ

0 (t),

where DΛ0 (t) is the process of service completions not followed by a cross-selling phase, i.e, DΛ

0 (t) =

DΛ1 (t)−∑k

i=1 DΛi (t). On [0, τΛ] the instantaneous rate of DΛ

0 (t) can be bounded from below by

µsZΛ1 (t)

k−1∑

i=1

λi

Λ(1− qi(0)) +

K∑

i=k

λi

Λ

,

which corresponds to the fact that, by the control (C), as long as the queue length is greater than

ηk all service completions of customers from type i ≥ k are followed by the admission to service of

13

a customer that is waiting in the queue. Fix now ε > 0 and δ > 0 and define the set

Ω(Λ) =

ω ∈ Ω :∥∥ZΛ

1 (·)−R∥∥

T≤ 2εΛ,

∥∥∥∥ZΛi,2(·)−

λiqi(0)µcs

i

∥∥∥∥T

≤ 2εΛ, i = 1, . . . , k

⋂Ω∗(Λ, T/Λ, δ),

where Ω∗ is as defined in (B21). Then, on Ω(Λ) and for t ≤ τΛ ∧ T/Λ,

QΛ(t) ≤ QΛ(0) + Λt−k∑

i=1

µcsi

∫ t

0ZΛ

i,2(u)du− µs

k−1∑

i=1

λi

Λ(1− qi(0)) +

K∑

i=k

λi

Λ

∫ t

0ZΛ

1 (u)du + δ,

Observe that for all t ≤ τΛ all servers are busy and consequently∑k

i=1 ZΛi,2 = NΛ − ZΛ

1 . Since, by

definition, NΛ −R =∑k

i=1 λiqi/µcsi , we have, after some straightforward algebra, that on Ω(Λ),

QΛ(t ∧ τΛ) ≤ QΛ(0) + CΛε(t ∧ τΛ)− λkqk(0) · t ∧ τΛ + δ, (B34)

for t ≤ T/Λ and for some constant C > 0. Equation (B34) is the crucial one. It reflects the fact

that once cross-selling to class k is stopped, the capacity of the system is large enough to attract

the aggregate queue back to ηk, and it will do so at a rate of approximately λkqk(0). We can now

choose ε small enough so that

QΛ(t ∧ τΛ) ≤ QΛ(0) + δ − 12Λλkqk(0) · t ∧ τΛ, (B35)

where for all i ≤ K, λi := λi/Λ. In particular, on Ω(Λ) we have that

ΛτΛ ≤ t∗ :=η + δ

12 λkqk(0)

.

Choosing δ ≤ η/2, the same argument shows that on Ω(Λ) and for all τΛ ≤ t ≤ t∗/Λ, QΛ(t) ≤QΛ(0) − η/2. Indeed, along the arguments leading to (B35) one can establish that on Ω(Λ), the

queue length is a linearly decreasing process as long as QΛ(t) > ηΛk. We conclude then that if

QΛ(0) ≥ ηΛk

+ M ,

Φ(QΛ(t∗/Λ))− Φ(QΛ(0)) ≤ −η/2 (B36)

on Ω(Λ). Since QΛ(t) ≤ QΛ(0) + AΛ(t), it is straightforward to show that

lim supΛ→∞

supξ∈X1

Eξ[Φ(QΛ(t∗/Λ))1Ωc]− Φ(QΛ(0)) = 0.

14

Consequently, we have from (B36) that for all Λ large enough

supξ∈X1:Φ(q(ξ))>M

Eξ[Φ(QΛ(t∗/Λ))]− Φ(QΛ(0)) ≤ −η

4.

Let

LΛ(t∗) := supξ∈X1

Φ−(q−2)(ξ)Eξ

[(Φ(QΛ(t∗/Λ))− Φ(q(ξ)))2(Φ(q(ξ)) + |Φ(QΛ(t∗/Λ))− Φ(q(ξ))|)q−2

].

(B37)

Using again the fact that QΛ(t) ≤ QΛ(0) + AΛ(t) as well as some basic properties of the Poisson

process we have that lim supΛ→∞ LΛ(t∗) ≤ C1 for some constant C1. Hence, we have by Lemma 8

in §B.4 that there exists a constant C2 such that

supξ∈X1:Φ(q(ξ))>C2

Eξ[Φ(QΛ(t∗/Λ))q]− Φ(q(ξ))q ≤ −ηq

8Φ(q(ξ))q−1. (B38)

Using again the fact that QΛ(t) ≤ QΛ(0) + AΛ(t), we have that

supξ∈X

Eξ

[Φ(QΛ(t∗/Λ))q

]− Φ(q(ξ))q

Φ(q(ξ))q≤ C3, (B39)

for some C3. In particular,

supξ∈X :Φ(q(ξ))≤C2

Eξ

[Φ(QΛ(t∗/Λ))q

] ≤ C4,

for some C4 > 0. We now adapt arguments from the proof of Theorem 5.1 in Gamarnik and Zeevi

[4], to establish bounds for (QΛ(∞) − ηΛk)+. Specifically, by the definition of stationarity we have

that

EπΛ

[Φ(QΛ(0))q

]= EπΛ

[Φ(QΛ(t∗/Λ))q

]. (B40)

and, in particular,

0 =∫

ξ∈ΞΛ

(Φ(q(ξ))q −Eξ[Φ(QΛ(t∗/Λ))q]

)πΛ(dξ). (B41)

Combining equations (B38) and (B39) we have that

Eξ[Φ(QΛ(t∗/Λ))q]− Φ(q(ξ))q ≤ −ηq8 Φ(q(ξ))q−11ξ ∈ X1+ C4 + C3Φ(q(ξ))q1ξ /∈ X1,

15

and, consequently, that for all ξ ∈ X ,

Φ(q(ξ))q−Eξ[Φ(QΛ(t∗/Λ))q] ≥ ηq

8Φ(q(ξ))q−1−C4−C3Φ(q(ξ))q1ξ /∈ X1+ηq

8Φ(q(ξ))q−11ξ /∈ X1.

Plugging back into (B41), we have that

∫

ξ∈ΞΛ

(ηq

8Φ(q(ξ))q−1 − C4 − C3Φ(q(ξ))q1ξ /∈ X1+

ηq

8Φ(q(ξ))q−11ξ /∈ X1

)πΛ(dξ) ≤ 0. (B42)

Now, using the bounds on the steady-state queue length from Proposition 5 and applying the

Cauchy-Schwarz inequality yields that both

EπΛ

[Φ(QΛ(0))q−1(1ΞΛ(0) /∈ X1

] → 0, as Λ →∞, (B43)

and

EπΛ

[Φ(QΛ(0))q(1ΞΛ(0) /∈ X1

] → 0, as Λ →∞. (B44)

Consequently, (B42) implies that for all Λ large enough

∫

ξ∈ΞΛ

(ηq

8Φ(q(ξ))q−1 − 2C4

)πΛ(dξ) ≤ 0.

so that

E[Φ(QΛ(∞))q−1] ≤ 16C2

ηq.

Note that with q = 2 we have that E[(QΛ(∞) − ηΛk)+] ≤ C4, for some C4 > 0 and for all Λ large

enough. Consequently, lim supΛ→∞E[(QΛ(∞) − ηΛk)+]/

√Λ = 0, so that the first part of (B29)

is established. Note that we have actually established that E[(QΛ(∞) − ηΛk)+] = O(1), which is

stronger than the statement of the Proposition which requires only that E[(QΛ(∞)−ηΛk)+] = o(

√Λ).

The statement about the steady-state waiting time now follows from Little’s law.

We now turn to the proof of equation (B31). To analyze the behavior of the queue length process

over the interval [0, T ], fix x > 0 and re-define the set

Ω(Λ) =

ω ∈ Ω :∥∥ZΛ

1 (·)−R∥∥

T≤ 2εΛ,

∥∥∥∥(ZΛ

i,2(·)− λiqi(0)µcs

i

)+∥∥∥∥

T

≤ 2εΛ, for all i = 1, . . . ,K

⋂Ω∗∗(Λ, T, δΛ),

16

where

Ω∗∗(Λ, T, δΛ) :=

ω ∈ Ω : sup

0≤η≤Tsup

t−s≤η(|B(mΛt)−B(mΛs)| − δΛ(t− s) + O(log(mΛt ∨ 2))) ≤ ε

4

√Λ

.

(B45)

One can easily show that PΩ∗∗(Λ, T, δΛ)c ≤ C5e−C6ε

√Λ for all Λ large enough and for some

constants C5 and C6. Indeed, this probability bound follows from a combination of equation (B18)

and a basic result for Brownian motion (see Example 4.3.12 in page 264 of [8]). We now define the

random times

τ ′Λ = sup

t ≤ T : QΛ(t) ≤ ηΛk + ε

√Λ

, and τ ′′Λ = inf

t ≥ τ ′Λ : QΛ(t) ≥ ηΛ

k + 2ε√

Λ

.

By the same arguments preceding equation (B34), we have for all t ≥ τ ′Λ and by choosing ε small

enough that

QΛ(t ∧ τ ′′Λ) ≤ QΛ(τ ′Λ)− 12λkqk(0) ·

(t ∧ τ ′′Λ − τ ′Λ

)+ δΛ

(t ∧ τ ′′Λ − τ ′Λ

)+

ε

4

√Λ. (B46)

Choosing small enough δ, we have by that on Ω(Λ), QΛ(·) is smaller than QΛ(τ ′Λ) + ε/4√

Λ for all

t ≥ τ ′Λ4. Consequently, τ ′′Λ > T on Ω(Λ) so that sup0≤t≤T QΛ(t) > ηΛk

+ 2ε√

Λ. Hence,

supξ∈X3

Pξ

sup

0≤t≤TQΛ(t) > ηΛ

k + 2ε√

Λ

≤ PΩ(Λ)c ≤ C7e

−C8

√Λ,

for all Λ large enough where the last inequality follows from Propositions 5 and 6 as well as the

probability bound for Ω∗∗(Λ, T, δΛ). This establishes equation (B31). The proof of (B32) uses the

following representation for the virtual waiting time:

WΛ(t) = infs ≥ 0 : DΛ(t + s)−DΛ(t) ≥ QΛ(t)

,

where DΛ(t) is the aggregate number of depletions of customers from the queue up to time t.

Let τΛ(t) = infs ≥ 0 : QΛ(t + s) = 0. Since while the queue is non-empty every cross-selling

completion is followed by an admission of a customer from the queue, we have that on [t, τΛ(t)],

DΛ(t + s)−DΛ(t) ≥k∑

i=1

DΛi,2(t + s)−DΛ

i,2(t) ≥k∑

i=1

µcsi

∫ t+s

tZΛ

i,2(u)du− |B(mΛ(t + s))−B(mΛt)|.

17

Then, on the set Ω(Λ) and for s ≤ τΛ(t),

DΛ(t + s)−DΛ(t) ≥k∑

i=1

µcsi

∫ t+s

tZΛ

i,2(u)du− δΛs ≥ sk∑

i=1

µcsi

λiqi(0)µcs

i

− C9εΛs− δΛs− ε

4

√Λ,

for some constant C9 > 0. Consequently, for all t ≥ 0,

WΛ(t) ≤ Cw√Λ

+‖QΛ(·)‖T∑k

i=1 λiqi(0)− CεΛ− δΛ,

for some constant Cw. The bound for the waiting time in (B32) now follows by choosing δ and ε

small enough and using the bound in (B31). Note that the virtual waiting time actually depends

on the behavior of the departures slightly after time T . This problem is easily overcome, however,

by re-defining Ω(Λ) using the interval [0, 2T ] instead of [0, T ]. ¥

In order to analyze the number of cross-sold customers from each class a finer analysis is required.

In particular, we need a handle of the waiting time of the customers that are present in the system

at time 0 (which is assumed to be distributed according to the stationary distribution). For the

following results, we let ZΛ(t) be the number of customers in the first phase of service or in queue

that found upon arrival a virtual waiting time that is longer than 2M2/√

Λ, where M2 is as given

in Proposition 7. Formally,

ZΛ(t) =∑

j∈S(t)

1wj(t) > 2M2

√Λ. (B47)

We then have the following Lemma where we use

X4 := ξ ∈ X3 : z(ξ) ≤ εΛ, (B48)

and X3 is as defined in (B29).

Lemma 7 Fix an integer q ≥ 2 and ε > 0. Then,

P

ZΛ(∞) > 2εΛ≤ c13

1√

Λ(q−1)

. (B49)

Moreover,

supξ∈X3

Pξ

‖ZΛ(·)‖T > 2εΛ

≤ c13e

−c14ε√

Λ, (B50)

for all Λ large enough and for some constant c13 and c14.

18

Proof: We define QΛ(t) to be the number of customers in queue that found a virtual waiting time

longer than 2ηΛk/Λ upon arrival. Then, focusing on the number of customers with waiting times

longer then 2ηΛk/Λ in queue or in service we have the equation,

QΛ(t) + ZΛ(t) = QΛ(0) + ZΛ(0) +∫ t

01WΛ(t−) > 2M2

√ΛdAΛ(t)

−DΛ

1 (t)∑

l=1

∑

j∈S(t−)

1es(S(t−), l) = j, wj(t−) > 2M2

√Λ. (B51)

Note that∑DΛ

1 (t)l=1

∑j∈S(t−) 1es(S(t−), l) = j, 2M2

√Λ is a non-homogeneous Poisson process with

instantaneous rate equal to µsZΛ(u).

Consider the differential equation (initialized at ZΛ(0)),

¯ZΛ(t) = ¯ZΛ(0)− µs

∫ t

0

¯ZΛ(u)du. (B52)

Since ZΛ(0) ≤ NΛ, there exists a finite time t∗ after which ¯ZΛ(t) ≤ εΛ. Fix the set

Ω(Λ) := Ω∗(Λ, T, εΛ)⋂

ω ∈ Ω : ‖WΛ(·)‖T ≤ M2 + ε√Λ

⋂ ω ∈ Ω : ‖QΛ(·)‖T ≤ ηΛ

k + 2ε√

Λ

.

Subtracting ZΛ(t) from ¯ZΛ(t), using the fact that∫ t0 1WΛ(t−) > 2ηΛ

k/ΛdAΛ(t) = 0 on Ω(Λ) and

finally applying Gronwall’s inequality (see, e.g., Problem 5.2.7 on page 287 of Karatzas and Shreve

[8]) we have that

‖ZΛ(·)− ¯ZΛ(·)‖T ≤ CeCT (‖B(mΛ·)‖T + ‖QΛ(·)|T ), (B53)

on Ω(Λ). We the have that

PπΛZΛ(t∗) > 2εΛ ≤ PπΛΞΛ(0) /∈ X3+ supξ∈X3

PξC ′Λ > εΛ,1Ω(Λ)

+ supξ∈X3

Pξ1Ω(Λ)c ≤ C13e−C14ε

√Λ, (B54)

where C ′Λ is the right hand side in equation (B53) and the last inequality follows from the definition

of Ω(Λ), Proposition 7 and the properties of Brownian Motion enlisted in §B.2.2. Since under the

steady state distribution ZΛ(t∗) has the same distribution as ZΛ(∞), we can use Proposition 7

and the bounds from §B.2.2 to establish equation (B49). The transient bound in equation (B50)

19

follows a similar argument in which one replaces in equation (B54) the initial distribution πΛ with

an arbitrary initial condition within the set X4. ¥

We now turn to the analysis of the number of agents busy cross-selling to type-i customers. We

first focus on types i ≤ k − 1. Type k requires a separate analysis which is given in Proposition 9.

We define

X5 :=

ξ ∈ X4 :∣∣∣∣zi,2(ξ)− λiqi(0)

µcsi

∣∣∣∣ ≤ εΛ, i = 1, . . . , k − 1

(B55)

and X4 is as defined in (B48).

Proposition 8 Fix an integer q ≥ 2 and ε > 0. Then,

P

∣∣∣∣ZΛi,2(∞)− λiqi(0)

µcsi

∣∣∣∣ ≥ εΛ≤ c15

1(√

Λ)(q−1), i = 1, . . . , k − 1, (B56)

and

E

[∣∣∣∣ZΛi,2(∞)− λiqi(0)

µcsi

∣∣∣∣]≤ εΛ, (B57)

for all Λ large enough and for some constant c15. Moreover,

supξ∈X5

Pξ

∥∥∥∥ZΛi,2(·)−

λiqi(0)µcs

i

∥∥∥∥T

≥ 2εΛ≤ c16e

−c17ε√

Λ, i = 1, . . . , k − 1, (B58)

for all Λ large enough and for some constant c16 and c17.

Remark 4 (Proof of Theorem 1 in [5]) Since ε is arbitrary, Proposition 8 implies that for

i ≤ k − 1, E[ZΛi,2(∞)] = λiqi/µcs

i + o(Λ). In particular, since E[ZΛ1 (∞)] = R + o(Λ) and N =

R +∑k

i=1 λiqiµcsi , it suffices to show that E[IΛ(∞)] = o(Λ), where IΛ(t) is the number of idle

agents at time t, to conclude that also for k:

E[ZΛk,2(∞)] =

λkqk

µcsk

+ o(Λ).

The required result for IΛ(∞) is established in Proposition 9, which together with Proposition 7

establish (A3) and complete the proof of Theorem 1 in the main paper [5].

Proposition 9

E[IΛ(∞)] = o(Λ), and E[ZΛi,2(∞)] =

λiqi

µcsi

+ o(Λ), ∀i ≤ k. (B59)

20

The proof of Proposition 9 is postponed until after the proof of Proposition 8.

Proof of Proposition 8: The argument is very similar in nature to the one used in the proof

of Lemma 7. Define first the set

Ω(Λ) := Ω∗(Λ, T, x√

Λ)⋂

ω ∈ Ω : ‖WΛ(·)‖ ≤ M2 + 2ε√

Λ√Λ

⋂ w ∈ Ω : ‖ZΛ(·)‖T ≤ (ε + x)Λ

⋂w ∈ Ω : ‖ZΛ

1 (·)−R‖T ≤ 2εΛ⋂

w ∈ Ω : ‖QΛ(·)‖T ≤ ηk + ε√

Λ

, (B60)

where we choose M so that M < (ηk−1 − ηk)/2. As ZΛi,2(t) = ZΛ

i,2(0)−DΛi,2(t) + DΛ

i (t), we have on

Ω(Λ) that

ZΛi,2(t) ≥ ZΛ

i,2(0)−µcsi

∫ t

0ZΛ

i,2(u)du+µs λi

Λqi

(2M2/

√Λ

)∫ t

0ZΛ(u)−ZΛ(u)du−‖B(mΛ·)‖T +O(log(mΛT∨2)).

(B61)

Here we used the definition of ZΛ as well as the fact that while QΛ(t) ≤ ηk−1 all service completions

with type-i customers are followed by a cross-selling offer if the customer agrees to listen. Also,

ZΛi,2(t) ≤ ZΛ

i,2(0)−µcsi

∫ t

0ZΛ

i,2(u)du+µs λi

Λqi(0)

∫ t

0ZΛ

1 (u)−ZΛ(u)du+‖B(mΛ·)‖T +O(log(mΛT∨2)).

(B62)

Consider now the differential equation (initialized at ZΛi,2(0)):

ZΛi,2(t) = ZΛ

i,2(0) + µs λi

Λqi(0)Rt− µcs

i

∫ t

0ZΛ

i,2(u)du. (B63)

Then, subtracting ZΛi,2(t) from ZΛ

i,2(t), using the inequalities (B61) and (B62) as well as equation

(B63) and applying Gronwall’s inequality we have that

‖ZΛi,2(·)− ZΛ

i,2(·)‖T ≤ CeCT [qi(‖WΛ(·)‖T )‖ZΛ1 (·)−R‖T ∨ ‖ZΛ(·)‖T

∨ (‖B(mΛ·)‖T + O(log(mΛT ∨ 2)))].(B64)

The argument now follows as in the proof of Lemma 7. Re-define the function Φ(x) = |x −λiqi(0)/µcs

i |. We first consider the differential equation (B63). Observe that since ZΛi,2(0) ≤ NΛ,

there exists t∗ so that

Φ(ZΛi,2(t

∗)) ≤ εΛ.

21

Hence,

PπΛΦ(ZΛi,2(t

∗)) > 2εΛ ≤ PπΛΞΛ(0) /∈ X4+ supξ∈X4

PξC ′′Λ > 2εΛ, Ω(Λ)

+ supξ∈X4

PξΩ(Λ)c ≤ C

(√

Λ)q−12

, (B65)

where C ′′Λ is the right hand side in (B64) and X4 is as defined in (B48). Since Φ(ZΛi,2(t

∗)) has the

same distribution as Φ(ZΛi,2(∞)) when starting with the steady distribution we have established

equation (B56). Equation (B57) follows a similar argument noting that

EπΛ [Φ(ZΛi,2(t

∗))] ≤ NΛPπΛΞΛ(0) /∈ X3+ supx∈X3

Eξ[Φ(ZΛi,2(t

∗))]

and applying the bounds we have from the previous propositions to show that

NΛPπΛΞΛ(0) /∈ X4 → 0, as Λ →∞

as well as

supξ∈X4

Eξ[Φ(ZΛi,2(t

∗))] ≤ NΛ supξ∈X4

PξΩc+ 2εΛ.

Replacing ε with ε/2 throughout one has the result. The transient bound is easily obtained using

similar arguments. ¥

Proof of Proposition 9: Re-define the set

Ω(Λ) = Ω∗∗(Λ, T, δΛ)⋂

ω ∈ Ω :∥∥ZΛ

1 (·)−R∥∥

T≤ 2εΛ

∥∥∥∥ZΛi,2(·)−

λiqi

µcsi

∥∥∥∥T

≤ 2εΛ, ∀i = 1, . . . , k − 1;

‖ZΛ(·)‖T ≤ 2εΛ; ‖WΛ(·)‖T ≤ (M2 + ε)/√

Λ

, (B66)

where Ω∗∗ is defined in (B45). Assume that IΛ(0) ≥ 2εΛ and set τΛ = inft ≥ 0 : IΛ(t) ≤

22

IΛ(0)− εΛ. Then, on Ω(Λ),

IΛ(t ∧ τΛ) ≤ IΛ(0)− Λ(t ∧ τΛ) +k−1∑

i=1

µcsi

∫ t∧τΛ

0ZΛ

i,2(u)du

+ µs(1− q

(2M2/

√Λ

)) ∫ t∧τΛ

0ZΛ(u)− ZΛ(u)du

+ µcsi

∫ t∧τΛ

0

N − IΛ(u)− ZΛ(u)−

k−1∑

i=1

ZΛi,2(u)

du + δΛt + ε

√Λ. (B67)

Some algebra yields that

IΛ(t ∧ τΛ) ≤ IΛ(0) + ε√

Λ− CΛt, (B68)

for some constant C > 0. Starting at IΛ(0) > 2εΛ, then, there exists t∗ at which IΛ(t∗) ≤ εΛ.

If, on the other hand, IΛ(0) ≤ 2εΛ, then we claim that IΛ(t) ≤ 3εΛ for all t ≤ T . Indeed let

τ ′Λ := inft ≥ t∗ : IΛ(t) ≥ εΛ and τ ′′Λ := inft ≥ τ ′Λ : IΛ(t) ≥ 2εΛ. Then, using (B68) we have

that on Ω(Λ), τ ′′Λ > T . Consequently, on Ω(Λ), there exists t∗ with IΛ(t∗) ≤ 3εΛ regardless of the

initial condition. We then have that

EπΛ [IΛ(t∗)] ≤ NΛPΞΛ(0) /∈ X4+ 3εΛ + supξ∈X5

PΩ(Λ)c,

and the proof is completed by noting that

NΛPΞΛ(0) /∈ X5+ supξ∈X5

PΩ(Λ)c → 0, as Λ →∞.

¥

As explained in Remark 4, with Proposition 9 we conclude the proof of Theorem 1 in the main

paper [5]. We conclude this appendix with Lemma 8 that was used in the proof of Proposition 7.

B.4 Auxiliary Results

In this section we prove Proposition 5 as well as one simple general result that we used in the proof

of Proposition 7. We begin with the latter.

The following Lemma is an adaptation of a result that and appears in [7] and is due to Gamarnik

and Zeevi. We state and prove it here for completeness. Fix Λ and consider the process Ξ(t) with

23

the domain X and Let Φ(x) be a function Φ(x) : X 7→ R+. We fix a subset X ⊂ X . We let

L(t) = supξ∈X

Φ−(q−2)(ξ)Eξ

[Φ(Ξ(t))− Φ(ξ))2(Φ(ξ) + |Φ(Ξ(t))− Φ(ξ)|)q−2

]. (B69)

Lemma 8 Fix an integer q ≥ 2. Assume that there exists K > 0, γ > 0 and t∗ > 0, so that

supξ∈X :Φ(ξ)>K

Eξ[Φ(Ξ(t∗))]− Φ(ξ) ≤ −γ, (B70)

and that L(t∗) is finite. Then,

supξ∈X :Φ(ξ)>K′

Eξ[Φ(Ξ(t∗))q]− Φ(ξ)q ≤ −γq

2Φ(ξ)q−1, (B71)

with K ′ = maxK,L(t∗)(q − 1)/γ.

Proof: Using second order Taylor’s expansion of the function xp around Φ(ξ) we obtain for every

ξ ∈ X such that Φ(ξ) > k,

Eξ[Φq(Ξ(t∗))]− Φq(ξ) = qΦq−1EξEξ[Φ(Ξ(t∗))− Φ(ξ)]

+q(q − 1)

2Eξ[(Φ(ξ) + Z(Φ(Ξ(t∗))− Φ(ξ)))q−2(Φ(Ξ(t∗))− Φ(ξ))2]

≤ −γqΦq−1(ξ) +q(q − 1)

2Eξ[(Φ(ξ) + |Φ(Ξ(t∗))− Φ(ξ)|)q−2(Φ(Ξ(t∗))− Φ(ξ))2]

≤ −γqΦq−1(ξ) +q(q − 1)

2L(t∗)Φq−2(ξ) (B72)

where Z is a random variable with support in [0, 1]. When Φ(ξ) > L(t∗)(q − 1)/γ, we obtain that

Eξ[Φq(Ξ(t∗))]− Φq(ξ) ≤ −γq

2Φq−1(ξ).

¥

Proof of Proposition 5: We now prove the estimates given in Proposition 5 for the number of

agents busy giving service and the queue length. Some of the steps are very similar to those in [2].

The similar parts will be abbreviated and the reader will be referred to the technical appendix of

[2] for the details. The first step is the following Lemma, the first part of which is an analogue of

Lemma B.1 in [2].

24

Lemma 9 Fix ε > 0. Then, there exists t0(ε) (independent of the initial conditions), such that

P

sup

t0(ε)≤t≤T

(ZΛ

1 (t)−R)−

> εΛ

≤ c−c10

√Λ

9 , (B73)

for all Λ large enough and for two positive constants c9 and c10. Consequently,

P(ZΛ

1 (∞)−R)− > εΛ ≤ c18e

−c19√

Λ,

and

supξ∈X1

Pξ

‖(ZΛ1 (·)−R)−‖T ≥ 2εΛ

≤ c9e−c10

√Λ. (B74)


Proof: The first part of the Lemma is proved just as in [2] and its proof is omitted. Since the

bound is independent of the initial state, we can in particular initialize the system with its steady

state distribution in which case we get the bound on this steady state distribution. ¥

The next step establishes a crude bound for the queue-length process. It shows that the queue

length is essentially of order Λ. This step is required to obtain later the more refined bound.

Lemma 10 Fix ε > 0. Then,

lim supΛ→∞

E

[exp

(QΛ(∞)

Λ

)]≤ CQ

for some constant CQ.

Proof: We use a Lyapunov type of argument along the lines of Gamarnik and Zeevi [4]. First,

assume that QΛ(0) > 2MΛ for some constant M > 0 and define the random time

τΛ = inft ≥ 0 : QΛ(t) ≤ QΛ(0)−MΛ.

Since the largest threshold, ηΛ1 satisfies ηΛ

1 = η1

√Λ, there exists Λ large enough such that MΛ > ηΛ

1 .

Consequently, on [0, τΛ], all service or cross-selling completions will be followed by an admission of

25

a customer from the head of the queue. The queue length process satisfies then the equation

QΛ(t ∧ τΛ) = QΛ(0) + AΛ(t)−DΛ1 (t)−

k∑

i=1

DΛi,2(t). (B75)

Using the strong approximation we have that

QΛ(t∧ τΛ) ≤ QΛ(0) + Λt− µs

∫ t

0ZΛ

1 (s)ds−k∑

i=1

µcsi

∫ t

0ZΛ

i,2(s)ds + ‖B(mΛ·)|T + O(log(mΛT ∨ 2)).

(B76)

Let

Ω(Λ) = Ω∗(Λ, T, εΛ)⋂ω ∈ Ω : ‖ZΛ

1 (·)−R‖T ≤ εΛ,

with Ω∗(Λ, T, εΛ) as defined in (B21). We then have after some algebraic manipulation that on

Ω(Λ),

QΛ(t ∧ τΛ) ≤ QΛ(0) + εΛ− CΛ(t ∧ τΛ). (B77)

We claim now that with t∗ = 3M/(2C) we have that

Eξ∈X :q(ξ)>εΛEξ[exp(QΛ(t∗)/Λ)]

exp(q(ξ)/Λ)≤ exp(−M/4). (B78)

We omit the simple argument. We now use equation (B78) to establish a bounds for the steady state

queue length using a result from [4]. Towards that end, note first that since QΛ(t) ≤ QΛ(t)+AΛ(t),

we have that

supξ∈X

Eξ

[eQΛ(t∗)/Λ

]

eq(ξ)/Λ≤ C1, (B79)

for some constant C1 > 0 and for all Λ. We can now apply Theorem 5 of [4] to obtain that

E

[exp

(QΛ(∞)

Λ

)]≤ CQ, (B80)

for some constant CQ and all Λ large enough. ¥

In the next Lemma we show that the queue hardly exceeds the greatest threshold, ηΛ1 , as stated in

Proposition 5.

26

Lemma 11 ε > 0 and q ≥ 2. Then,

lim supΛ→∞

E

[((QΛ(∞)− ηΛ

1 )+)q−1

]≤ CQ (B81)

for some strictly positive constant CQ. Moreover,

supξ∈X1

Pξ

∥∥(QΛ(·)− ηΛ1 )+

∥∥T≥ 2ε

√Λ

≤ c9e

−c10ε√

Λ (B82)


Proof: The proof follows almost exactly the proof of the Lemma 10 up to equation (B77). The

main difference is that here, like in the proof of Proposition 7, one replaces the time interval

[0, T ] with the time interval [0, T/Λ]. Once the dynamics of the system above the level of ηΛ1 are

established, the proof follows almost exactly as the proof of Proposition 7 with two exceptions:

First, we replace X1 there with

X ′1 := ξ ∈ X : (z1(ξ)−R)− ≤ εΛ.

Then, the modification of equations (B43) and (B44) is established here using Lemma 10 (rather

than 5). Specifically, by the Cauchy-Schwarz inequality that

EπΛ

[Φ(QΛ(0))q−1(1ΞΛ(0) /∈ X ′

1] ≤

√EπΛ [(Φ(QΛ(∞)))2(q − 1)]

√PπΛΞΛ(0) /∈ X ′

1. (B83)

By Lemma 10,

lim supΛ→∞

E[(Φ(QΛ(∞)))2(q−1)] ≤ CΛq−1,

for some constant C > 0. By Lemma 9

PπΛΞΛ(0) /∈ X ′1 ≤ c18e

−c19√

Λ.

Plugging back into (B83) we have that

EπΛ

[Φ(QΛ(0))q−1(1ΞΛ(0) /∈ X ′

1] → 0, as Λ →∞.

27

The modification of equation (B44) follows similarly. The remainder of the proof follows almost

exactly the remainder of proof of Proposition 7 with the obvious modifications required by the

replacement of the smallest threshold ηΛk

with the largest threshold ηΛ1 . ¥

The last component required to complete the proof of Proposition 5 is to establish a two-sided

bound for the difference ZΛ1 (·)−R, rather then the one sided bound from Lemma 9. The result is

given in the following Lemma.

Lemma 12 Fix ε > 0 and q ≥ 2. Then,

P∣∣ZΛ

1 (∞)−R∣∣ > εΛ

≤ c7

(√

Λ)q−1, (B84)

and

supξ∈X1

Pξ

∥∥ZΛ1 (·)−R

∥∥T≥ 2εΛ

≤ c9e−c10

√Λ, (B85)


Proof: The proof is very similar in nature to the proof of Proposition 8 and is actually simpler.

First, note that QΛ(t) and ZΛ1 (t) satisfy the equation

QΛ(t) + ZΛ(t) = QΛ(0) + ZΛ(0) + AΛ(t)−DΛ1 (t),

or using the strong approximation decomposition,

QΛ(t)+ZΛ(t) = QΛ(0)+ZΛ(0)−Λt−µs

∫ t

0ZΛ

1 (s)ds+B

(Λt + µs

∫ t

0ZΛ

1 (s)ds

)+O(log(mλt∨2)).

Consider the differential equation (initialized at ZΛ1 (0))

ZΛ1 (0) = ZΛ

1 (0) + Λt− µs

∫ t

0ZΛ

1 (s)ds.

Then, subtracting ZΛ1 (t)− ZΛ

1 (t) and using Gronwall’s inequality we have that

‖ZΛ1 (·)− ZΛ

1 (·)‖T ≤ CeCT[‖QΛ(·)‖T + ‖B(mΛ·)|T + O(log(mΛT ∨ 2))

].

28

The proof now proceeds as in the proof of Proposition 8 using the bounds for the Brownian motion

from §B.2.2 and the bound for QΛ(·) from Lemma 11 above. ¥

References

[1] M. Armony. Dynamic routing in large-scale service systems with heterogeneous servers. Queue-ing Systems, 51(3-4):287–329, 2005.

[2] M. Armony and I. Gurvich. When promotions meet operations: Cross-selling and its effect oncall-center performance. Working Paper. New York University and Columbia University, NewYork, 2006.

[3] J. G. Dai. On positive Harris recurrence of multiclass queueing networks: A unified approachvia fluid limit models. Ann. Appl. Prob., 5:49–77, 1995.

[4] D. Gamarnik and A. Zeevi. Validity of heavy traffic steady-state approximations in generalizedJackson networks. Ann. Appl. Prob., 16:56–90, 2006.

[5] I. Gurvich, M. Armony, and C. Maglaras, Cross-Selling in a Call Center with a HeterogeneousCustomer Population. Preprint, 2006.

[6] Whitt W. Martingale proofs of many-server heavy-traffic limits for Markovian queues. Proba-bility Surveys forthcoming, 2007.

[7] I. Gurvich and A. Zeevi. Validity of heavy-traffic steady-state approximations in open queue-ing networks: sufficient conditions involving state-space collapse. Working Paper. ColumbiaUniversity, New York, 2007.

[8] Karatzas I., S.E. Shreve. 1991. Brownian Motion and Stochastic Calculus, 2nd ed. Springer-Verlag, New York.

[9] A. Mandelbaum, Massey W. and Reiman M. Strong approximations for Markovian servicenetworks. Queueing Systems 30:149–201, 1998.

[10] Puhalskii A. On the Invariance Principle For the First Passage Time. Math. Oper. Res.19(4):946–954, 1994.

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