staffing call centers with uncertain arrival rates and co&#x2010

Staffing Call Centers with Uncertain Arrival Rates andCo-sourcing

Yas�ar Levent Koc�a�gaSy Syms School of Business, Yeshiva University, Belfer Hall 403/A, New York City, New York 10033, USA, [email protected]

Mor ArmonyStern School of Business, New York University, KMC 8–62, New York City, New York 10012, USA, [email protected]

Amy R. WardMarshall School of Business, University of Southern California, Bridge Hall 401H, Los Angeles, California 90089, USA

[email protected]

I n a call center, staffing decisions must be made before the call arrival rate is known with certainty. Once the arrivalrate becomes known, the call center may be over-staffed, in which case staff are being paid to be idle, or under-

staffed, in which case many callers hang-up in the face of long wait times. Firms that have chosen to keep their call centeroperations in-house can mitigate this problem by co-sourcing; that is, by sometimes outsourcing calls. Then, the requiredstaffing N depends on how the firm chooses which calls to outsource in real time, after the arrival rate realizes and thecall center operates as a M/M/N + M queue with an outsourcing option. Our objective is to find a joint policy for staffingand call outsourcing that minimizes the long-run average cost of this two-stage stochastic program when there is a linearstaffing cost per unit time and linear costs associated with abandonments and outsourcing. We propose a policy that usesa square-root safety staffing rule, and outsources calls in accordance with a threshold rule that characterizes when the sys-tem is “too crowded.” Analytically, we establish that our proposed policy is asymptotically optimal, as the mean arrivalrate becomes large, when the level of uncertainty in the arrival rate is of the same order as the inherent system fluctua-tions in the number of waiting customers for a known arrival rate. Through an extensive numerical study, we establishthat our policy is extremely robust. In particular, our policy performs remarkably well over a wide range of parameters,and far beyond where it is proved to be asymptotically optimal.

Key words: call center operations; co-sourcing; staffing; overflow routing; parameter uncertaintyHistory: Received: July 2013; Accepted: September 2014 by Michael Pinedo, after 2 revisions.

1. Introduction

Call centers have become ubiquitous in business.Today, every Fortune 500 company has at least onecall center, and the average Fortune 500 companyemploys 4500 call center agents (who may be distrib-uted across more than one site) (Gilson and Khandel-wal 2005). For many companies, the call center is aprimary point-of-contact with their customers. Hence,a well-run call center promotes good customer rela-tions, and a poorly managed one hurts them. But callcenter management is difficult.A call center manager faces the classical operational

challenge of determining appropriate staffing levelsthroughout the day and week in order to meet a ran-dom and time-varying call volume. This is extremelydifficult especially when call arrival rate is itself ran-dom, as was empirically shown in Brown et al. (2005)and Maman (2009), among others. When staffing lev-els are too low, customers are put on hold, and many

hang up in frustration while waiting for an agent totake their call. But when staffing levels are too high,the call center manager ends up paying staff to beidle.One option in managing this uncertainty is for a

company to outsource its call center operations. Then,the challenges of call center management can be han-dled by a vendor firm whose primary focus is callcenter operations. That vendor can pool demandamongst various companies, thereby lowering vari-ability, which should allow for more accurate demandforecasts, and so better staffing decisions. However, itis also true that many companies are reluctant torelinquish control of their call center operations. Thisis evidenced by a recent survey from the IncomingCall Management Institute (ICMI 2006): only 7.9% of279 call center professionals used an outside vendorto handle most or all of their calls. One reason for thatare the “hidden costs of outsourcing” (Kharif 2003),which include service quality costs that are hard to

Vol. 0, No. 0, xxxx–xxxx 2015, pp. 1–17 DOI 10.1111/poms.12332ISSN 1059-1478|EISSN 1937-5956|15|00|0001 © 2014 Production and Operations Management Society

Please Cite this article in press as: Koc�a�ga, Y. L., et al. Staffing Call Centers with Uncertain Arrival Rates and Co-sourcing. Productionand Operations Management (2015), doi 10.1111/poms.12332

info:doi/10.1111/poms.12332

explicitly quantify. As a result, many of these compa-nies prefer to co-source; that is, to outsource some,but not all, of their calls.We study a co-sourcing structure in which the ven-

dor charges the company a fee per call outsourced,which is consistent with the pay-per-call (PPC) co-sourcing structure analyzed in Aks�in et al. (2008). Weassume the company can decide on a call-by-call basiswhich calls to answer in-house and which calls toroute to the vendor. This is helpful because call cen-ters typically make their daily staffing decisions atleast a week in advance, before the actual arrival rateto the call center for a given day is known. If theplanned staffing is sufficient to handle the mean arri-val rate, then the company needs to outsource only asmall fraction of calls in order to handle the inherentvariability that results in congestion every so often.On the other hand, if the planned staffing is insuffi-cient to handle the mean arrival rate, then the com-pany can outsource a large fraction of its calls,thereby preventing high congestion levels.The relevant question for this study is: how do we

decide on staffing levels when the arrival rate isuncertain and the aforementioned co-sourcing optionis present? To answer this question, we begin withone simple and widely used queueing model of a callcenter, the Erlang A orM/M/N + M (see, e.g., section4.2.2 in Gans et al. 2003), and add uncertainty in thearrival rate and an outsourcing option. Then, ourmodel for the call center is a multi-server queue witha doubly stochastic time-homogeneous Poisson arri-val process, exponential service times, and exponen-tial times to abandonment. Although this modelignores the time-varying nature of the arrival rateover the course of each day, there is call center litera-ture that discusses how to use the Erlang A model tomake staffing decisions for time-varying arrival rates,using the “stationary independent period by periodapproach” (SIPP); see Green et al. (2001), Gans et al.(2003), Aks�in et al. (2007), and Liu and Whitt (2012)for more discussion. We suppose that a similarapproach can be adopted here to accommodate theadded feature of arrival rate uncertainty.The control decisions in our model are (i) an

upfront staffing decision and (ii) real-time call out-sourcing (routing) decisions. Recall that staffing deci-sions are made on a much longer time horizon andwell before the timing of the control decisions. In par-ticular, these decisions are made on two different timescales. This means that we have a two stage stochasticprogram: The staffing decisions are made in the firststage, before the arrival rate is known, and the out-sourcing decisions are made in the second stage, afterthe arrival rate is known. Then, the outsourcing deci-sions can depend on the actual arrival rate eventhough the staffing decisions cannot.

Our objective is to propose a policy for staffing andoutsourcing under the assumption of linear staffingcost and linear abandonment and outsourcing costs.For each arriving customer that cannot be immedi-ately served, there is a tension between choosing tooutsource that customer (and paying the outsourcingfee) or having that customer wait for an in-houseagent (and risking incurring an abandonment cost).In summary, we are solving a joint staffing and rout-ing control problem for a (modified) Erlang A modelwith an uncertain arrival rate and an outsourcingoption.The three main contributions in this study are:

• The modeling contribution is the formulation ofa joint staffing and outsourcing problem for acall center that has access to co-sourcing, andmust make staffing decisions when there isarrival rate uncertainty. This modeling frame-work can be used to study more general jointstaffing and control problems in call centersthat have been previously studied in the litera-ture under the assumption of a known arrivalrate.

• The application contribution is the developmentof a square-root safety staffing and thresholdoutsourcing policy that we numerically showto be extremely robust over the entire parame-ter space. This robustness may come as no sur-prise for readers who are familiar with relatedliterature such as Borst et al. (2004) and Gur-vich et al. (2014). However, the existing litera-ture has not addressed the issue of robustnessin the context of random arrival rates anddynamic control, nor can this robustness bereadily explained using existing results.

• The technical contribution is the proof that ourproposed square-root safety staffing andthreshold outsourcing policy is asymptoticallyoptimal, as the mean arrival rate becomeslarge, when the level of uncertainty in the arri-val rate is of the same order as the inherentsystem stochasticity (which is of the order ofthe square-root of the mean of the arrival rate).

The remainder of this paper is organized as follows.First, we review the most relevant literature. Next, insection 2, we describe our model in detail. In section3, we present the exact (non-asymptotic) analysiswhich leads to an algorithm to compute the optimalpolicy numerically. However, that algorithm does notprovide insight into the structure of an optimal policy,and so, in section 4, we perform an asymptotic analy-sis under the assumption that the level of uncertaintyin the arrival rate is of the same order as the inherentsystem stochasticity. That asymptotic analysis moti-vates us to propose, in section 5, a square-root safety

Koc�a�ga, Armony, and Ward: Staffing and Co-sourcing for Call Centers2 Production and Operations Management 0(0), pp. 1–17, © 2014 Production and Operations Management Society



staffing and threshold outsourcing policy that is uni-versal in the sense that there is no assumption on thelevel of uncertainty in the arrival rate. We evaluatethe performance of our universal policy numericallyin section 6. We make concluding remarks in section7. All proofs and additional numerical results can befound in the electronic companion (EC).

1.1. Literature ReviewPrevious work on joint staffing and routing problemsin call centers includes Gurvich et al. (2008) whostudy staffing and dynamic routing in call centerswith multiple customer classes and a single serverpool, Armony and Mandelbaum (2011) who considerthe symmetric case of a single customer type and aheterogeneous server pool, and Gurvich and Whitt(2010) who consider multiple customer classes and aheterogeneous server pool. These papers study thestaffing and dynamic routing problems within theHalfin–Whitt heavy traffic regime, pioneered byHalfin and Whitt (1981), and extended to includeabandonments by Garnett et al. (2002). This is alsoknown as the quality and efficiency driven (QED)heavy traffic regime. The key idea is to approximatethe behavior of call centers that are modeled as multi-server queues with that of their limiting diffusions.The limiting diffusion arises from a specific relation-ship between the arrival rate and the staffing level asboth grow large without bound. Our work is differentfrom the aforementioned papers in that our model ispertinent to situations where the arrival rate is notknown when staffing decisions are made, and thushas to be inferred or forecasted using availablehistorical data.Given a staffing level and a realized arrival rate,

our dynamic outsourcing decision is equivalent to theadmission control problem studied in Koc�a�ga andWard (2010). The equivalence follows because we donot explicitly model the vendor firm and assume ithas ample capacity to handle the outsourced calls.Koc�a�ga and Ward (2010) show that a thresholdadmission control policy is optimal, and characterizea simple form for the threshold level that is asymptot-ically optimal when the staffing level is assumed to besuch that the system operates in the QED regime. Incontrast, this study explicitly models the staffing deci-sions and has a random arrival rate.There is a growing body of literature that studies

staffing for call centers with uncertain arrival ratesincluding (in chronological order) Chen and Hender-son (2001), Jongbloed and Koole (2001), Ross (2001),Bassamboo et al. (2005), Whitt (2006), Bassambooet al. (2006), Steckley et al. (2009), Maman (2009),Bassamboo and Zeevi (2009), Gurvich et al. (2010),Robbins and Harrison (2010), Bassamboo et al. (2010),Mehrotra et al. (2010), Gans et al. (2012), and Zan

et al. (2014). The two works most closely related toours are Maman (2009) and Bassamboo et al. (2010),and we discuss each in turn.The focus of Maman (2009) is to extend the QED

staffing formula under a general form of arrival uncer-tainty. Our asymptotic optimality result assumes aspecial case of the form of the arrival rate uncertaintypresented in that paper. However, that paper does notexplicitly study the cost minimizing staffing and doesnot model routing decisions, as we do.Bassamboo et al. (2010) propose a staffing policy

for an M/M/N + M queue in which the arrival rate israndom, and there is no outsourcing option. Theyestablish that a simple newsvendor based staffing pol-icy performs extremely well when the order of uncer-tainty in the arrival rate exceeds the order of theinherent system stochasticity. In contrast, we establishthe asymptotic optimality of our proposed policywhen the aforementioned two magnitudes are thesame. In our numeric study, we adapt their policy toour setting with outsourcing, in order to evaluatepolicy robustness.In relation to the literature on call center outsourc-

ing (see, e.g., Zhou and Ren 2011), our paper is mostsimilar to Aks�in et al. (2008). In contrast to mostpapers in this literature, which assumes all calls willbe outsourced, Aks�in et al. (2008) considers the con-tract design problem of a company which faces anuncertain call volume, and can outsource part of itscalls by choosing between a capacity-based and vol-ume-based contract that is pay-per-call. Although bothAks�in et al. (2008) and our model study co-sourcingdecisions which are driven by call volume uncertainty,Aks�in et al. (2008) focuses on the optimal contractchoice, whereas we focus on the in-house staffing anddynamic routing decisions and assume the contract.

2. Model Description

We model the in-house call center (which we hence-forth refer to as “the call center” or “the system”) asan M/M/N + M queueing system in which the arri-val rate is uncertain. We let Λ denote the random arri-val rate with known cdf FK, and we let l denote aparticular realization of Λ. We assume that Λ is a non-negative random variable with mean E[Λ] = k. Forease of exposition, we assume the mean of the expo-nential service time is 1, so that we can think of mea-suring time in terms of the mean time to serve anarrival. Each customer call put on hold has an expo-nential patience time with mean 1/c and abandonsthe system if not answered within this time. Weassume that Λ is independent of the service andpatience times, and that, given a realization l of Λ, theinter-arrival, service, and patience times are indepen-dent and identically distributed.

Koc�a�ga, Armony, and Ward: Staffing and Co-sourcing for Call CentersProduction and Operations Management 0(0), pp. 1–17, © 2014 Production and Operations Management Society 3



The call center manager must make two decisions:the upfront staffing level N, and the dynamic out-sourcing decision. The staffing level N :¼ NðFKÞmust be set before the arrival rate Λ is realized, basedon the knowledge of its distribution. After the arrivalrate Λ is realized as l, every arriving call can be eitheraccepted into the system, or routed to the outsourcingvendor. The routing control policy p := p(N,l) is ingeneral a function of the staffing level N and l. Thenotation p(N,Λ) refers to a routing policy that maydepend on the actual realization l of Λ. Any stationaryrouting control policy p ¼ ðpn : n 2 f0; 1; . . .gÞ is avector, where pn 2 ½0; 1� denotes the probability that acustomer is accepted into the system when there are ncustomers currently present there. We let Π be the setof all such vectors. An admissible policy

u :¼ ðN; pðN;KÞÞ ¼ ðN; ðpnðN;KÞ : n 2 f0; 1; . . .gÞÞsets the staffing level as a non-negative integer N,and, after the arrival rate Λ realizes as l, controlsoutsourcing decisions dynamically by routing callsaccording to the policy p(N,l) 2 Π.After the arrival rate l realizes, the system operates

as a birth and death process with birth rate lpn anddeath rates

ln ¼ minðN; nÞ þ c½n�N�þ:It is straightforward to solve the balance equationsfor the steady-state distribution for the number-in-system process, from which it follows that the fol-lowing performance measures are well defined:

Ppðab;N; lÞ ¼ the probability an entering customer

abandons;

Ppðout;N; lÞ ¼ the probability an arriving customer

is routed to the outsourcer:

The objective of the system manager is to minimizethe expected long-run average cost, when there arecosts due to customer abandonment, routing custom-ers to the outsourcing vendor, and staffing costs.Every customer that abandons the system beforereceiving service costs a and the per call cost of rout-ing to the outside vendor is p (which can also includeindirect costs such as the hidden costs of outsourc-ing). The long-run average operating cost associatedwith p 2 Π when the arrival rate realizes as l and thestaffing level is N is

zp :¼ zpðN; lÞ ¼ plPpðout;N; lÞ þ alPpðab;N; lÞ: ð1ÞThis is expressed as a random variable by replacingthe realized arrival rate l in Equation (1) with therandom arrival rate Λ. For a given realization l of Λ,

zoptðN; lÞ denotes the minimum cost, andpoptðN; lÞ 2 P is a policy that achieves that minimumcost. We let zoptðN;KÞ denote the random variableassociated with the minimum cost and let poptðN;KÞdenote an optimal routing policy that may dependon the actual realization l of Λ. The expected long-run average cost under the policy u = (N,p) = (N,p(N,Λ)), with respect to the random arrival rate Λ, is

CðuÞ ¼ cN þ E½zpðN;KÞ�: ð2Þ

We would like to find a staffing level Nopt and arouting control policy poptðN;KÞ that achieves theminimum long-run average cost

Copt :¼ infuCðuÞ ¼ min

N2f0;1;2;...gcN þ E zoptðN;KÞ� �

: ð3Þ

REMARK 1. (INCLUDING WAITING COSTS). The objectivefunction in Equation (2) can be modified to includea customer waiting cost at the in-house call centerby modifying Equation (1) as follows. Suppose thecost for one customer to wait one time unit is w ≥ 0.Then, Equation (1) becomes

zp :¼ zpðN; lÞ¼ plPpðout;N; lÞ þ alPpðab;N; lÞþ wlð1� Ppðout;N; lÞÞWpðlÞ;

where WpðN; lÞ is the steady-state average waitingtime, including both abandoning and served cus-tomers. Letting QpðN; lÞ denote the steady-stateaverage number of customers waiting in queue tobe served, it follows from Little’s law that

lð1� Ppðout;N; lÞÞWpðN; lÞ ¼ QpðN; lÞ;and so

zp ¼ plPpðout;N; lÞ þ alPpðab;N; lÞ þ wQpðN; lÞ:Also, since the steady-state rate at which abandon-ing customers arrive must equal the steady-stateabandonment rate

lPpðab;N; lÞ ¼ cQpðN; lÞ;and so

zp ¼ plPpðout;N; lÞ þ aþ w

c

� �lPpðab;N; lÞ:

The analysis in this paper is valid with a replacedby a0 :¼ a þ w=c. Therefore, to include a customerwaiting cost, the only change is to replace a in Equa-tion (1) by a0.




It makes intuitive sense that if c ≥ p the systemmanager will not invest in any in-house capacitybecause serving an arrival is more costly than routingthat arrival to the outsourcing vendor (recall thatl = 1, so c, p and a are comparable). Continuing withsuch intuitive comparisons (see Table 1), if c ≥ min(a,p), then the system manager will either route everycall to the outsourcer (a > p) or will let every callabandon (p ≥ a). This suggests that it is only whenc < min(a,p) that the system manager will invest in in-house capacity. Furthermore, he will not route calls tothe outsourcer if a ≤ p. In summary, we expect thesystem manager to invest in capacity and route somecalls to the outsourcing vendor only if c < p < a. Thefollowing proposition confirms this observation.

PROPOSITION 1. Characterizing the parameter regimes

(i) Suppose that c ≥ min(a,p). Then, Nopt ¼ 0 solvesEquation (3).

(a) In addition, if a > p, then poptðN; lÞ ¼ð0; 0; 0; . . .Þ, so that all calls are routed to theoutsourcing vendor.

(b) Otherwise, if a ≤ p, then poptðN; lÞ ¼ð1; 1; 1; . . .Þ, so that all calls are left to abandon.

(ii) If c < min(a,p) and a ≤ p, then the optimal controlpolicy is poptðN; lÞ ¼ ð1; 1; 1; . . .Þ for any givenN ≥ 0.

From Proposition 1, is follows that the only caseswith a non-trivial optimal staffing are when c <min(a,p). Hence, for the remainder of the paper weassume that c < min (a,p).

3. Exact Analysis

For a fixed staffing level N and realized arrival rate l,the problem of minimizing zp is a Markov decisionproblem (MDP), having solution zoptðN; lÞ. This MDPhas been solved in Koc�a�ga and Ward (2010) in thecontext of an admission control problem. It followsfrom Theorems 3.1, 3.2, and 3.3 in Koc�a�ga and Ward(2010) that the optimal policy is a deterministicthreshold policy (with a potentially infinite thresholdlevel). Hence, we can restrict ourselves to the class ofthreshold control policies

sðTÞ ¼ ðsnðTÞ : n 2 f0; 1; 2; . . .gÞ 2 P;

defined for threshold level T:=T(N,l) 2 [0,∞] as

snðTÞ :¼ 1 if n\T;0 if n�T:

�

Under the threshold policy s(T), after the arrival raterealizes, an arriving call will be accepted into the sys-tem if and only if the number of customers currentlyin the system is less than the threshold. Hence, the sys-tem operates as an M/M/N/T + M queue and theprocess tracking the number of customers in the sys-tem is a birth-and-death process on {0,1,. . .,N � 1,N,N + 1,. . .,T}with birth rate l and death rate in state n

ln ¼ minðn;NÞ þ c½n�N�þ:Then, we can solve the balance equations to find thesteady-state probabilities

hkðN; lÞ ¼Yki¼1

l

li

!h0ðN; lÞ

for

h0ðN; lÞ ¼ 1PTk¼0

Qki¼1

lli

� � ;

and develop the expressions for the performancemeasures1

PsðTÞðout;N; lÞ ¼hTðN; lÞ

QsðTÞðN; lÞ ¼XTk¼0½k�N�þhkðN; lÞ:

Next, recalling from Remark 1 that lPpðab;N; lÞ ¼cQpðN; lÞ, we can express the long-run average costin terms of the steady-state probabilities as

zsðTÞ ¼ plhTðN; lÞ þ acQsðTÞðN; lÞ:

Hence, we can optimize over T to find

Topt :¼ argminT2f0;1;2;...g

zsðTÞ;

for which

zsðToptÞðN; lÞ ¼ zoptðN; lÞ:

Unfortunately, the resulting expression for zsðTÞ isnot simple, and so the above minimization over T tofind Topt must be performed numerically. Further-more, it still remains to take expectations, and mini-mize over the staffing level to numerically solve for

Nopt ¼ argminN2f0;1;2;...g

cN þ E zsðToptÞðN;KÞ� �;

Table 1 Optimal In-House Staffing and Outsourcing Decisions

In-house capacity investment

c < min(a,p) c ≥ min(a,p)

Outsourcing leverage a > p Co-sourcing Complete outsourcinga ≤ p No outsourcing No operation




and the associated minimum cost Copt. To do this,we must perform an exhaustive search over N. Thereason an exhaustive search should be performed isthat it is very difficult to establish, in general, thatthe cost in Equation (2) is convex in the staffinglevel N. In fact, even for a system where the arrivalrate is known and there is no outsourcing option,convexity results are yet to be established whenl < c (Armony et al. 2009, Koole and Pot 2011).The exhaustive search to find Nopt involves a

numeric integration to calculate the expectation withrespect to the arrival rate Λ, and, for each value l usedin the numeric integration, there is another search thatmust be performed to find zoptðN; lÞ. In other words,the exhaustive search algorithm is not a simple linesearch as it includes three nested layers of enumera-tion that correspond to the staffing level, the arrivalrate used in numeric integration and the outsourcingthreshold. The exhaustive search algorithm is for-mally described as follows:

Initialization: Set N0 ¼ 0, C0 ¼ Cðð0; poptð0;KÞÞÞ ¼minða; pÞ�, and N = 1. Decide on themaximum possible staffing level toallow, Nmax.

Step 1: Compute CððN; poptðN;KÞÞÞ ¼ cN þ E zopt½N;Kð Þ� via numeric integration. For eachpossible arrival rate realization l in thenumeric integration, initialize2T = N, decideon the stopping criterion,3 and then com-pute Topt ¼ ToptðN; lÞ and zopt ¼ zoptðN; lÞ asfollows.

(A) Solve for zsðTÞ from the steady-state probabil-ities fhnðTÞ : n ¼ 0; 1; 2; . . .;Tg.

(B) If zsðTþ1Þ � zsðTÞ,4 or if the stopping criterion

holds, set Topt ¼ T, zsðTÞ ¼ zopt and stop.Otherwise, increase T by 1 and go to step (A).

Step 2: If CððN; poptðN;KÞÞÞ\ C0, then N0 N andC0 CððN; poptðN;KÞÞÞ

Step 3: If N ¼ Nmax, then set Nopt ¼ N0 andCopt ¼ C0 and stop. Otherwise, increase N by1 and go to Step 1.

Although the algorithm above can compute theoptimal policy numerically, it does not provide anyinsight with regards to the structure of the optimalpolicy. Furthermore, the computation time to obtainan optimal policy can be several hours for large sys-tem sizes. (We implemented the algorithm in Matlab.)Therefore, we take the following approach to developour proposed policy: we evaluate the performance ofthe family of square-root safety staffing policies com-bined with threshold routing. To do this, we firstassume that the form of the arrival rate uncertainty ison the order of the square-root of the mean arrival

rate, and then show that square-root safety staffingcombined with threshold routing is asymptoticallyoptimal (section 4). Second, we propose a universalpolicy U that is based on that asymptotic optimalityresult (section 5), and show numerically that not onlyits computation time is in the order of seconds, it alsohas a very good performance even outside of theregime in which we proved its asymptotic optimality(section 6).

4. Asymptotic Analysis

In this section of the paper only, we assume that theorder of uncertainty in the arrival rate is the same asthe square-root of the mean of the arrival rate. To dothis, we consider a sequence of systems indexed bythe mean arrival rate k, and let k?∞. We assume thatthe random arrival rate Λ throughout this sequence ofsystems can be expressed as

K ¼ � þ Xffiffiffi�p

; ð4Þwhere X is a random variable with mean zero andhas E|X| < ∞. For this section only, the expectationoperator is with respect to X (instead of Λ). We notethat this form for the arrival rate is a special case ofthe model assumed in Maman (2009). Our conventionis to use the superscript k to denote a process or quan-tity associated with the system having random arrivalrate K ¼ K�ðXÞ given in Equation (4). The notation

l�ðxÞ ¼ �þ xffiffiffi�p

denotes the realized arrival rate in the system hav-ing mean arrival rate k; the k superscript shouldremind the reader that l�ðxÞ ! 1 as k?∞ for anyx 2 (�∞,∞).

An admissible policy u ¼ ðN;pÞ :¼ fðN�; p�Þ :� � 0g refers to an entire sequence that specifies anadmissible policy for each k. In particular, N� is a non-negative integer and p� ¼ p�ðN�; l�ðxÞÞ 2 P for each kand any realization x of X (so that the system arrivalrate is l�ðxÞ). The notation z�pðN; l�ðxÞÞ is the long-runaverage operating cost, as defined in Equation (1), forthe system with realized arrival rate l�ðxÞ, andz�pðN;K�ðXÞÞ is the associated random variable. Simi-larly, P�

pðout;N; l�ðxÞÞ (P�pðab;N; l�ðxÞÞ) is the steady-

state probability an arriving customer is routed to theoutsourcer (probability of abandonment) when therealized arrival rate is l�ðxÞ, and P�

pðout;N;K�ðXÞÞ(P�

pðab;N;K�ðXÞÞ) is the associated random variable.In this section, we first define what we mean by

asymptotic optimality (section 4.1). Then, we performan asymptotic analysis in order to understand thebehavior of the family of square-root staffing policiescombined with threshold routing (section 4.2).Finally, we optimize over the aforementioned policy




class to obtain our proposed policy (section 4.3), andwe establish its asymptotic optimality.

4.1. The Asymptotic Optimality DefinitionOur definition of asymptotic optimality is motivatedby first observing that the lowest achievable cost onfluid scale is ck + o(k), where the notation f� ¼ oðg�Þmeans that lim�!1 f�=g� ¼ 0.

PROPOSITION 2. Fluid-scaled cost. Under the assumption(4) we have that:

(i) Any admissible policy u = (N,p) has

lim inf�!1

cN� þ E z�pðN;K�ðXÞÞ� ��

� c:

(ii) If N� ¼ � þ bffiffiffi�p þ oð�Þ, then, under the routing

policy s(∞) that outsources no customers,

lim�!1

cN� þ E z�sð1ÞðNK�ðXÞÞ

h i�

¼ c:

The following refined and diffusion scaled cost func-tion (defined for the any admissible policy u = (N,p))

C�ðuÞ :¼ffiffiffi�p cN� þ E z�pðNK�ðXÞÞ� �

�� c

!� 0 ð5Þ

captures both the cost of additional staffing (abovethe offered load level k) and the cost of the routingcontrol.

DEFINITION 1. Asymptotic optimality. An admissiblepolicy uH ¼ ðNH; pHÞ ¼ fðN�;H; p�;HðN�;H;K�ðXÞÞÞ :� � 0g is asymptotically optimal if

lim sup�!1

C�ðuÞ\1

and

lim sup�!1

C�ðuHÞ� lim inf�!1

C�ðuÞ;

for any admissible policy u.

4.2. The Asymptotic Behavior of Square-RootSafety Staffing Combined with Threshold RoutingIt has been shown in the extensive literature onstaffing in large-scale service systems (e.g., Borstet al. 2004, Halfin and Whitt 1981, Mandelbaumand Zeltyn 2009) that when the arrival rate isdeterministic, square-root safety staffing performsextremely well in minimizing both the staffing plusdelay costs as well as staffing costs subject to

performance constraints. When the arrival rate k islarge, under square-root safety staffing, the waitingtimes are small (at the order of 1=

ffiffiffi�p

), so that thepercentage of customers that should be routed tothe outsourcer (Koc�a�ga and Ward 2010), as well asthe percentage of customers that abandon (Garnettet al. 2002) are both small. This suggests thatsquare-root safety staffing should be also relevantwhen the arrival rate is random. Similarly toKoc�a�ga and Ward (2010), to route calls, we use athreshold routing policy, s ¼ fsðT�Þ : � � 0g, asdefined in section 3. The threshold level T� ¼T�ðN�;K�ðXÞÞ is determined after the arrival raterealizes as l�ðxÞ.The following lemma establishes the asymptotic

behavior of square-root safety staffing combined withthreshold routing for a fixed realization x of X. Let /and Φ be the standard normal pdf and cdf, respectively.

LEMMA 1. Asymptotic behavior with deterministic arri-val rate.

Suppose the random variable X realizes as the valuex 2 (�∞,∞). Assume the policy u = (N,s) is such that

N� ¼ �þ bffiffiffi�pþ oð

ffiffiffi�pÞ ð6Þ

T� ¼ N� þ Tffiffiffiffiffiffiffiffiffiffil�ðxÞ

q; where T :¼ Tðb; xÞ 2 ½0;1Þ:

ð7Þ

Suppose the initial number of customers in the system Y�0

is such thatY�0�N�ffiffi�p ) Yð0Þ as k?∞, for some random

variable Yð0Þ that is finite with probability 1. Then,

1ffiffiffi�p z�s ðN; l�ðxÞÞ ! zðb� x; TÞ; as �!1;

where

zðm; TÞ :¼ Aðm; TÞBðm; TÞ ð8Þ

for

Aðm; TÞ :¼p/ ffiffifficp

T þm

c

� �� þ aþ w

c

� �

� /m

c

� �� /

ffiffifficp

T þm

c

� ��

þ mffiffifficp U

mffiffifficp� �

� Uffiffifficp

T þm

c

� ��

Bðm; TÞ :¼/ mffiffi

cp �/ðmÞ UðmÞ

þ 1ffiffifficp U

ffiffifficp

T þm

c

� �� U

mffiffifficp� ��




The appearance of b � x as an argument in z inLemma 1 occurs because under Equation (4) the staff-ing N� in Equation (6) is such that the system operatesin the QED regime regardless of the realization x of X;in particular,

N� � l�ðxÞffiffiffi�p ¼ N� � �ffiffiffi

�p � x! b� x as �!1:

Furthermore, the following Corollary to Lemma 1highlights that the dependence of the thresholdlevel on the realized arrival rate is through thedefinition of T, and not through its multiplier(which is always of order

ffiffiffi�p

under the assump-tion (4)).

COROLLARY 1. Lemma 1 continues to hold when T� inEquation (7) is re-defined as

T� ¼ N� þ Tffiffiffi�p

:

The issue is that in order to analyze the perfor-mance of square-root safety staffing combined withthreshold routing, we require that Lemma 1 and Cor-ollary 1 hold when the fixed value x is replaced by therandom variable X.

THEOREM 1. Asymptotic cost convergence. Assume thepolicy u = (N,s) is as defined by the Equations (6) and(7). Then, under the conditions of Lemma 1,

C�ðuÞ ! CðuÞ :¼ cbþ E zðb� X; TÞh i

; as �!1:

4.3. The Proposed PolicyIt is sensible to set the parameters b and T ofLemma 1 in order to minimize the limiting costCðuÞ of Theorem 1. The first step is to observethat, for p < a and any given b, Proposition 4.1 inKoc�a�ga and Ward (2010) shows that for therealized arrival rate l�ðxÞ, the unique TH ¼ TH

ðb � xÞ\1 that solves

ða� pÞcT � zðb� x; TÞ ¼ pðb� xÞ ð9Þ

has the property that

zðb� x; THÞ� zðb� x; TÞ ð10Þ

for any other T � 0. Otherwise, for a ≤ p, TH ¼ 1and

zðb� x;1Þ :¼ limT!1

zðb� x; TÞ� zðb� x; T0Þ ð11Þ

for any finite T0 � 0. The second step is to plug TH

into the limiting expression in Theorem 1, and tooptimize over b to find

bH :¼ arg minb

cbþ E zðb� X; THðb� XÞÞh in o

: ð12Þ

It is important to observe that bH is well defined inthe sense that bH is finite and

cbH þ E zðbH � X; THðbH � XÞÞh i

¼ infb2ð�1;1Þ

cbþ E zðb� X; THðb� XÞÞh i

\1:

This follows from the next two propositions.

PROPOSITION 3. For any b 2 ( � ∞,∞), cbþE zðb� X; THðb � XÞÞh i

\1.

PROPOSITION 4. The infimum in infb2ð�1;1Þ

cb þ E zðb�½X; THðb � XÞÞ� is attained by a finite b 2 (�∞,∞).

We are now in a position to define our proposedpolicy: We let

uH ¼ ðNH; sHÞ :¼ fðN�;H; sðT�;HÞÞ : �� 0g ð13Þthat has the staffing level

N�;H ¼ �þ bHffiffiffi�p

ð14Þand sets the threshold level T�;H ¼ T�;HðN�;H;K�ðXÞÞ when the arrival rate realizes as l�ðxÞ as

T�;H ¼ N�;H þ THðbH � xÞ �ffiffiffiffiffiffiffiffiffiffil�ðxÞ

q; ð15Þ

for THðbH � xÞ defined by Equation (9) with bH

replacing b.Theorem 1 is valid for uH, and so

C�ðuHÞ ! CH :¼ cbH þ E zðbH � X; THðbH � XÞÞh i

Our next result confirms that CH is the minimumachievable cost, meaning that the policy uH isasymptotically optimal.

THEOREM 2. Asymptotic optimality of our proposed pol-icy. The policy uH, defined through Equations (12), (13),(14), and (15) is asymptotically optimal under Equation(14); that is, under any other admissible policy u

lim inf�!1

C�ðuÞ� CH:




Furthermore, it follows that our proposed policy has asso-ciated cost that is oð ffiffiffi�p Þ higher than the minimumachievable cost for a given k; that is, that

cN�;H þ E z�sH

NHK�ðXÞ� � �� C�;optffiffiffi�p ! 0; as �!1;

where C�;opt :¼ Copt for Copt defined in Equation (3) forthe system with mean arrival rate k.

REMARK 2. Performance under the optimal thresh-old. Another asymptotically optimal policy

ðNH; poptÞ :¼ fðN�;H; p�;optÞ : �� 0g

has staffing levels N�;H defined in Equation (14),and, after the arrival rate K�ðXÞ realizes as l�ðxÞ,solves the relevant MDP for the routing control pol-icy p�;opt ¼ p�;optðN�;H; l�ðxÞÞ that achieves the mini-mum long-run average operating cost z�;optðNH;l�ðxÞÞ. To see this, it is enough to observe that

z�sHðNH; l�ðxÞÞ� z�;optðNH; l�ðxÞÞ;

for every k and any realization x of X.In the following, f� ¼ Oðg�Þ means that

lim sup�!1 jf�=g�j\1.

REMARK 3. Comparison to Bassamboo et al. 2010.When a < p (in addition to our assumption thatc < min(a,p)), it follows from Proposition 1 part (ii)that the optimal control policy does not outsourceany calls. Then, the cost minimization problem (2) isa pure staffing problem (instead of a joint staffingand routing problem), which is equivalent to theproblem solved in Bassamboo et al. (2010). Theorem1 part (c) of that paper, adapted to our setting,shows that a policy based on a newsvendor pre-scription can have associated cost that is Oð ffiffiffi�p Þhigher than the minimum achievable cost for agiven k. In comparison, our proposed policy hasassociated cost that is oð ffiffiffi�p Þ higher than the mini-mum achievable cost for a given k by Theorem 2.Hence, we expect our policy to provide significantimprovements over that of Bassamboo et al. (2010),as the arrival rate uncertainty decreases.

5. The Proposed Universal Policy

For models that do not assume uncertain arrivalrates, square-root safety staffing is known in theliterature to be very robust. For an M/M/N queuewith no abandonments, no dynamic routing deci-sions, and known arrival rate, Borst et al. (2004)show that square-root safety staffing performs

extremely well, both inside and outside of the param-eter regime (linear staffing and waiting costs) inwhich they prove it to be asymptotically optimal(see their numerical experiments in section 10). In amore recent paper, Gurvich et al. (2014) prove thatperformance approximations that are based on thepremise that the staffing is of a square-root safetyform are asymptotically universally accurate, as thearrival rate becomes large. This latter paper is alsolimited to the case of deterministic arrival rates andno dynamic control.This leads us to propose the universal policy U

when there are no restrictions on the form of the arrivalrate uncertainty, as in Equation (4). We define U forthe model as specified in section 2, and analyzedexactly in section 3, without considering a sequenceof systems as in section 4. To do this, we begin withthe non-negative random variable Λ that representsthe system arrival rate and has mean E[Λ] = k. Then,we make the transformation

X :¼ K� �ffiffiffi�p ; ð16Þ

and use the random variable X to define U

U ¼ ðNU; pUðNU;KÞÞ:The proposed staffing level is

NU ¼ �þ bHffiffiffi�ph i

;

for bH that satisfies Equation (12), with X in thatexpression defined by Equation (16), and the func-tion [�] rounds the expression inside the brackets tothe nearest integer. The proposed routing policywhen the arrival rate Λ realizes as l is the thresholdrouting policy

pUðN; lÞ ¼ sðTUÞfor

TU ¼ NU þ THffiffilp

;

and TH defined by Equation (9), with x in thatexpression replaced by ðl � �Þ= ffiffiffi

�p

.The universal U policy “pretends” that the magni-

tude of the uncertainty in the arrival rate Λ is on theorder of

ffiffiffi�p

, as in Equation (4), and sets bH and TH

accordingly. In contrast to the policy defined in sec-tion 4.3 under assumption (4), the magnitude of thesecond order term appearing in the definitions of NU

and TU may not be of orderffiffiffi�p

. In particular, depend-ing on the distribution of Λ, the value of bH may endup being of the same order of

ffiffiffi�p

, so that the secondterm in NU is of order k (see discussion in section 6.5).This flexibility suggests that U may perform well,




even outside of the regime in which it is proved to beasymptotically optimal, as we indeed observe in thenext section.

6. Numerical Evaluation of theProposed Policy

Theorems 1 and 2 establish when the order of uncer-tainty in the arrival rate is the same as the square-rootof the mean arrival rate, so that Equation (4) holds, Ustaffs and routes in a way that achieves minimum costfor large enough k. However, Theorems 1 and 2 donot provide guidance on: how large k must be, whathappens when Equation (4) does not hold, or how Uperforms in comparison to alternative benchmark pol-icies. In this section, we show that U generallyachieves within 0.1% of the minimum cost even whenthese assumptions are relaxed, and its robustness incomparison to two benchmark policies is the highest.To do this, we first vary the system size expressed bythe mean arrival rate k (section 6.1) and the level ofuncertainty in Λ (section 6.2) to gain an initial conclu-sion that U performs remarkably well. Then, we showthat this conclusion is, to a large degree, insensitive tochanges in the cost parameters (section 6.4) and theasymmetry of the arrival rate distribution (section6.4). In summary, U performs extremely well, evenwhen the system is far away from the regime in whichit is proved to be asymptotically optimal.Throughout our numerical examples, we set the

mean service time and the mean patience time equalto 1 and fix the cost parameters at c = 0.1, p = 1 anda = 5 unless specified otherwise. It follows from Prop-osition 1 that our choice of cost parameters is suchthat it is optimal for the system manager to set anonzero staffing level and route some calls to theoutsourcer.

6.1. Finite System SizeHaving established that U is asymptotically optimalas the system size grows without bound, under theform of uncertainty in Λ as in Equation (4), weproceed to evaluate its performance for finite size

systems. This evaluation is done by comparing U tothe numerically computed optimal staffing policy. Wecompute the optimal staffing level Nopt via an exhaus-tive search, as described in section 3.Table 2 illustrates the performance of our proposed

staffing policy with respect to the optimal staffinglevel by varying the system size and letting the distri-bution of the arrival rate Λ be in accordance withEquation (4). Specifically, we assume that X follows aUniform distribution on [�1,1], and increase the meanarrival rate k from 1 to 1600. This implies that Λ fol-lows a Uniform distribution with its support intervalincreasing from [0,2] to [1560,1640]. This is consistentwith our assumption in Theorems 1 and 2 that proveasymptotic optimality of U as k becomes large underEquation (4). The first and second columns in Table 2show the resulting distribution for Λ. The third andfourth columns in Table 2 show the optimal staffinglevel and the associated optimal average cost, whilecolumns five and six show our proposed approximatestaffing policy, along with its average cost. Column 7displays the staffing error which is the differencebetween the optimal staffing level and our approxi-mate staffing level. Finally, column eight displays thepercentage cost error with respect to the optimalpolicy.We see from Table 2 that U performs extremely

well for all system sizes, that are consistent with theassumption that the uncertainty in the arrival rate isof the same order as the square-root of the mean arri-val rate. Notice that the percentage cost error may benonzero even when the staffing error equals zero.This is because U sets the threshold level according toEquation (15) which may not equal to the optimalthreshold. In light of Theorems 1 and 2, that establishasymptotic optimality, it is not surprising that ourpolicy performs extremely well for large k. The lessexpected numerical insight is that U also performsextremely well for small k. (We note that there is achance that the rounding can go the wrong way,and subsequently may cause a large cost error inextremely small system size. However, such smallsystems sizes are not realistic for most call center

Table 2 Performance of U: Increasing System Size

k Distribution of Λ

Optimal policy U Difference

Nopt Copt NU CðNU Þ Nopt � NUCðNU Þ�Copt

Copt

1 U[0,2] 3 0.4149 3 0.4188 0 0.9400%9 U[6,12] 16 1.7702 15 1.7786 1 0.4745%25 U[20,30] 36 3.8979 36 3.8998 0 0.0487%100 U[90,110] 121 12.7131 121 12.7149 0 0.0142%226 U[210,240] 257 26.5227 257 26.5236 0 0.0034%400 U[380,420] 443 45.3338 442 45.3355 1 0.0037%625 U[600,650] 678 69.1435 678 69.1441 0 0.0009%900 U[870,930] 964 97.9536 963 97.9553 1 0.0017%1600 U[1560,1640] 1685 170.5732 1684 170.5750 1 0.0011%




applications). In summary, U is very robust to systemsize, provided the order of uncertainty in the arrivalrate is as assumed in Equation (4).

6.2. Varying Arrival Rate UncertaintyNext, we evaluate the robustness of U with respect tochanges in the level of uncertainty in the arrival rate.This is important because the proof of asymptoticoptimality of U requires the assumption that the levelof uncertainty in the arrival rate is of the same orderas the square-root of the mean arrival rate (i.e., thatEquation (4) holds). We measure the level of uncer-tainty in the arrival rate through its coefficient of vari-

ation CV :¼ CVK ¼ffiffiffiffiffiffiffiffiffiffiVar½K�p

E½K� . We are interested in bothcases where the level of uncertainty in the arrival rateis lower than that assumed in Equation (4) and whereit is higher.In this subsection, we keep the mean arrival rate

fixed at k = 100, and we assume that Λ follows a Uni-form distribution with support [a,b], so that

CV ¼ 1ffiffiffi3p b� a

aþ b� 1ffiffiffi

3p ¼ 0:5774:

In comparison, under assumption (4), when X fol-lows a uniform distribution with support [�1,1] asin section 6.1, the coefficient of variation of the arri-val rate Λ = Λ(X) is

CVKðXÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar X½ �pffiffiffi�p ¼ 1

10ffiffiffi3p ¼ 0:0577: ð17Þ

By varying CV from 0 to approximately 1/2, wecover both the cases where the level of uncertaintyin the arrival rate is lower that Equation (4) andhigher.It is sensible to compare the performance of U to

two other possible staffing policies: one that isexpected to perform well when the level of uncer-tainty in the arrival rate is low and the other that isexpected to perform well when the level of uncer-tainty in the arrival rate is high. The first alternativepolicy we consider is D, a square-root safety staffingpolicy that has the same form as U, but chooses thecoefficient of

ffiffiffi�p

differently by assuming that the arri-val rate is deterministic and fixed at the mean arrivalrate k = 100. Specifically, when the mean arrival rateis k, D staffs

ND :¼ �þ bH1ffiffiffi�ph i

;

where

bH1 :¼ arg minb

cbþ zðb; THðbÞÞ

for z as defined in Equation (8) and THðbÞ that satis-fies Equation (9). Note that D is exactly the pro-

posed policy U in the case P(X = 0) = 1. It isintuitive to expect that the performance of D deteri-orates significantly as CV increases.The second alternative policy we consider is NV, a

newsvendor based prescription that is a modificationof the policy proposed in Bassamboo et al. (2010) toinclude co-sourcing. The NV policy follows a fluidapproximation which ignores stochastic queueingeffects and, as a result, when a > p, the abandonmentcost is irrelevant. That is, in the fluid scale, all custom-ers who cannot be served in-house immediately uponarrival will be outsourced. Similarly, when a ≤ p nocalls will be outsourced. In particular, in newsvendorterminology the overage cost is c (because of extrastaffing) and the underage cost is min{a,p} � c(because we incur the cost of routing or cost of aban-donment but do not incur the cost of an additionalperson for staffing). Then, the critical ratio is minfa;pg� c

minfa;pg ,and the newsvendor based staffing prescription is

NNV :¼ F�1Kminfa; pg � c

minfa; pg� ��

:

We observe that when Equation (4) holds, NNV canalso be written as NNV :¼ � þ bH2

ffiffiffi�p� �

, wherebH2 :¼ F�1X ðp� c

p Þ and FX is the cumulative distributionfunction of X. Notice that, in sharp contrast to D,which disregards the uncertainty in the arrival rate,NV disregards the inherent stochasticity of the sys-tem that produces queueing. Hence, we expect theperformance of the NV policy to deteriorate whenCV decreases.We have specified the staffing rules, ND and NNV,

of two alternative policies D and NV. There is still thequestion of what should be the routing policy. Forthis, we recall that after the arrival rate realizes, theoptimal routing policy can be found by solving therelevant MDP (see Koc�a�ga and Ward 2010). Hence, inour numerical experiments, after the arrival rate real-izes, we operate both comparison policies under theoptimal routing policy. The U policy follows thethreshold routing policy sðT�;HÞ where T�;H ¼T�;HðN�;H;KðXÞÞ is as defined in section (although weobserve that the performance of the diffusion basedthreshold routing policy and the exact solution to therelevant MDP are almost indistinguishable).Figure 1 plots the relative percentage cost error and

staffing error of U, NV and D. The staffing error andpercentage cost error for U is defined as in columnsseven and eight in Table 2, and is defined similarlyfor D and NV. Table B.1 in EC contains further detailsregarding this study, such as the exact costs and staff-ing levels. We see from Figure 1 that U staffs veryclose to the optimal staffing level and therefore per-forms well even for very high CV values. We also seethat U outperforms NV for lower CV values and




outperforms D for higher CV values. Furthermore, inboth cases, the staffing and percentage cost error canbe arbitrarily large. Hence, we conclude that U isrobust and performs extremely well even in parame-ter settings beyond which it has been proven to beasymptotically optimal.Figure 1 is a first step in concluding that U is very

robust, and performs extremely well over a largerange of parameter settings much beyond where The-orems 1 and 2 establish its asymptotic optimality. Thenext step in establishing the aforementioned conclu-sion is to explore the effect of varying other parame-ters, for example, the staffing cost.

6.3. Varying Staffing CostsNext, we explore the effect of the staffing cost, whichdetermines the associated critical ratio of the news-vendor policy. To do this, we change the staffing costc while holding the other parameters constant. Weperform three separate studies by fixing the arrivalrate distribution at three separate levels of uncer-tainty; low CV, moderate CV and high CV. In particu-lar, we assume K�U½90; 110� to produce low CV(Figures 2a and 3a), K�U[50,150] to produce

moderate CV (Figures 2b and 3b), and K�U[10,190]to produce high CV (Figures 2c and 3c). We plot thepercentage cost errors in Figure 2 and the staffingerrors in Figure 3 for U, D and NV. We refer thereader to Tables B.2–B.4 in EC for further details(exact costs and staffing levels).We first observe that for all staffing costs and

across all levels of CV, U staffs very close to theoptimal policy and thus performs extremely well.On the other hand, NV performs poorly when thestaffing cost is low although the effect gets less pro-nounced for higher CV values. This is because NVtends to understaff for low staffing costs when theCV is low. As a result, when cp < a, NV incurshigher routing control and abandonment costs. Asthe in-house staffing cost c increases to p = 1, NVtends to overstaff, although the adverse effects ofoverstaffing are not as detrimental. We see that Dperforms very poorly as the CV level increasesbecause it fails to capture the effect of randomness.Overall, we see that U is robust and performs wellacross different critical ratio and CV combinationswhile the alternative policies can perform arbitrarilypoorly.

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

0 0.1 0.2 0.3 0.4 0.5

Coefficient of Variation

% cost error of U vs other policies

NV

D

U

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.1 0.2 0.3 0.4 0.5

Coefficient of Variation

staffing error of U vs other policies

NV

D

U

(a) (b)

Figure 1 Performance of U and Other Policies for Increasing CV

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

0 0.2 0.4 0.6 0.8 1staffing cost (c)

NV

D

U

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

0 0.2 0.4 0.6 0.8 1staffing cost (c)

NV

D

U

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

140.00%

160.00%

0 0.2 0.4 0.6 0.8 1staffing cost (c)

NV

D

U

(a) (b) (c)

Figure 2 % Cost Error for Changing Staffing Costs at Three Levels of Arrival Rate Uncertainty




6.4. Effect of Distribution AsymmetryOur numerical result thus far have assumed symmet-ric Uniform arrival rate distributions. Next, we gener-alize our results by considering arrival ratedistributions that are asymmetric and follow a Betadistribution to study the effect of skewness on the per-formance of U and the other policies. Specifically, weassume thatK�Betaða1; a2; � � b

ffiffiffi�p

; � þ bffiffiffi�p Þ, where

the first two arguments are the scale parameters ofthe distribution and the last two arguments are thelower and upper bounds of the support. We let b andb be arbitrarily large so that the arrival rate may notrealize in the QED regime (i.e., the assumption (4) isnot necessarily satisfied). Our proposed policy U isdefined for X�Betaða1; a2; b; bÞ from Equation (16).We keep the mean arrival rate fixed at E

[Λ] = k = 100 (i.e., E[X] = 0) throughout this sectionand we consider three cases where we keep the vari-ance of the arrival rate fixed at three levels: The lowCV case keeps the variance of Λ fixed and equal tothat of a U½90; 110� random variable (i.e.,Var Xð Þ ¼ Var U �1; 1½ �ð Þ), the moderate CV case keepsthe variance of Λ fixed and equal to that of a U½50; 150�random variable (i.e., Var Xð Þ ¼ Var U �5; 5½ �ð Þ), andthe high CV case keeps the variance of Λ fixed andequal to that of a U½10; 190� random variable (i.e.,Var Xð Þ ¼ Var U �9; 9½ �ð Þ).We study the effect of asymmetry by changing the

skewness of the Beta distribution through its scaleparameters a1 and a2. In particular, we set

E½X� ¼ a1bþ a2ba1 þ a2

¼ 0 and Var Xð Þ ¼ a1a2 b� bð Þ2a1 þ a2ð Þ2 a1 þ a2 þ 1ð Þ ¼

r2, where r2 denotes the variance of the associated CVlevel, and we choose5a1 and a2 such that a1 þ a2 ¼ 2.We start with a negative-skewed (left-skewed) Betadistribution with scale parameters a1 ¼ 1:5 anda2 ¼ 0:5 with a corresponding skewness of �1. Then,we decrease a1 and increase a2 so that the skewness ofthe Beta distribution increases.6 The mid-point wherea1 ¼ a2 ¼ 1 and so the skewness equals 0, corre-sponds to the symmetric Uniform distribution. Aftera1 ¼ a2 ¼ 1, the distribution becomes positive-

skewed (right-skewed) as we decrease a1 and increasea2 and we continue until a1 ¼ 0:5 and a2 ¼ 1:5 whichcorresponds to a skewness of +1. We plot the percent-age cost error and staffing errors of U and the otherpolicies in Figures 4 and 5, respectively. Tables B.6–B.8 in EC provide further details (the exact costs andstaffing levels as well as the shape parameters of theBeta distribution).From Figures 4 and 5, we first see that U continues

to perform well under asymmetric arrival rate distri-butions and across varying levels of skewness. In linewith our observations for the uniform distribution, Ddoes not perform well except for low CV values. Onthe other hand, the newsvendor policy performs wellfor high levels of variability while its performancedeteriorates for lower levels of variability and in par-ticular left-skewed distribution. This is because thenewsvendor staffing is given by NNV :¼ � þ bH2

ffiffiffi�p

for bH2 :¼ F�1X ðp� cp Þ, which decreases as the skewness

decreases. Hence, the newsvendor staffs less asskewness decreases and the distribution gets moreleft-skewed, as seen in Figure 5. Therefore, thenewsvendor performs worse when the distribution isleft-skewed because its understaffing is more severe,resulting in higher abandonment and routing controlcosts.

6.5. DiscussionOur numerical results in sections 6.1–6.4 show that Uis extremely robust, and achieves close to minimumcost over a large range of parameters and assump-tions on the amount of the arrival rate uncertainty,and its distribution form. In fact, in virtually all of ourexperiments U outperformed D and NV and achieveda cost that was very close to the true optimal cost. Tobetter understand the reason for the extremely robustperformance of U, we compare the actual expectedcost to the diffusion approximation of the expectedcost, for a wide range of staffing levels. We keep thestaffing level N fixed, and approximate the cost usingthe expression that appears in the limit in Theorem 1.However, we do not assume that the form of the arri-

-40

-20

0

20

40

60

80

100

0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99

staffing cost (c)

NV

D

U

-60

-40

-20

0

20

40

60

0 0.2 0.4 0.6 0.8 1

staffing cost (c)

NV

D

U-80

-60

-40

-20

0

20

40

60

0 0.2 0.4 0.6 0.8 1

staffing cost (c)

NV

D

U

(a) (b) (c)

Figure 3 Staffing Error (i.e., Nopt � D where D ‰fNU ;ND ;NNV g) for Changing Staffing Costs at Three Levels of Arrival Rate Uncertainty




val rate uncertainty is consistent with Equation (4) (asassumed by Theorem 1). Specifically, for

b ¼ N � �ffiffiffi�p and X ¼ K� �ffiffiffi

�p ;

it follows that the limiting diffusion costcb þ E zðb � X; THðb � XÞÞ

h iwhen re-scaled gives

the following approximation for the actual cost:

cN þ E zopt N;Kð Þ� � c�þ

ffiffiffi�p

cbþ E zðb� X; THðb� XÞÞh i �

¼ cN þffiffiffi�p

E zðb� X; THðb� XÞÞh i

:

ð18Þ

Figure 6 demonstrates numerically that the approx-imation Equation (18) is very accurate, far beyondwhat is proven in Theorem 1. Specifically, Figure 6plots the actual expected cost (the left-hand side ofEquation (18)) and the re-scaled diffusion cost (theright-hand side of Equation (18)), and the differencebetween the two. It is clear that the approximation inEquation (18) is extremely accurate, over a wide rangeof staffing levels. This helps explain the robustness inthe performance of U.Figure 6 suggests that the performance of square-

root safety staffing policies can be approximatedwell without making special assumptions on thelimit regime; that is, there is a “universal” approxi-mation. This is because we do not restrict b valuesto a particular range which can also be evidenced

from the bH values that we observed in our numeri-cal studies in sections 6.1–6.4. In particular, the bH

values in Figures 2–4 come from a wide range from�12 to 10 (see Table B.5 for details). Recalling thatffiffiffi�p ¼ 10 in Figures 2–4, we observe that such extre-mely low or high values of bH that are essentially ofthe same order of magnitude as

ffiffiffi�p

(so that theresulting safety staffing is in fact of order k) allowus to capture heavily overloaded or underloadedsystems, and thus allows us to approximate param-eter regimes beyond what is assumed in our asymp-totic analysis in section 4.Our observation is also consistent with the universal

approximation result of Gurvich et al. (2014) for a M/M/N + M model with deterministic arrival rate and norouting control. Although it is tempting to think thatGurvich et al. (2014) can be used to explain Figure 6,the modeling generalization from a deterministic to arandom arrival rate is not immediate, even if we donot allow for outsourcing. Our goal is to show that

E zsð1ÞðN;KÞ� �� ffiffiffi�p

E zN � Kffiffiffi

�p ;1

� �� ¼ Oð1Þ; ð19Þ

where zðm;1Þ is as defined in Lemma 1, when thelimit as T !1 is taken. Under the policy s(∞) thatoutsources no customers (and so besides staffing onlyincurs costs through customer abandonment), thenzsð1ÞðN; lÞ ¼ acQsð1ÞðlÞ, and so it follows from Corol-lary 1 in Gurvich et al. (2014) and algebraic manipula-tion that

-16

-14

-12

-10

-8

-6

-4

-2

0-1 -0.5 0 0.5 1

Skewness

NV

D

U

-35

-30

-25

-20

-15

-10

-5

0-1 -0.5 0 0.5 1

Skewness

NV

D

U

-70

-60

-50

-40

-30

-20

-10

0-0.2 0 0.2 0.4 0.6 0.8 1

Skewness

NV

D

U

(a) (b) (c)

Figure 5 Staffing Error (i.e., Nopt � D where D ‰fNU ;ND ;NNV g) for Varying Skewness at Three Levels of Arrival Rate Uncertainty

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

20.00%

-1 -0.5 0 0.5 1Skewness

NV

D

U0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

-1 -0.5 0 0.5 1

Skewness

NV

D

U

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

-0.2 0 0.2 0.4 0.6 0.8 1

Skewness

NV

D

U

(a) (b) (c)

Figure 4 % Cost Error for Varying Skewness Levels at Three Levels of Arrival Rate Uncertainty




zsð1ÞðN; lÞ �ffiffilp

zN � lffiffi

lp ;1

� �¼ Oð1Þ: ð20Þ

Although it seems reasonable that a technical argu-ment would enable us to show that Equation (20)implies

E zsð1ÞðN;KÞ� �� EffiffiffiffiKp

zN � Kffiffiffiffi

Kp ;1

� �� ¼ Oð1Þ;

it is not clear how to establish that

ffiffiffi�p

E zN � Kffiffiffi

�p ;1

� �� E

ffiffiffiffiKp

zN � Kffiffiffiffi

Kp ;1

� �� ¼ Oð1Þ;

ð21ÞIntuitively, we expect something like Equation (21)to be true because as the system becomes moreunderstaffed zð�;1Þ starts to look linear and as thesystem becomes more overstaffed zð�;1Þ should benegligible (and when the staffing is neitherseverely understaffed or overstaffed we can appealto Theorem 1). However, the analytic argument isnot straightforward, and remains as an open ques-tion.

7. Conclusions

This study deals with the joint staffing and dynamicoutsourcing decisions for a co-sourced customer callcenter. The key feature in our model is the time scalesdifferentiation, where staffing decisions are madeupfront when the arrival rate is uncertain, while theoutsourcing decisions are made dynamically in realtime with full knowledge of the arrival rate realiza-tion as well as the system state. We propose the policyU, which staffs the call center with the mean offeredload plus a safety staffing that is of the order of thesquare-root of this mean offered load. The policy alsooutsources calls whenever the number of customersin line exceeds a threshold. Because the policy U isanalytically tractable, it can be used to help call

centers negotiate outsourcing contracts; in particular,our analysis shows the implications of various valuesof the per call outsourcing cost.When the order of magnitude of arrival rate ran-

domness is the same order as the inherent systemfluctuations in the queue length (which is on the orderof the square-root of the realized arrival rate), weshow that our proposed policy is asymptotically opti-mal as the mean arrival rate grows large. Then, weperform an extensive numerical experiment to studythe performance of U beyond the regime for which itis proved to be asymptotically optimal. In all of ournumerical experiments in which the system has morethan a few servers, we did not encounter even onecase in which the performance of U was not superb.In contrast, our two benchmark policies each haveparameter regimes in which their performance can bearbitrarily bad. It is, therefore, the robustness of theperformance of U that we would like to stress as themain takeaway from this paper. One does not need toidentify the “right” operating regime in order todetermine which policy to use. The U policy appearsto be a “one-size-fits-all.”Several important extensions are worth pursuing.

One is to model the staffing decisions of the outsidevendor more explicitly, as in Gurvich and Perry(2012) in the case of known arrival rate. In our model,we assume that a call outsourced incurs a fixed costno matter how many calls are sent to the outside ven-dor. This is consistent with the assumption made inAks�in et al. (2008), and is equivalent to assumingeither that the outside service provider has ample ser-vice capacity or that it pools demand from a largeenough client base so that the calls the company sendsto the outside vendor do not have much impact. Forthe case where the outsourcing is not preferred ordoes not exist, we see that the performance of Uremains superb. (We do not report these results dueto space limitations.) However, if the outsourcingcapacity is positive and limited we do not know howU will perform, or how it would need to be modifiedto maintain this superior performance.

0

50

100

150

200

250

300

350

400

0 100 200 300 400 500 600 700 800 900 1000

Expe

cted

Cos

t

Staffing Level (N)

Diffusion Approximation

Actual Expected Cost

-1

-0.5

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700 800 900 1000

paGtsoC

detcep xE

Staffing Level (N)

Figure 6 Expected Cost Difference: Actual vs. Diffusion (K�U½20; 780�, k = 400, c = 0.1, p = 1 and a = 5.)




Beyond the co-sourcing application considered inthis study, one might use a similar framework toexamine joint staffing and control decisions withother system topologies and other types of controlsuch as the problems considered in Gurvich et al.(2008), Dai and Tezcan (2008) Tezcan and Dai (2010),Gurvich and Whitt (2010), and Armony and Mandel-baum (2011). In particular, it will sometimes be possi-ble to incorporate arrival rate uncertainty withoutchanging the control that has been proven to beasymptotically optimal when the arrival rate isknown. Ideally, the robustness of the policy perfor-mance with respect to the assumptions on the arrivalrate uncertainty in this model will be true in muchmore generality.Finally, our numerical results with respect to the

remarkable robustness of U suggest that there mightbe an underlying theoretical justification to thisrobustness, in the spirit of the universal approxima-tion of Gurvich et al. (2014). In section 6.5, we havediscussed why their results in a framework thatassumes a known arrival rate and no dynamic controlmay not be readily applied to our framework. But onewonders whether similar universal approximationprinciples apply when a random arrival rate anddynamic control are incorporated into the model.

Notes

1See also section 7 in Whitt (2005) for exact expressionsfor other performance measures of interest, such as theexpected wait time conditioned on an arrival being served,in a more general model that allows for state-dependentabandonment rates.2Lemma 3.2 in Koc�a�ga and Ward (2010) implies Topt � N.3For example, Theorem 3.4 in Koc�a�ga and Ward (2010)provides a bound on the difference between the currentcost and the minimum cost.4If zsðTþ1Þ � zsðTÞ, Theorem 3.2 in Koc�a�ga and Ward (2010)

implies that TH ¼ T.5Note that setting E[X] = 0 yields b

b ¼ � a2a1, which together

with Var Xð ÞÞ ¼ r2 yields b2 ¼ r2a1ða1 þ a2 þ 1Þa2

. Hence, the

values of a1, a2, b, and b are not fully determined and we

arbitrarily set a1 þ a2 ¼ 2 and change a1 and a2 accord-

ingly, which also changes b and b.6Recall that the skewness of Beta distribution is given by

2ða2 � a1Þðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 þ a2 þ 1p Þ

ða1 þ a2 þ 2Þð ffiffiffiffiffiffiffia1a2p Þ .

ReferencesAks�in, Z., M. Armony, V. Mehrotra. 2007. The modern call center:

A multi-disciplinary perspective on operations managementresearch. Prod. Oper. Manag. 16(6): 665–688.

Aks�in, Z., F. de V�ericourt, F. Karaesmen. 2008. Call center outsourc-ing contract analysis and choice.Manage. Sci. 54(2): 354–368.

Armony, M., A. Mandelbaum. 2011. Routing and staffing inlarge-scale service systems: The case of homogeneous

impatient customers and heterogeneous servers. Oper. Res.59(1): 50–65.

Armony, M., E. Plambeck, S. Seshadri. 2009. Sensitivity of optimalcapacity to customer impatience in an unobservable M=M=Squeue (Why you shouldn’t shout at the DMV). Manuf. Serv.Oper. Manag. 11(1): 19–32.

Bassamboo, A., A. Zeevi. 2009. On a data-driven method for staff-ing large call centers. Oper. Res. 57(3): 714–726.

Bassamboo, A., J. M. Harrison, A. Zeevi. 2005. Dynamic routingand admission control in high volume service systems:Asymptotic analysis via multi-scale uid limits. Queueing Syst.51(3): 249–285.

Bassamboo, A., J. M. Harrison, A. Zeevi. 2006. Design and controlof a large call center: Asymptotic analysis of an LP-basedmethod. Oper. Res. 54(3): 419–435.

Bassamboo, A., R. S. Randhawa, A. Zeevi. 2010. Capacity sizingunder parameter uncertainty: Safety staffing principles revis-ited. Manage. Sci. 56(10): 1668–1686.

Borst, S., A. Mandelbaum, M. I. Reiman. 2004. Dimensioning largecall centers. Oper. Res. 52(1): 17–34.

Brown, L., N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zel-tyn, L. Zhao. 2005. Statistical analysis of a telephone call cen-ter. J. Am. Stat. Assoc. 100(469): 36–50.

Chen, B. P. K., S. G. Henderson. 2001. Two issues in setting callcentre staffing levels. Ann. Oper. Res. 108(1): 175–192.

Dai, J. G., T. Tezcan. 2008. Optimal control of parallel server sys-tems with many servers in heavy traffic. Queueing Syst. 59(2):95–134.

Gans, N., G. Koole, A. Mandelbaum. 2003. Telephone call centers:Tutorial, review, and research prospects. Manuf. Serv. Oper.Manag. 5(2): 79–141.

Gans, N., H. Shen, Y.-P. Zhou, K. Korolev, A. McCord, H. Ristock.2012. Parametric stochastic programming models for call-center workforce scheduling. Working paper, University ofPennsylvania, Philadelphia.

Garnett, O., A. Mandelbaum, M. Reiman. 2002. Designing a callcenter with impatient customers. Manuf. Serv. Oper. Manag.4(3): 208–227.

Gilson, K. A., D. K. Khandelwal. 2005. Getting more from callcenters. The McKinsey Quaterly.

Green, L. V., P. J. Kolesar, J. Soares. 2001. Improving the SIPPapproach for staffing service systems that have cyclicdemands. Oper. Res. 49(4): 549–564.

Gurvich, I., O. Perry. 2012. Over ow networks: Approximationsand implications to call-center outsourcing. Oper. Res. 60(4):996–1009.

Gurvich, I., W. Whitt. 2010. Service-level differentiation in many-server service systems via queueratio routing. Oper. Res. 58(2):316–328.

Gurvich, I., M. Armony, A. Mandelbaum. 2008. Service-level dif-ferentiation in call centers with fully exible servers. Manage.Sci. 54(2): 279–294.

Gurvich, I., J. Luedtke, T. Tezcan. 2010. Staffing call centers withuncertain demand forecasts: A chance-constrained optimiza-tion approach. Manage. Sci. 56(7): 1093–1115.

Gurvich, I., J. Huang, A. Mandelbaum. 2014. Excursion-based uni-versal approximations for the erlang-a queue in steady-state.Math. Oper. Res.. 39(2): 325–373.

Halfin, S., W. Whitt. 1981. Heavy-traffic limits for queues withmany exponential servers. Oper. Res. 29(3): 567–588.

ICMI. 2006. ICMI’s contact center outsourcing report. Available athttp://www.icmi.com/files/ICMI/members/ccmr/ccmr2006/ccmr06/June2006_issue.pdf (accessed date December 1, 2014).




Jongbloed, G., G. Koole. 2001. Managing uncertainty in call cen-tres using Poisson mixtures. Appl. Stoch. Mod. Bus. Ind. 17(4):307–318.

Kharif, O. 2003. The hidden costs of it outsourcing. BusinessWeek(October 26, 2003).

Koc�a�ga, Y. L., A. R. Ward. 2010. Admission control for amulti-server queue with abandonment. Queueing Syst. 65(3):275–323.

Koole, G., A. Pot. 2011. A note on profit maximization and mono-tonicity for inbound call centers. Oper. Res. 59(5): 1304–1308.

Liu, Y., W. Whitt. 2012. Stabilizing customer abandonment inmany-server queues with time varying arrivals. Oper. Res.60(6): 1551–1564.

Maman, S. 2009. Uncertainty in the demand for service: The caseof call centers and emergency departments. Master’s thesis,Technion-Israel Institute of Technology, Haifa, Israel.

Mandelbaum, A., S. Zeltyn. 2009. Staffing many-server queueswith impatient customers: Constraint satisfaction in call cen-ters. Oper. Res. 57(5): 1189–1205.

Mehrotra, V., €O. €Ozl€uk, R. Saltzman. 2010. Intelligent proceduresfor intra-day updating of call center agent schedules. Prod.Oper. Manag. 19(3): 353–367.

Robbins, T. R., T. P. Harrison. 2010. A stochastic programmingmodel for scheduling call centers with global service levelagreements. Eur. J. Oper. Res. 207(3): 1608–1619.

Ross, A. M. 2001. Queueing systems with daily cycles and sto-chastic demand with uncertain parameters. Ph.D. thesis, Uni-versity of California, Berkeley.

Steckley, S. G., S. G. Henderson, V. Mehrotra. 2009. Forecasterrors in service systems. Probab. Eng. Inform. Sci. 23(2): 305–332.

Tezcan, T., J. G. Dai. 2010. Dynamic control of n-systems withmany servers: Asymptotic optimality of a static priority policyin heavy traffic. Oper. Res. 58(1): 94–110.

Whitt, W. 2005. Engineering solution of a basic call-center model.Manage. Sci. 51(2): 221–235.

Whitt, W. 2006. Staffing a call center with uncertain arrival rateand absenteeism. Prod. Oper. Manag. 15(1): 88–102.

Zan, J., J. J. Hasenbein, D. P. Morton. 2014. Asymptotically opti-mal staffing of service systems with joint QoS constraints.Queueing Syst. 78(4): 359–386.

Zhou, Y.-P., Z. J. Ren. 2011. Service outsourcing. Wiley Encyclope-dia of Operations Research and Management Science.

Supporting InformationAdditional Supporting Information may be found in theElectronic Companion available online:

Appendix S1: Proofs and Supporting Numerical Tables.




staffing call centers with uncertain arrival rates and co&#x2010

Documents