capacity allocation and pricing strategies for new...

Capacity Allocation and Pricing Strategies for NewWireless Services

Lingjie DuanEngineering Systems and Design, Singapore University of Technology and Design, Singapore 487372, [email protected]

Biying ShouDepartment of Management Sciences, College of Business, City University of Hong Kong, Hong Kong, [email protected]

Jianwei HuangDepartment of Information Engineering, The Chinese University of Hong Kong, Hong Kong, [email protected]

I ndoor cell phone users often suffer poor connectivity. One promising solution to this issue, femtocell technology, hasbeen rapidly developed and deployed over the past few years. One of the biggest challenges facing femtocell deploy-

ment is the lack of a clear business model. This study investigates the economic incentive for cellular operators (alsocalled macrocell operators) to enable femtocell service by leasing spectrum resources to independent femtocell operators.We model the interactions between a macrocell operator, a femtocell operator, and end-users as a three-stage dynamicgame, and derive the equilibrium pricing and capacity allocation decisions. We show that when spectrum resources arevery limited, the macrocell operator has more incentive to lease spectrum to the femtocell operator, as femtocell servicescan help cover more users and improve the utilization efficiency of the limited spectrum resource. However, when thetotal spectrum resource is large, femtocell service offers significant competition to macrocell service and, as a result, themacrocell operator has less incentive to enable femtocell service. We also show the impact of the additional operationalcosts and limited coverage of femtocell service on equilibrium decisions, consumer surplus, and social welfare.

Key words: wireless service; capacity allocation; pricing; game theoryHistory: Received: March 2013; Accepted: August 2015 by Chris Tang, after 3 revisions.

1. Introduction

Today, there are over 4.5 billion mobile phone usersin the world (eMarketer 2014), and many of themexperience poor indoor reception at home or theoffice. This is because, in the current cellular network(also called the macrocell network), a base station cov-ers an area of a radius from several hundred metersto several kilometers. High-frequency and low-powerwireless signals, however, often have difficulty ineffectively traveling from an outdoor macrocell basestation to indoor cell phones through walls. As aresult, indoor cell phone users often experiencedropped calls and reduced wireless data rates (San-dler 2009).As one promising solution to the indoor reception

problem, femtocell technology, has been rapidly devel-oped and deployed over the past few years. Figure 1provides an illustration of four homes covered by onemacrocell base station, three of which have installedfemtocell base stations. Femtocell is a small base sta-tion with a size similar to a wireless router. An indoor

femtocell base station is much closer to users’ indoorcell phones, hence, there is little channel attenuationor fading. It can pick up the cell phones’ signals muchmore effectively, and then deliver voice and data traf-fic to the cellular network through users’ home wire-line Internet connection. Femtocell technology cansignificantly increase the quality of voice calls andimprove the speed of data communications (Shettyet al. 2009).Major operators worldwide are enthusiastic about

femtocell technology due to its capability of improv-ing customers’ experiences. In the United States,AT&T, Sprint Nextel, and Verizon Wireless arealready offering femtocell services to their customers.T-Mobile and Vodafone in Europe, NTT DoCoMoand Softbank in Japan, and Unicom in China havebeen testing the technology and planning to imple-ment nationwide femtocell services. The UK researchfirm Informa Telecoms & Media reported that femto-cell deployments had more than doubled from 2009to 2010, with substantially increasing numbers oftier-one operators joining this trend. Global femtocell

Please Cite this article in press as: Duan, L., et al. Capacity Allocation and Pricing Strategies for New Wireless Services. Production andOperations Management (2015), doi 10.1111/poms.12510

Vol. 0, No. 0, xxxx–xxxx 2015, pp. 1–17 DOI 10.1111/poms.12510ISSN 1059-1478|EISSN 1937-5956|15|00|0001 © 2015 Production and Operations Management Society

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info:doi/10.1111/poms.12510

deployments is expected to increase from 2.1 millionunits in 2011 to 87.3 million units in 2016 (Informa,Telecoms & Media 2011, 2012).However, one of the biggest challenges to an opera-

tor’s large-scale femtocell deployment is the lack of aclear business model. As Emin Gurdenli, Chief Tech-nology Officer of Deutsche Telekom AG’s T-MobileUnited Kingdom, put it (The Wall Street Journal2009): “The rationale for femtocells is well-established, buta quantitative business case with a clear business model interms of how we go to market is not there yet.”The purpose of this study is to develop a quantita-

tive model to examine the business trade-off of femto-cell deployments. In particular, we examine acommonly used distributed approach to deployingfemtocell service, in which a macrocell operator maylease some of its spectrum resources to a femtocelloperator. The femtocell operator serves as a virtualoperator, and determines the service provision andpricing independently. There are many examples ofdistributed deployment of femtocell service in thewireless industry. For example, Sprint leases spec-trum to Virgin Mobile USA to provide femtocell ser-vice (Gabriel 2009), and BT Mobile uses Vodafone’sresource to provide femtocell service (Chambers2010).We investigate the following research questions:

• Should a macrocell operator support femtocell ser-vice? How would the operator allocate bandwidth(capacity) resources and make pricing decisions?The key trade-off here is between additionalrevenue source and increased market competi-tion: a macrocell operator may increase its rev-

enue by leasing resources to the femtocelloperator, meanwhile it will have fewerresources for its own services and faceincreased market competition from the femto-cell operator.

• How would users choose between femtocell andmacrocell services? By using the indoor femto-cell base station, users can avoid the poorreception problem and achieve better qualityof service. In contrast, when users choose touse the outdoor macrocell base stations, theservice quality highly depends on the userlocations and the communication environ-ments (which can be described by a user-dependent spectrum efficiency parameter). Asthe prices of femtocell and macrocell servicesmay be very different, a user needs to evalu-ate the trade-off between cost and quality ofservice.

Different from the studies of femtocell technologyin the wireless telecommunications literature, whichfocus on the technical operations of femtocell services,our study focuses on the business analysis of capacityand pricing decisions as well as the economic trade-off between cooperation and competition. Further-more, compared to the dual channels studies in theoperations management and marketing science litera-ture which generally assumes unlimited supply, weincorporate the constraint of limited supply of spec-trum resources. The capacity constraint better reflectsthe reality of the wireless market – wireless spectrumhas become a very scarce resource, because wirelessdemand has grown dramatically during recent years,

Macrocell coverage

Wireline central office to address cellular traffics Femtocell user

Macrocell user

Wireless home Internet connec�ons

Femtocell base sta�on

Macrocell base sta�on

Figure 1 Coexistence of Femtocell Service and Macrocell Service, in Which a Macrocell and Three out of Four Femtocells are Deployed

Duan, Shou, and Huang: Strategies for New Wireless Services2 Production and Operations Management 0(0), pp. 1–17, © 2015 Production and Operations Management Society


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but the spectrum resources remain limited due tovarious physical constraints. The capacity constraint,as we will show, significantly affects the operator’sdecisions (and complicates our analysis). Finally, wemodel the heterogeneity and the utility functions ofusers based on the unique characteristics of wirelesscommunications, which makes our model very differ-ent from the typical setup in the dual channel litera-ture.Our main results are summarized as follows:

• Characterization of equilibrium decisions: Wederive a threshold structure in terms of thespectrum efficiency parameter, which separatesusers who prefer femtocell or macrocell ser-vice. We further characterize the femtocelloperator’s femtocell price and the macrocelloperator’s capacity allocation and pricing deci-sions at equilibrium.

• Sensitivity analysis of macrocell’s total capacity:We analyze how the spectrum capacity con-straint affects the equilibrium decisions. Onekey (and perhaps counter-intuitive) finding isthat the macrocell operator has more incentiveto lease spectrum to the femtocell operatorwhen its spectrum capacity is small.

• Examination of consumer surplus and social wel-fare: We show that, in general, femtocell servicecan increase both the total consumer surplusand social welfare. However, some users whooriginally receive good service in macrocellmight experience a payoff drop after the fem-tocell service is introduced, due to fewerresources being allocated to the macrocell ser-vice and a higher macrocell price.

In addition, we examine two extensions of the basicmodel, with additional femtocell operational cost andwith limited femtocell coverage. The rest of the studyis organized as follows. Section 3 introduces the net-work model of macrocell service, which serves as abenchmark for later analysis. Section 4 presents thenetwork model of dual services and analyzes how theoperators make optimal capacity allocation and pric-ing decisions. Sections 5 and 6 extend the results insection 4 by examining the various effects of addi-tional femtocell operational cost and limited femtocellcoverage. Section 7 concludes our study and discussessome future research directions.

2. Literature Review

Our work is closely related to two main streams ofliterature: (i) studies of femtocell deployment in thetelecommunication literature; and (ii) studies of dualchannel competition in the operations managementand marketing science literature.

Most existing work on femtocell deployment in thetelecommunication literature (e.g., Chandrasekharand Andrews 2009) focuses on various technicalissues in service provisions, such as access control,resource management, and interference management.Only a few papers discuss the economic issues of fem-tocells (e.g., Claussen et al. 2007, Yun et al. 2011,Shetty et al. 2009, Chen et al. 2012, Duan et al. 2013),examining the impact of network deployment costsand femtocells’ openness to macrocell users. The keydifference between our study and the existing litera-ture is that we study the provision of dual services interms of both spectrum allocations and pricing deci-sions. Our focus is on the business analysis of thetrade-off between cooperation and competition.Our work is also related to studies on dual channel

management in the operations management and mar-keting science literature. Tsay and Agrawal (2004a)provide a comprehensive review of earlier studies onthe competition and coordination of dual channel dis-tribution systems. The authors grouped the relatedstudies into two categories based on whether themanufacturer owns the direct channel or the retailstore. Our study would belong to the former category.Representative papers in this category are Chianget al. (2003) and Tsay and Agrawal (2004b). Chianget al. (2003) consider whether a manufacturer shouldsell through a direct channel, an exclusive retailer, ora hybrid of both. The key finding is that, at equilib-rium, the manufacturer will price in the direct chan-nel, such that all customers will purchase from theretail channel – the direct channel exists not to sell theproduct, but to influence the independent retailer’sprice to reduce double marginalization. Tsay andAgrawal (2004b) consider a similar problem setting asin Chiang et al. (2003); however, they assume that themanufacturer and the retailer can control not only theprices, but also the sales efforts. They further exploitseveral means whereby the manufacturer can mitigatechannel conflict between the direct channel and theretail channel, including adjustments of wholesaleprice, paying a commission to a retailer, and entirelyconceding demand fulfillment on the part of theretailer.The past few years continued to see a good

amount of interest in the dual-channel system. Chenet al. (2008) study a setting in which a manufactureroperates both a direct online channel and an inde-pendent retail channel. The two channels competein service, measured as delivery time for the onlinechannel and stocking rate for the retail channel. Theprices for the two channels are assumed to beexogenous and the same. The authors incorporate adetailed consumer channel choice model and iden-tify optimal dual channel strategies, that is, whenthe manufacturer should establish a direct channel,

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a retail channel, or both. Huang and Swaminathan(2009) propose a stylized deterministic demandmodel, in which each channel relies on prices,degree of substitution across channels, and the over-all market potential. Dumrongsiri et al. (2008) inves-tigate the influence of demand variability on pricesand manufacturer’s incentive to open the directchannel. David and Adida (2014) study competitionand coordination in a supply chain, in which a sup-plier operates a direct channel and also sellsthrough multiple retailers. The direct channel andthe retail channels engage in a quantity competition,with the market prices determined by the total orderquantities. They show that the supplier benefitsfrom having more retailers in the market and thatlinear quantity discount contracts can mitigate sup-ply chain inefficiency.In the context of wireless service deployment, our

paper studies the decision of a macrocell wireless ser-vice provider (manufacturer) to sell through a directchannel, an independent femtocell operator (retailer),or both.However, our study has several key differences

from the existing dual channel literature. First, weexplicitly incorporate the limited capacity constraintinto our model, while all of the prior studies dis-cussed above have assumed unlimited capacity.Second, we model the heterogeneity of users basedon the unique characteristics of wireless communi-cations. In particular, users have different channelconditions (and thus different evaluations of thesame resource allocation) under the macrocell ser-vice (direct channel), but have the same maximumchannel condition under the femtocell service (re-tailer channel). In contrast, prior literature on dualchannel either assumes that users are homogenous,or users are different in terms of their willingnessto pay or sensitivities to delivery time, and suchheterogeneity is often independent of the channelchoice. Moreover, the users’ utility function in ourstudy is also motivated by today’s wireless commu-nication technologies, which renders much of theprior simplified models (e.g., linear form of user-specific coefficients) and generic analysis inapplica-ble. Finally, we characterize the impact of limitedfemtocell coverage on the new service provision.To the best of our knowledge, none of the priordual-channel studies have considered such a con-straint.

Figure 2 Two-Stage Stackelberg Game Between the Macrocell Operator and Users

3. Benchmark Scenario: MacrocellService Only

In this study, we consider the interaction between onemacrocell operator and one femtocell operator. This isa simplified assumption, as in many countries thereoften exists intensive competition among many wire-less service providers. However, given that there islittle business research about the multi-tiered struc-ture and the trade-off between competition andcooperation in the wireless service market, such aone-to-one model serves as an important first step.(In Appendix S9, we generalize our results to a com-petitive femtocell market, under the assumption thatusers have a fixed positive reservation payoff due tothe existence of an alternative service.) A moregeneral study on an oligopoly market is one of theinteresting directions for future research.We first examine the case in which the macrocell

operator is the only service provider in the market.The operator determines its optimal service chargein order to maximize total profit. This case servesas a benchmark for our later analysis, in which amacrocell operator may lease partial resources toanother operator to provide new femtocell serviceto the market.The macrocell operator owns limited resources of

wireless spectrum (also called bandwidth). Wirelessusers need to access the bandwidth in order to con-duct their wireless communications (e.g., voice calls,video streaming, data transfer). A larger bandwidthmeans more resources for the user and thus bettercommunication quality of service; however, it alsomeans higher cost for the user.We focus on the operation of a single macrocell and

derive the optimal service charge under limitedcapacity. As shown in Figure 2, we model the interac-tions between the macrocell operator and end-usersas a two-stage Stackelberg game. In Stage I, themacrocell operator determines the macrocell price pMper unit bandwidth. Note that due to the exponentialgrowth of wireless data traffic and the scarce spec-trum resource, the usage-based pricing is becoming amain trend in the macrocell service market (Goldstein2011). In Stage II, each user decides how much band-width to purchase. The users may be heterogeneouswith their efficiencies in using wireless spectrum. Theoperator wants to maximize its profit, while the userswant to maximize their individual payoffs. We solve



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this two-stage Stackelberg game by backward induc-tion (Myerson 2002, Duan et al. 2012).

3.1. Stage II: Users’ Purchasing DecisionsThe quality of service that a user receives dependsnot only on the size of the spectrum resource that heor she obtains, but also on the condition of the wire-less channel. The channel condition is determined byboth the location and the surrounding environment.For example, consider the transmissions from theusers’ mobile phones to the common single macro-cell base station (as in Figure 1). The channel condi-tion generally decreases with the distance betweenthe user and the base station, and can become veryweak if the user is inside a house with thick walls. Auser with a bad channel condition will not be able toachieve a high data rate, even with a large band-width allocation.We model the users’ channel heterogeneity by a

macrocell spectrum efficiency h, which is assumed tobe uniformly distributed in [0, 1]. A larger h means abetter channel condition and a higher spectrum effi-ciency when using the macrocell service. The uniformdistribution is widely used to approximate wirelessusers’ fluid population considering their locations(e.g., Altman et al. 2009, Inaltekin et al. 2007, andLeung and Huang 2011). This assumption also facili-tates our analysis. However, we can show that usingother continuous distributions, for example, normaldistribution, will not change the main insightsobtained in this study.For a user with a macrocell spectrum efficiency

h < 1, if the acquired macrocell bandwidth is b, thenthe user’s effective resource allocation is hb. Similar toChiang (2005), Huang et al. (2006), Sengupta andChatterjee (2009), Song and Li (2005), and Wang andLi (2005), we model user’s utility u(h, b) as:

uðh; bÞ ¼ lnð1þ hbÞ;which is concave in b, representing the diminishingreturn in bandwidth consumption. Take watching aYouTube video as an example. It is very importantto guarantee the basic level of bandwidth to supportthe video playback at the lowest quality of 360p.When the bandwidth allocation increases, a user canwatch a higher resolution of the video (e.g., 480p,720p, or even 1080p). However, the additionalincrease of satisfaction is sublinear with respect tothe increase of data rate. The logarithmic functioncaptures such a principle of diminishing returns.The “1” in the logarithmic function represents thefact that the user will experience zero benefit if thespectrum allocation is zero. The utility function canbe interpreted as the data rate that the user obtains.The more bandwidth that a user acquires, the higher

the data rate and better quality of service he or shereceives (in the latest 4G cellular system based onOFDMA technology, different users communicatingwith the same macrocell or femtocell base stationwill be allocated different spectrum bands; Once thespectrum allocation is fixed, the communication per-formance among different users is decoupled andindependent of each other). The product form of hand b represents the fact that different users havedifferent capabilities of using the spectrum resource.For an outdoor user who has a very good connec-tion with the macrocell base station, he or she canfully utilize the bandwidth and achieve the maxi-mum data transmission rate (the so-called “ShannonCapacity”) in communications with proper codingand modulation schemes. However, for an indooruser behind thick walls, the connection between thecellular phone and the outdoor macrocell base sta-tion is weak, and the user obtains a much smallerdata rate with the same amount of allocated band-width, compared with an outdoor user. A usefulanalogy to understand this is to imagine spectrumallocation b as the width of a road, and the spec-trum efficiency h as the type (speed) of the vehicle.How much traffic per unit time that the road canaccommodate depends both on the width of theroad and the types of the vehicles.The user needs to pay a linear payment pMb to the

macrocell operator, where the price pM is announcedby the macrocell operator in Stage I. The user’s payoffis the difference between the utility and payment:

rMðh; b; pMÞ ¼ lnð1þ hbÞ � pMb: ð1ÞThe optimal amount of bandwidth that maximizesthe user’s payoff with the macrocell service is:

b�ðh; pMÞ ¼ 1=pM � 1=h; if pM � h;0; if pM [ h:

�ð2Þ

b�ðh; pMÞ is decreasing in pM and increasing in h,when pM � h. As shown in Figure 3, the users aresegmented into two groups – those who purchasethe macrocell service and those who do not. Accord-ingly, a user’s maximum payoff with macrocellservice is:

rMðh; b�ðh; pMÞ; pMÞ ¼ lnhpM

� �� 1þ pM

h; ð3Þ

if pM � h, and 0 otherwise.

Figure 3 Segmentation of Users Based on Macrocell Spectrum Effi-ciency



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3.2. Stage I: Macrocell Operator’s Pricing pMNext, we consider the macrocell operator’s optimalchoice of price pM in Stage I. To achieve a positiveprofit, the macrocell operator needs to setpM � maxh2½0;1� h ¼ 1; otherwise, no user will request

any bandwidth in Stage II.We denote the total user population as N. The frac-

tion of users choosing macrocell service is 1 � pM, asshown in Figure 3. The total user demand is as fol-lows:

QMðpMÞ ¼ N

Z 1

pM

1

pM� 1

h

� �dh ¼ N

1

pM� 1þ ln pM

� �;

ð4Þwhich is a decreasing function of pM.The macrocell operator has a limited bandwidth

capacity B, and thus can only satisfy a total demandno larger than B. The macrocell operator optimizes pMto maximize profit:

max0\pM � 1

pmacroðpMÞ ¼ pM min B;N1

pM� 1þ ln pM

� �� :

ð5ÞTheorem 1 characterizes the unique optimal solu-

tion to Problem (5).

THEOREM 1. The equilibrium macrocell price pbenchM thatmaximizes the macrocell operator’s profit in the two-stageStackelberg game is the unique solution to the followingequation:

B ¼ N1

pM� 1þ ln pM

� �: ð6Þ

That is, the total user demand equals the maximumcapacity at equilibrium.

It is easy to show that the equilibrium price pbenchM

decreases in B, and the macrocell operator’s equilib-

rium profit pmacroðpbenchM Þ increases in B. Furthermore,when the total bandwidth B is small, the equilibrium

macrocell price pbenchM becomes close to 1, and thusmost users will not get service. This motivates themacrocell operator to consider facilitating the femto-cell service, to serve more users and obtain a higherprofit.

Figure 4 Three-Stage Dynamic Game with the Macrocell Operator, the Femtocell Operator, and the End-Users

4. Femtocell Deployment

We now turn to the question of when and how femto-cell service may improve the macrocell operator’sprofit. We are interested in understanding the follow-ing issues:

• Strategic decision: Is it economically beneficialfor the macrocell operator to lease spectrum toa femtocell operator? How to evaluate thetrade-off between cooperation and competi-tion?

• Operational decisions: If the answer to the previ-ous question is yes, how should the macrocelloperator allocate and price the spectrumresources?

The analysis in this section is based on two simpli-fied assumptions: (1) The femtocell service does notincur any additional operational cost compared to themacrocell service. This assumption will be relaxed insection 5. (2) The femtocell service has the same cover-age as the macrocell service, so that users can freelychoose between two services. The full femtocell cover-age can be a good approximation when we look at adensely populated residential area with manydeployed femtocells. This assumption will be relaxedin section 6.We study a three-stage dynamic game, as shown in

Figure 4. In Stage I, the macrocell operator decidesbandwidth allocations to femtocell service BF andmacrocell service BM, such that BF þ BM ¼ B. In thisstudy, we consider the “separate carriers” scheme,where the macro and femtocell services operate onnon-overlapping spectrum bands. Such a scheme iseasy to manage and can avoid interferences betweenmacrocells and femtocells. For example, China Uni-com, one of the top three wireless service providers inChina, has deployed femtocell service with thisscheme since 2009 (China Femtocell Symposium2011). The macrocell operator also decides the macro-cell price pM in Stage I, which is charged to both thefemtocell operator and end-users who choose macro-cell services. (In Appendix S6, we show that even ifthe macrocell operator is allowed to charge two differ-ent prices to the femtocell operator and the end-users,it is optimal for the macrocell operator to charge thesame price to maximize his or her profit. Hence, wefocus on the uniform pricing scheme in the study.)



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In Stage II, the femtocell operator decides howmuch bandwidth BR to lease from the macrocell oper-ator, such that BR � BF. It also determines the femto-cell retail price pF. In Stage III, each user decideswhich service to choose and how much bandwidth torequest. If a user’s preferred service is not available,the user will seek the other service. We assume thatthere is a large group of users, where a single user’sdemand is infinitesimal to the total demand. Thus, wecan ignore cases in which a user purchases bandwidthfrom both services.Note that our model assumes that the macrocell

operator is the leader and makes decisions first inStage I, while the femtocell operator is the followerand reacts in Stage II. This is because, in practice, themacrocell operator is usually a market incumbentwith much better network coverage and greater mar-ket power; whereas, the femtocell operator is usuallya new market entrant with less market power (e.g.,the Sprint-Virgin Mobile interaction in the UnitedStates and the BT-Vodafone interaction in the UnitedKingdom).We analyze this three-stage game using backward

induction. To distinguish from the benchmark case(macrocell service only) in section 3, we refer to thesetup in this section as dual services. Note that dualservices may degenerate to the benchmark case whenthe macrocell operator decides not to lease spectrumto the femtocell operator. Table 1 summarizes the keynotation used in the model and analysis. All of thedetailed proofs are provided in the Appendix.

4.1. Stage III: Users’ Service Choices andBandwidth PurchasesGiven both macrocell and femtocell services, a userneeds to decide which service to choose and howmuch bandwidth to request. If a user has a macrocell

spectrum efficiency h, his or her maximum payoff byusing the macrocell service is given by Equation (3).Now we consider a user’s payoff by using the femto-cell service.Since femtocell base stations are deployed indoors

and are very close to users’ cell phones, it is reason-able to assume that all users using the femtocellservice have equally good channel conditions andachieve the maximum spectrum efficiency 1. Thismeans, independent of the macrocell spectrum effi-ciency h, that each user achieves the same payoffrFðb; pFÞ when using the femtocell service with abandwidth of b:

rFðb; pFÞ ¼ lnð1þ bÞ � pFb: ð7ÞAccordingly, the optimal amount of bandwidth torequest is:

b�ðpFÞ ¼ 1

pF� 1; ð8Þ

if pF � 1 and 0 otherwise. A user’s maximum payoffunder the femtocell service is:

rFðb�ðpFÞ; pFÞ ¼ ln1

pF

� �� 1þ pF; ð9Þ

if pF � 1, and 0 otherwise.We will show that pF [ pM at equilibrium, that is,

the equilibrium femtocell price is always larger thanthe equilibrium macrocell price. By comparing theuser’s payoffs in Equations (3) and (9), it is clear that auser with h = 1 always chooses macrocell service tomaximize the payoff. On the other hand, a user with asmall h would prefer to use femtocell service to

improve the payoff. Let hprth denote the threshold of

end-users’ preferred (initial) choices between themacrocell and femtocell services. That is, users with

h 2 ½0; hprthÞ prefer to use the femtocell service, and

users with h 2 ½hprth ; 1� prefer to use the macrocell

service.Comparing a user’s optimal payoff from using

macrocell and femtocell services in Equations (3) and(9), we obtain the following lemma:

LEMMA 1. hprth ¼ pM=pF.

Because the macrocell and femtocell operators havelimited bandwidth capacity, a user may not alwaysobtain the preferred service. We assume that a user’sservice selection and payment process is as follows:

(i) Given prices of the femtocell and macrocell ser-vices, a user first computes the maximum pay-off achievable under each service, and choosesthe service with a higher payoff as his or her

Table 1 Notation

pM The retail price charged to end-users by the macrocelloperator

pF The retail price charged to end-users by the femtocell operatorpW The wholesale price charged to femtocell operator by the

macrocell operatorBR The bandwidth requested by the femtocell operator from the

macrocell operatorBF The bandwidth reserved by the macrocell operator for the

femtocell operatorrM An end-user’s payoff from using the macrocell servicerF An end-user’s payoff from using the femtocell serviceh The macrocell spectrum efficiency of an end-user, h in [0, 1]hprth The threshold of end-users’ preferred (initial) choices between

the two serviceshth The threshold of end-users’ final choices between the two

servicespMacro The macrocell’s total profitpFemto The femtocell’s total profit



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first (preferred) choice, or no service at all ifthe maximum payoffs are zero.

(ii) If the first choice request (e.g., the request touse femtocell service) is accepted, then the userwould be charged based on the usage of thatservice. If the request is rejected, then the usercan request the alternative service.

(iii) If the second choice request is accepted, thenthe user would be charged based on the usageof that service. If the second choice request isalso rejected, then the user gets no service andpays nothing.

Next, we introduce the definition of Final Demand:If a user’s demand from his or her preferred service issatisfied, then his or her final demand is the same asthe preferred demand. Otherwise, the user wouldrequest the alternative service, and the new demandbecomes the final demand. Note that a user’s finaldemand may not always be satisfied, if the total finaldemand for the alternative service is larger than itsspectrum capacity. Nevertheless, as the user ischarged based on the actual service that he or sheeventually receives, there is no uncertainty for theuser.When the macrocell operator receives more

demand than its capacity, we assume that users withlarger h will be served first. This helps the macrocelloperator to maximize the overall spectrum efficiency,as well as the revenue (see the proof in Appendix S1).It is also consistent with the current engineering prac-tice, in which cell phones receiving stronger pilot sig-nals from the macrocell base station will be served bythe base station first.Let hth denote the threshold of end-users’ final

choices between the macrocell and femtocell services.That is, users with h 2 ½hth; 1� receive the macrocellservice, while users with h 2 ½0; hthÞ receive either thefemtocell service or no service. In general, hth may be

different from hprth , depending on the service capaci-

ties. For example, a user who wants to subscribe tofemtocell service may be rejected because the femto-cell service is short of spectrum capacity. However,one of the key results that we will show later is that atequilibrium, all users’ initial demands for their pre-

ferred services are satisfied, which means hprth ¼ hth.

4.2. Stage II: Femtocell Operator’s Leasing andPricing DecisionsWe now analyze Stage II, where the femtocell opera-tor determines BR and pF to maximize its profit. In thisstage, the macrocell operator’s decisions on pM and BF

(and BM ¼ B � BF) are already determined andknown to the femtocell operator. We denote the fem-tocell operator’s equilibrium decisions as B�

RðpM; BFÞand p�FðpM; BFÞ, which are functions of pM and BF.

To maximize profit, the femtocell operator needs toknow which users will choose femtocell service andtheir characteristics. Users with macrocell spectrumefficiency h 2 ½0; hprthÞ will choose femtocell servicefirst. Some other users may also choose femtocell ser-vice if their preferred demands are not satisfied bythe macrocell services. The following lemma, how-ever, shows that the macrocell operator will reserveenough bandwidth BM during Stage I, such that allusers who prefer to use macrocell service will be satis-fied.

LEMMA 2. At equilibrium, the macrocell operator satis-fies all preferred demands from users with h 2 ½hprth ; 1�.

Since the femtocell operator only serves users withh 2 ½0; pM=pFÞ, its profit is:

pFemtoðpF;BRÞ¼pFmin BR;N

Z pMpF

0

1

pF�1

� �dh

!�pMBR

¼min ðpF�pMÞBR;Nð1�pFÞpMpF

�pMBR

� �:

ð10ÞThe femtocell operator’s profit-maximization prob-lem is:

max0\pF � 1;BR � 0

pFemtoðpF;BRÞ

subject to BR �BF:ð11Þ

By solving Problem (11), we have the followingresult:

LEMMA 3. In Stage II, the femtocell operator’s equili-brium femtocell price is:

p�FðpM;BFÞ¼max2pM1þpM

;�NpMþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðNpMÞ2þ4NBFpM

q2BF

0@

1A:

ð12Þ

The equilibrium femtocell bandwidth purchasing amount is:

B�RðpM;BFÞ ¼ min

Nð1� p2MÞ4pM

;BF

� �; ð13Þ

which equals users’ total preferred demand for femtocellservice. As a result, hprth ¼ hth.

4.3. Stage I: Macrocell Operator’s SpectrumAllocation and Pricing DecisionsNow, we come back to Stage I, in which the macrocelloperator determines the optimal pM, BF, and BM tomaximize its profit. Note that Lemma 2 shows that it isoptimal for the macrocell operator to serve all users



doi.org/


with h 2 ½pM=p�FðpM;BFÞ; 1� by macrocell service,where p�FðpM; BFÞ is the equilibrium femtocell pricegiven in Lemma 2. This means that the macrocell oper-ator does not want users with large macrocell spectrumefficiency h to choose its competitor (i.e., the femtocelloperator). Intuitively, users with a large h demandmore bandwidth in macrocell service than in femtocellservice; thus, this leads to a larger profit for the macro-cell operator if they choose macrocell service.The macrocell operator’s profit-maximization prob-

lem is:

max0\pM � 1;BF � 0

pMacroðpM;BFÞ ¼ pMB�RðpM;BFÞ

þNpM

Z 1

pMp�FðpM ;BFÞ

1

pM� 1

h

� �dh:

subject to N

Z 1

pMp�FðpM ;BFÞ

1

pM� 1

h

� �dh�B� BF;

ð14Þ

where p�FðpM; BFÞ and B�RðpM; BFÞ are given in Equa-

tions (12) and (13), respectively. The constraint statesthat the total macrocell demand cannot exceed themacrocell service capacity. Problem (14) is complex,and the following result can help us to simplify it.

LEMMA 4. At equilibrium, we have B�F �

Nð1 � ðp�MÞ2Þ=ð4p�MÞ, which means that:

B�RðpM;BFÞ ¼ BF;

p�FðpM;BFÞ ¼�NpM þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðNpMÞ2 þ 4NBFpM

q2BF

:

Furthermore, we can show that the equilibrium macrocellprice satisfies the following:

BF þN

Z 1

p�M

p�Fðp�M

;BFÞ

1

p�M� 1

h

� �dh ¼ B:

Lemma 4 means that, at equilibrium, the macrocelloperator will not reserve excessive bandwidth for thefemtocell service, and the total spectrum demand equalsthe supply. With this, we can simplify Problem (14) to:

max0\pM�1;BF�0

pMacro¼pMB

subject to BFþN1

pM�pMþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2Mþ4BFpM=N

q2pM

0@

þlnpMþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2Mþ4BFpM=N

q2

0@

1A!

¼B;BF�Nð1�p2MÞ4pM

: ð15Þ

Next, we define y as:

yðpM;BFÞ ¼pM þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2M þ 4BFpM=N

q2

:

With further analysis (see Appendix S4), we canshow that Problem (15) can eventually be simplifiedto the following one-variable optimization problem:

max ypMacroðyÞ ¼ B � y2 � yþ 1

y� ln yþ B=N

subject to 2y� 1� y2 � yþ 1

y� ln yþ B=N� 0:

ð16Þ

Problem (16) is still a non-convex optimization pro-blem; thus, it is generally difficult to obtain the opti-mal solution in closed-form. Nevertheless, there aremany efficient numerical methods to search the opti-mal solution. Hence, in the following, we use exten-sive numerical studies to demonstrate theequilibrium capacity allocation and pricing decisions.

4.4. Numerical ResultsFigure 5 shows the macrocell operator’s equilibriumbandwidth allocation. The x-axis is the total band-width capacity divided by the user population B/N,and the y-axis is the equilibrium bandwidth alloca-tion. We also use a 1-0 binary value to denote whetheror not the macrocell operator leases sufficient spec-trum to the femtocell operator (the green curve). Wesee that when the bandwidth capacity is small, themacrocell operator would lease sufficient spectrum

(B�F ¼ ð1�p�MÞN

4p�M

) to the femtocell operator; however,

when the total bandwidth capacity is large, only a

1 2 3 4 5 60

1

2

3

4

5

6

Bandwidth capacity versus user population B/N

Mac

roce

ll op

erat

or’s

ban

d al

loca

tions Femtocell band B

F*

Macrocell band BM*

BF* = or < N(1−p

M* )/(4p

M* )

Figure 5 Equilibrium Bands B�F and B�

M as Functions of NormalizedCapacity B/N



http://blogs.wsj.com/digits/2009/12/09/att-to-new-york-and-san-francisco-were-working-on-it/


very small amount of bandwidth is allocated to thefemtocell operator and most users are served via themacrocell service (i.e., B�

F is close to zero and B�M is

almost the same as B).The intuition behind this is as follows: with large

bandwidth capacity, the macrocell operator canalready serve most users by macrocell service. Thus,the macrocell operator would lease only a very smallamount of spectrum to the femtocell to serve userswith very small h. On the other hand, when the capac-ity is small, the macrocell service price p�M is high, andthus many users, with h 2 ½0; p�MÞ, would not requestmacrocell service. By leasing enough bandwidth tothe femtocell operator, the macrocell operator canobtain a larger profit from serving more users (indi-rectly through the femtocell operator). Figure 5 showsthat it is when B/N is around 4.77 that the change ofthe macrocell operator’s leasing strategies occurs.Figure 6 shows how the femtocell and macrocell

prices p�F and p�M change in the total bandwidth capac-ity. It also shows the macrocell price of the benchmarkcase pbenchM where there is macrocell service only. First,we observe that when the bandwidth capacity B/Nbecomes large, the femtocell price p�F becomes close to1, and p�M approaches to pbenchM . This essentially meansthat most users would be served by macrocell service(except for those with very small h), which is consis-tent with the observation from Figure 5. Second, theequilibrium macrocell price p�M is always no less thanthe benchmark price pbenchM . This means that the macro-cell operator can obtain a larger profit with femtocelldeployment.Recall that hth ¼ p�M

p�Fis the equilibrium user segmen-

tation threshold, i.e., users with h 2 ½hth; 1� choose themacrocell service, and the rest choose the femtocellservice. We can show that if the macrocell operator

had full control of both the macrocell and femtocellprices, then it would prefer the segmentation thresh-

old to be ~hth ¼ 11

p�M� 1

p�Fþ1, i.e., users with

h 2 0; 11

pM� 1

p�FðpM ;BFÞ

þ1

� �choosing femtocell service, and

the rest choosing macrocell service. The differencebetween these two thresholds is due to the marketcompetition between the macrocell and femtocelloperators. Figure 7 compares these two threshold val-ues, and shows that the gap increases in the totalcapacity B/N initially, which means that the femtocelloperator gradually attracts more users from macrocellservice to femtocell service, and thus market competi-tion becomes more intense. Nevertheless, when B/Nbecomes sufficiently large, only users with very smallh are served by the femtocell service, and the competi-tion between the macrocell a nd femtocell servicebecomes much less.By summarizing the results in Figures 5–7, we have

the following observation:

OBSERVATION 1. When the macrocell operator’s totalcapacity is small, it will lease enough spectrum to thefemtocell operator and thus increase the total users servedby both operators. When the total capacity is large, themacrocell operator will only lease a very small amount ofspectrum to the femtocell operator to reduce market com-petition.

In practice, we observe that many macrocell opera-tors in big cities do not have a large enough capacityto satisfy the fast growing wireless data demand,which leads to poor quality of service (LaVallee 2009,

1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Equ

ilibr

ium

pri

ces

Femtocell price pF*

Macrocell price pM*

Macrocell price pMbench

Figure 6 p�F and p�M in Dual Services, and Benchmark Price pbenchM asFunctions of B/N

1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Use

rs’ p

artit

ion

thre

shol

ds in

dua

l ser

vice

s

Users’ finalized partition pM* /p

F*

Macrocell operator’s preferred partition

Figure 7 The Equilibrium Final Threshold hth ¼ p�M=p�F and the

Macrocell Operator’s Preferred ~hth ¼ 11p�M� 1

p�Fþ1

as Functionsof B/N



http://www.wsj.com/articles/SB123447740993079445


WNN Wi-Fi Net News 2008). One good example isthe heavy congestion observed within the AT&T net-work in New York City and San Francisco, since theiPhone was introduced to the network in 2007. Therecent “Global Mobile Data Traffic Forecast” pub-lished by Cisco in February 2013 has predicted thatthe global mobile data traffic will increase at anannual rate of 66% during the next few years, reach-ing 11.2 exabytes/month in 2017, a 13-fold increasecompared to 2012. On the other hand, the global wire-less spectrum available for macrocell networks onlygrew at an annual rate of 8% between 2007 and 2012.The spectrum will remain limited in the near futuredue to various physical constraints. This means thatB/N will become relatively small and, as a result, weexpect to see more femtocell services emerging.Next, we investigate how the introduction of femto-

cell service affects the macrocell operator’s profit, con-sumer surplus (i.e., users’ aggregate payoff), andsocial welfare (i.e., summation of the profits of bothoperators and the payoffs of all users). In each figure,we compare the dual-service case with the macrocell-service-only benchmark case. Our discussions focuson the dual-service region.Figure 8 shows the profits of the macrocell operator

when both services are provided vs. when onlymacrocell service is provided. Both profits increase inthe normalized capacity B/N. The provision of femto-cell service can substantially increase the macrocelloperator’s profit, when the capacity is relatively small.When the capacity is sufficiently large, femtocell ser-vice is only used to serve users with very small h;thus, the benefit to the macrocell operator is muchless.

Figure 9 shows that the total consumer surplusincreases with the deployment of femtocell service.This is mainly because users with low spectrum effi-ciency (small values of h) will be able to obtain higherquality service by using femtocell. Nevertheless, someconsumers could actually become worse off in thedual-service case. Figure 10 shows users’ surplus(payoff) with respect to their macrocell spectrum effi-ciency h, given the total capacity B/N = 2.1. It showsthat a large h user (e.g., h > 0.6) actually becomesworse off with the dual services. This is because themacrocell price increases under the dual service case,compared to the macrocell service only case. In

1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

Bandwidth capacity B/N

Mac

roce

ll op

erat

or’s

equ

ilibr

ium

pro

fits

Macrocell service onlyDual services

Figure 8 The Macrocell Operator’s Profits in the Macrocell ServiceOnly Case and the Dual Service Case as Functions of Capa-city B/N

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Bandwidth capacity B

Tota

l con

sum

er s

urpl

us


Figure 9 Comparison of Total Consumer Surplus Between Dual Ser-vices and Macrocell Service Only Benchmark as Functionsof B/N

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Macrocell spectrum efficiency θ

Any

use

r’s p

ayof

f


Figure 10 Comparison of a Consumer’s Payoff Between Dual Servicesand Macrocell Service Only Benchmark as Functions of h



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contrast, users with very small h benefit substantiallyfrom the dual service provision. Based on Figures 8–10, we have the following observation:

OBSERVATION 2. After introducing the femtocell service,the macrocell operator’s profit increases as more usersreceive service. Similarly, the total consumer surplusincreases, although some users’ payoffs can decrease.Overall, the total social welfare increases.

We also investigate how the decisions of the macro-cell and femtocell operators are affected by the changeof the active user numberN. The first subfigure of Fig-ure 11 demonstrates how the average number of end-users changes during 24 hours of a typical weekday,based on the measurements reported by Willkommet al. (2009) (for proprietary reasons, Willkomm et al.(2009) show the relative user (call) arrival rates of24 hours for every 5-minute interval, instead of theactual arrival rates; we approximate the arrival ratesin a weekday using a one-hour interval and assumethat the maximum arrival rate per hour is 100). Thesecond subfigure shows how the service prices changein the day as the number of users changes, based onour analysis. Recall that pbenchM is the macrocell serviceprice in the macrocell-only scenario, and p�F and p�M arethe equilibrium femtocell and macrocell prices in thedual services scenario. When N is small (between 1-6 am), the bandwidth capacity B is relatively ade-quate, in which case the macrocell operator will notlease much spectrum to the femtocell operator and p�Fis close to 1. When N becomes larger (after 7 am), allprices become non-decreasing functions of N. Also,

similar to Figure 6, the equilibrium price p�M in thedual service scenario is not smaller than the bench-mark price pbenchM . The last subfigure of Figure 11 showsthat users’ payoffs decrease with N when N is large,due to increased congestion level and higher prices.

5. Extension I: With FemtocellOperational Cost

In section 4, we consider a model in which femtocellservice does not incur any additional cost comparedwith the macrocell service. As shown in Figure 1, fem-tocell users’ traffic will first go through wirelinebroadband connection before reaching the controlcenter of the cellular network. The broadband connec-tion is owned by an Internet Service Provider (ISP).When the femtocell operator and the ISP belong to thesame entity (e.g., both belonging to AT&T) or the ISPis sharing-friendly (NetShare Speakeasy 2002), thereis no additional cost for broadband access. Otherwise,there is usually an access charge. In this section, westudy the more general case in which the ISP chargesthe femtocell operator fees for using the wirelineInternet connection in sending the femtocell users’traffic from the Internet. We examine how this opera-tional cost affects the provision of femtocell service.If the operation cost is a flat fee, then it is straight-

forward to see that the femtocell operator’s equilib-rium decisions in section 4 would remain unchangedif its revenue can compensate for the flat cost; other-wise, it would just not provide the femtocell service.In the following analysis, we assume that the totaloperational cost is linearly proportional to femtocell

5 10 15 200

20406080

Use

r po

pula

tion

5 10 15 20

0.20.40.60.8

1

Equ

ilibr

ium

pri

ces

2 4 6 8 10 12 14 16 18 20 22 24

0.51

1.52

2.5

Time t (in hours)

Use

rs’

payo

ffs

pMbench

pM*

pF*

θ=0 (femtocell user)

θ=0.5 (macrocell or femtocell)

θ=1 (macrocell user)

Figure 11 Changes of User Population N, Equilibrium Prices (pbenchM in Macrocell Only Service, and p�F and p�M in Dual Services), and RepresentativeUsers’ Payoffs Throughout a Weekday



doi.org/


bandwidth with a coefficient C. This is motivated bythe fact that femtocell traffic uses ISP’s broadbandresource, and many ISPs have adopted usage-basedpricing (Deleon 2011). Recall that in section 4 we haveshown that a femtocell user’s data rate and traffic vol-ume are both linear in the demand for bandwidth;thus, femtocell users’ aggregate traffic volume is lin-ear in their total bandwidth demand (which equalsthe femtocell capacity).We will focus on the case of C 2 (0, 1) in this sec-

tion, since if C ≥ 1, we can show that the femtocelloperator will charge a femtocell price pF [ C � 1,and no user will choose the femtocell service. Thethree-stage decision process is similar to that shownin Figure 4. The analysis of Stage III is the same as insubsection 4.3, hence, we focus on Stages II and I inthe following analysis. In this section we assume thatthe macrocell operator charges the same price to boththe macrocell users and the femtocell operator. Theanalysis for the differentiated pricing setting is pro-vided in Appendix S7.In Stage II, the femtocell operator determines BR

and pF to maximize its profit. We still use B�RðpM; BFÞ

and p�FðpM; BFÞ to denote the equilibrium decisions ofthe femtocell operator. Without loss of generality, wenormalize the value of N to 1. By following a similaranalysis as of Lemma 2, we can obtain the followingresult:

LEMMA 5. At equilibrium, the macrocell operator willsatisfy all preferred demands from users withh 2 ½hprth ; 1�.

Accordingly, the femtocell operator’s profit-maxi-mization problem is:

maxpF � 0;BR � 0

pFemtoðpF;BRÞ

¼ ðpF � CÞmin BR;

Z pMpF

0

1

pF� 1

� �dh

!� pMBR

subject to BR �BF;

pM þ C� pF � 1;

ð17Þ

where the second constraint means that the femto-cell price pF should at least cover the total cost(pM þ C) for the femtocell operator. Solving Problem(17), we have the following result:


p�FðpM;BFÞ¼max2

1þ 1pMþC

;�pMþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpMÞ2þ4BFpM

q2BF

0@

1A;

ð18Þ

and its equilibrium femtocell bandwidth purchase is:

B�RðpM;BFÞ ¼ min pM

1ðpMþCÞ2 � 1

4;BF

!; ð19Þ

which equals users’ total preferred demand for femtocellservice. As a result, hth ¼ hprth.

We now study Stage I, in which the macrocell oper-ator determines pM, BF, and BM to maximize its profit.The equilibrium decisions are denoted as p�M, B�

F, andB�M. Lemma 5 states that the macrocell operator

should serve all users with h 2 pMp�FðpM;BFÞ ; 1

h iby the

macrocell service. The macrocell operator’s profit-maximization problem is:

maxBF � 0;pM

pMacroðpM;BFÞ ¼ pMB�RðBF; pMÞ

þ pM

Z 1

pMp�FðBF ;pMÞ

1

pM� 1

h

� �dh

subject to 0�BF þZ 1

pMp�FðBF ;pMÞ

1

pM� 1

h

� �dh�B;

0\pM � 1� C;

ð20Þ

where B�RðBF; pMÞ and p�FðBF; pMÞ are given in Equa-

tions (18) and (19), respectively. The second con-straint shows that the total cost for the femtocelloperator, C þ pM, should be less than 1; otherwise,no user will request the femtocell service.

5.1. Numerical ResultsProblem (20) is not convex and is difficult to solve inclosed-form, but can be solved easily using numericalmethods. We are interested in investigating how thecost C affects the equilibrium capacity and pricingdecisions.We have shown in section 4 that, when the total

spectrum capacity B is small, the amount of spectrumused for the macrocell and the femtocell services arecomparable and both increase as B increases. How-ever, when the macrocell’s spectrum capacity reachesa certain threshold, the amount of spectrum leased tothe femtocell operator drops dramatically. We definethe corresponding value of spectrum capacity whenthis change occurs as the boundary between the lowcapacity and high capacity regimes.Figure 12 shows how this boundary value changes

with the femtocell operational cost. For example, theboundary value is 4.77 when C = 0 (also see Figure 5),and it becomes 3.58 when C = 0.1, and becomes 5.22when C = 0.4. When C increases, the femtocell pricep�F increases and the demand for femtocell servicedecreases. This makes it less attractive to provide fem-tocell service. On the other hand, the increase of price



doi.org/


p�F also reduces the market competition, which makesthe macrocell operator more willing to lease spectrumto the femtocell operator. The interactions of thesetwo factors determine the boundary of the two capac-ity regimes. More specifically, with a cost C ≤ 0.12,the decrease of femtocell demands dominates,and the boundary decreases. With a larger costC > 0.12, the decrease of competition dominates, andthe boundary increases.Figure 13 illustrates that, as C increases, the gap

between the final threshold hth ¼ p�Mp�Fand the thresh-

old ~hth ¼ 11

p�M� 1

p�Fþ1

that the macrocell operator prefers

decreases, which means less service competition. Fig-ures 14 and 15 show that the profits of the macrocelloperator and the femtocell operator both decrease asthe cost C increases. When C is smaller, the macrocelloperator benefits more from enabling femtocell ser-vice; as C becomes larger, the benefit decreases andeventually the femtocell service becomes no longereconomical and only the macrocell service is offered.Furthermore, for the macrocell operator, the benefit ofhaving femtocell service is larger when the band-width capacity B is smaller, since under such scenar-ios the femtocell service can help the macrocelloperator serve more users without introducing toomuch competition. This is consistent with what weobserved when the cost C is zero.

6. Extension II: Limited FemtocellCoverage

In section 4, we assume that femtocell service has thesame ubiquitous coverage as the macrocell service. Inthis section, we look at the general case in which thefemtocell service only covers g 2 (0, 1) portion ofthe user population, and 1 � g portion of users can

only access the macrocell service. We call the g frac-tion users overlapped users, and the rest 1 � g non-over-lapped users. We are interested in understanding howthe limited coverage affects the provision of femtocellservice. In this setting, users only have macrocell ser-vice when they are out of femtocell coverage; how-ever, they can still choose between the femtocell andmacrocell services within the femtocell coverage.Thus, users’ segmentation is characterized by macro-cell spectrum efficiency h and femtocell coverage g,and their decisions within the femtocell coverageareas are the same as those in section 4.The three-stage decision process is similar to that

depicted in Figure 4. The analysis of Stage III is thesame as subsection 4.3. Without loss of generality,we normalize the user population N to 1. For Stage II,

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.43.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

Cost C

Bou

ndar

y be

twee

n tw

o ca

paci

ty r

egim

es

high capacity regime

low capacity regime

Figure 12 The Boundary Between Low and High Capacity RegimesChanges with Femtocell Operational Cost C

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

Cost C

Diff

bet

wee

n p M*

/ p F* a

nd 1

/ (1/

pM*

− 1

/ pF* +1

)

B=1.1B=2.1

Figure 13 The Equilibrium Threshold hth ¼ p�Mp�Fand Macrocell Operator’s

Preferred Threshold ~hth ¼ 11p�M� 1

p�Fþ1

with Respect to C

0 0.05 0.1 0.15 0.2 0.25 0.30.3

0.35

0.4

0.45

0.5

0.55

0.6

Cost C

Mac

roce

ll op

erat

or’s

equ

ilibr

ium

pro

fit

B=1: Dual servicesB=1: Macrocell onlyB=2: Dual servicesB=2: Macrocell onlyB=3: Dual servicesB=3: Macrocell only

Figure 14 The Macrocell Operator’s Equilibrium Profit as a Functionof Cost C and Capacity B



doi.org/


following a similar analysis as of Lemma 2, we canalso conclude that the overlapped users withh 2 ½hprth ; 1� will be served by macrocell service, andthe other overlapped users will be served by femtocellservice, that is, hth ¼ hprth . We derive the followingresult similar to Lemma 3:


p�FðpM;BFÞ¼max2

1pMþ1

;�pMgþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpMgÞ2þ4pMBFg

q2BF

0@

1A;

ð21Þ

and its leased bandwidth from the macrocell operator is:

B�RðpM;BFÞ ¼ min gpM

1ðpMÞ2 � 1

4

!;BF

!; ð22Þ

which equals overlapped users’ total preferred demand infemtocell service.

Back to Stage I, the macrocell operator’s profit-max-imization problem is:

max0\pM�1;BF�0

pMacroðpM;BFÞ¼pMB�RðpM;BFÞ

þgpM

Z 1

pMp�FðpM ;BFÞ

1

pM�1

h

� �dhþð1�gÞpM

Z 1

pM

1

pM�1

h

� �dh;

subject to, BFþgZ 1

pMp�FðpM ;BFÞ

1

pM�1

h

� �dh

þð1�gÞZ 1

pM

1

pM�1

h

� �dh�B;

ð23Þ

where p�FðpM; BFÞ and B�RðpM; BFÞ are respectively

given in Equations (21) and (22).

6.1. Numerical ResultsFigure 16 shows that as g increases, it is more attrac-tive to provide femtocell service, and the equilibriumfemtocell (macrocell) band B�

F (B�M) increases (de-

creases). Yet both prices p�F and p�M increase in g. Theintuition is, as g increases, more users are served andthe total femtocell demand increases, which leads to alarger p�F. The overall wireless service (macrocell plusfemtocell) becomes more efficient, and the total userdemand of both services increases, which leads to alarger p�M. Since the macrocell operator can sell itstotal capacity at a higher price, its profit increases ing. Fig 17 shows that the total consumer surplus in the

0 0.05 0.1 0.15 0.2 0.25 0.30.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Cost C

Fem

toce

ll op

erat

or’s

equ

ilibr

ium

pro

fit B=1B=2B=3

Figure 15 The Femtocell Operator’s Equilibrium Profit as a Functionof Cost C and Capacity B

0.2 0.4 0.6 0.8 10.35

0.4

0.45

0.5

0.55

Femtocell coverage η

p F∗ B=1.1B=2.1

0.2 0.4 0.6 0.8 10.2

0.25

0.3

0.35


p M*

B=1.1B=2.1

Figure 16 Equilibrium Prices p�F and p�M as Functions of g and B

0.2 0.4 0.6 0.80.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

0.2


Tota

l con

sum

er s

urpl

us


Figure 17 Comparison of Total Consumer Surplus Between DualServices and Macrocell Benchmark Under B = 1.1



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dual-service case increases in g, compared to themacrocell service only benchmark case.

OBSERVATION 3. As femtocell coverage expands, theoverall wireless service (macrocell plus femtocell) becomesmore efficient. The macrocell operator’s profit, totalconsumer surplus, and social welfare increase in g.

7. Conclusion

This paper studies the economic incentives for amacrocell operator to deploy new femtocell service inaddition to its existing macrocell service, by leasingspectrum to an independent femtocell operator. Wemodel the interactions among the macrocell operator,the femtocell operator, and the end-users as a three-stage dynamic game.We characterize the equilibrium decisions for the

end-users’ choices between the macrocell and thefemtocell services, as well as the capacity allocationand pricing decisions for the macrocell and femtocelloperators. Our analysis shows that the macrocelloperator has more incentive to enable both macrocelland femtocell services when its total bandwidth is rel-atively tight, since femtocell service can help servemore users with small macrocell spectrum efficiencywithout introducing too much market competition.However, when the total bandwidth is large, femto-cell service becomes a severe competitor to macrocellservice, and thus the macrocell operator has lessincentive to lease its bandwidth to the femtocell oper-ator. In general, the introduction of femtocell servicecan increase the macrocell’s profit, the total consumersurplus, and social welfare, although some macrocellusers may experience a payoff decrease due to fewerresources and higher price. We also extend the mainresults to the more general cases, considering theadditional femtocell operational cost and the limitedfemtocell coverage.Our study represents an initial step to better under-

stand the efficient management of new wireless ser-vices. There are several interesting directions forfuture research:

• We can further consider the “shared carriers”scheme besides “separate carriers,” wherefemtocell service and macrocell service sharepart of or the whole spectrum. We can alsoconsider frequency spectrum reuse, in whichmultiple femtocells can reuse the same spec-trum if they do not overlap with each other.We need to optimize the pricing and spec-trum allocation decisions by trading off theincreased spectrum efficiency and mutualinterferences between macrocell and femtocellservices.

• We may study the case where both the macro-cell and the femtocell services are provided byan integrated service provider, for example, amacrocell operator directly provides bothmacrocell and femtocell services to users, andit fully controls bandwidth resource allocationand service prices. We can also study the moregeneral case of an oligopoly market, in whichmultiple macrocell/femtocell service providerscompete with each other. Intuitively, we canenvision that the macrocell and femtocellprices would go down due to new horizontalcompetitions. However, the technical analysiswould be much more complex in the oligopolycase.

• We can consider the more general scenario inwhich a user travels through multiple loca-tions over a long time horizon (e.g., over ayear’s time). If the macrocell and femtocelloperators’ prices are fixed at different loca-tions, then a user’s subscription decision inthis general scenario can be decomposed intoseveral static decision problems, as describedin this study. However, if prices are dynamic,then it requests a new, comprehensive analy-sis to determine the optimal bandwidth allo-cation and pricing strategies, consideringusers’ mobility patterns and spectrum efficien-cies, as well as femtocell’s deployment loca-tions.

Acknowledgments

The authors thank Prof. Chris Tang, the Senior Editorand the referees for their valuable comments and sugges-tions which substantially improved the study. Theauthors also thank Dr. Lin Gao for his helpful discus-sions. This work is supported by the General ResearchFunds (Project Number CityU 149812; CUHK 412511 and412713) established under the University Grant Commit-tee of the Hong Kong Special Administrative Region,China. It is also supported by SUTD-MIT InternationalDesign Center (IDC) Grant (Project Number IDS-F1200106OH).

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Supporting InformationAdditional Supporting Information may be found in theonline version of this article:

Appendix S1: Proof of Lemma 2Appendix S2: Proof of Lemma 3Appendix S3: Proof of Lemma 4Appendix S4: Further Anaylsis and Simplification of

Problem (15) of the Main PaperAppendix S5: Proof of Lemma 6Appendix S6: Extension of Section 4 Results to Differen-

tiated Pricing CaseAppendix S7: Differential Pricing Case with Femtocell

Operational CostAppendix S8: Proof of Lemma 2 with Differentiated

PricingAppendix S9: Femtocell Service Provision in a Competitive

Market







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