1

Spectrum and Infrastructure Sharing in MillimeterWave Cellular Networks: An Economic Perspective

Fraida Fund, Shahram Shahsavari, Shivendra S. Panwar, Elza Erkip, Sundeep Rangan

Abstract—The licensing model for millimeter wave bands hasbeen the subject of considerable debate, with some industryplayers advocating for unlicensed use and others for traditionalgeographic area exclusive use licenses. Meanwhile, the massivebandwidth, highly directional antennas, high penetration lossand susceptibility to shadowing in these bands suggest certainadvantages to spectrum and infrastructure sharing. However,even when sharing is technically beneficial (as recent researchin this area suggests that it is), it may not be profitable. In thispaper, both the technical and economic implications of resourcesharing in millimeter wave networks are studied. Millimeterwave service is considered in the economic framework of anetwork good, where consumers’ utility depends on the size ofthe network, and the strategic decisions of consumers and serviceproviders are connected to detailed network simulations. Theresults suggest that “open” deployments of neutral small cellsthat serve subscribers of any service provider encourage marketentry by making it easier for networks to reach critical mass,more than “open” (unlicensed) spectrum would. The conditionsunder which competitive service providers would prefer to shareresources or not are also described.

I. INTRODUCTION

The millimeter wave (mmWave) bands represent one ofthe largest unlicensed bandwidths ever allocated, presentinga tremendous opportunity for both technical and policy inno-vation. The appropriate licensing model for this band remainsthe subject of considerable debate. Replies to an FCC noticeof inquiry [1] requesting comments on usage of bands greaterthan 24 GHz in the United States reveal disagreement on howto best utilize this spectrum, with economic considerationsplaying a significant role. Major industry players argued infavor of exclusive use licensing on a geographic service areabasis, primarily on the grounds that this offers sufficientcertainty to motivate major capital investment. Several of theseexplicitly asked the FCC to reject licensing mechanisms thatrequire spectrum sharing on some bands [2]–[8]. Others arguedthat unlicensed use maximizes efficient spectrum use, andencourages innovation and competition by lowering barriers toentry [9], [10]. A recent notice of proposed rulemaking [11]for these bands involves 3,850 MHz of spectrum, but doesnot move on an additional 12,500 MHz of potentially usefulspectrum in bands above 24 GHz.

Beyond these business concerns, technical properties ofmmWave bands favor spectrum and infrastructure (base sta-tion) sharing. While cellular frequencies have traditionallybeen allocated with geographic area exclusive use licenses,the physical characteristics of mmWave signals suggest that

All of the authors are with the Department of Electrical and ComputerEngineering, NYU Tandon School of Engineering.

exclusive use licenses would be sub-optimal in these bands.Specifically, in the mmWave space, the massive bandwidthand spatial degrees of freedom are unlikely to be fully usedby any one cellular operator. The use of high-dimensionalantenna arrays implies that spectrum can be shared, not justin time, but also in space. Furthermore, mmWave signalsobserve high penetration loss through brick and glass [12],and are highly susceptible to shadowing. This implies thatmany more base stations are likely to be needed for wide areacoverage, significantly increasing the cost of deployment, thusmotivating infrastructure sharing.

However, technological justification for resouce sharingdoes not always translate to economic benefits, for serviceproviders or for consumers. Network service providers aremainly concerned with increasing profit, which is a function ofdemand, price, and cost. Even when resource sharing improvesconsumers’ quality of service, it may have a negative effect onthe service provider’s profits if it shifts demand to a competingservice provider, or if it changes the market dynamics in away that forces down the price. Similarly, consumers prefera higher quality of service, but they are also concernedwith service availability and price, which could potentiallybe negatively affected by resource sharing. To gain a fullerunderstanding of the benefits of resource sharing in mmWavenetworks, we need to identify the specific impact on qualityof service, and then understand how this affects the demand,price, and cost of service.

A. Contributions

The goal of this work is to model the strategic decisions ofwireless service providers in building out mmWave networkswith or without sharing of spectrum and/or infrastructure. Weapply economic models of network goods [13] - productswhose value to consumers depends on the number of unitssold - to mmWave cellular networks, where the value of thenetwork to the consumer depends on the size of the network(in terms of base station and spectrum resources), and theinvestment of the service provider in base station and spectrumresources depends on its expected market share. We quantifythe positive and negative effects associated with increasingnetwork size, i.e., how a subscriber’s data rate changes asthe mmWave service provider increases its spectrum holdings,base station deployments, and market share. Using the conceptof critical mass [14], we investigate the growth of demandfor a mmWave network service under three circumstances:increasing a network from zero size by deploying base stationsand licensing spectrum, licensing spectrum but utilizing an

arX

iv:1

605.

0460

2v1

[cs

.NI]

15

May

201

6

2

existing deployment of “open” small cells, and deployingbase stations but using unlicensed spectrum. We further modelthe resource sharing decision of competing mmWave serviceproviders as a compatibility problem [15], [16].

The contributions of this work are primarily to connect theperformance of mmWave networks to economic models ofdemand for these network services, as follows:• We quantify the positive and negative effects of subscrib-

ing to a large service provider - one with greater basestation density, bandwidth, and number of subscribers -as well as interactions between these. We find a strongpositive effect with increasing base station density in alow-SNR regime, but only a weak positive effect withincreasing bandwidth. In high-SNR regimes, there is onlya weak positive effect with increasing base station density,and this is partially mitigated by a very slight negativeinterference effect. Moderate-SNR subscribers gain themost from increasing bandwidth and base station density.

• Given these network effects, we consider the effect of“open” resources on new service offerings, where theprimary concern of the provider is to establish a stablepresence in the market. We find that the slow initialgrowth of demand at small network sizes makes it dif-ficult for a new provider to reach critical mass. With anexisting deployment of “open” small cells, there is robustdemand even at small network sizes, which encouragesgrowth. Open spectrum (i.e., unlicensed) does not haveas encouraging an effect on market entry.

• We separately consider resource sharing betweenmmWave service providers who are already establishedin the market, and are mainly concerned with profit. Weapply the economic model of compatibility of networkgoods to resource sharing in established mmWave cellularnetworks. We describe a duopoly game involving twovertically differentiated mmWave service providers withand without resource sharing, and quantify the serviceprovider profits and market coverage in each case. Wefind that providers may prefer to share spectrum andbase stations when the market is highly segmented, sharebase stations when the spectrum is unlicensed, and sharespectrum when base stations are all “open”. Otherwisethe high-end service provider will prefer not to shareresources.

B. Related Work

The idea of resource sharing is, of course, not new; agreat deal of research effort has been devoted to quantifyingthe benefits of base station and spectrum sharing in cellularnetworks. In [17], the authors consider several sharing optionsfor LTE networks, and conclude that an arrangment similar toa traditional roaming agreement offers the best performancewith the least complexity for inter-operator sharing. The au-thors in [18] assess the benefit of sharing both infrastructureand spectrum in the context of a proposed merger betweentwo major cellular operators in the United States, using realbase station deployment data to support their claims. In [19]the authors investigate the trade-offs between infrastructure

sharing (which improves coverage and has a small positiveeffect on data rate) and spectrum sharing (which has a positiveeffect on data rate but reduces coverage probability), and findthat combining both kinds of sharing offers the best data ratewhile partially mitigating the reduced coverage of spectrumsharing. All of these, as well as others [20], [21], conclude thatunder some conditions, resource sharing increases the capacityof traditional cellular networks, but some find that spectrumsharing without coordination in traditional cellular networkscreates interference and degrades performance relative to ex-clusive use of spectrum by one operator.

Given the unique propagation characteristics of mmWavenetworks, there has been renewed interest in resource sharingin these bands. mmWave networks will require a denserdeployment of base stations than conventional frequencies,increasing the appeal of base station sharing. Industry per-spectives on 5G cellular networks [22] suggest a favorableview of neutral small cells owned by a third party and sharedby multiple operators. With respect to spectrum sharing, asopposed to conventional cellular frequencies, the inter-cellinterference in mmWave bands can be controlled by directionaltransmissions [23]–[25], potentially allowing much greaterspectrum reuse. In [23], it is shown that with sufficient beamdirectionality in the transmission pattern, the inter-cell inter-ference is low enough to favor resource sharing even withoutinter-operator coordination. These conclusions are supportedby [26], [27] for different channel models. However, in [25],the authors show that inter-operator coordination is importantto users with poor data rates, especially in dense deployments.Similarly, [28] proposes a spectrum sharing scheme with inter-cell coordination to avoid inter-cell interference and increasesharing gain.

With the exception of [19], all of these have consid-ered resource sharing only in the case of symmetric serviceproviders (those with equivalent spectrum and base stationresources, and equal market share). On conventional cellularfrequencies, [19] shows that resource sharing benefits a largeservice provider more than a small service provider, becauseless interference is imposed by the small service provider.However, in mmWave frequencies, where the effect of intercellinterference is much smaller, it is not clear whether this resultremains relevant.

Furthermore, even when full sharing is strictly beneficialfrom a technical perspective, competitive dynamics betweenservice providers may discourage sharing unless there areexternal incentives. With full resource sharing, subscribersof all service providers have exactly the same quality ofservice, making it difficult for service providers to distinguishthemselves in the market and gain market share. None of [19],[23], [25], [26] consider this effect. In [23], the authors claimthat with resource sharing, a network operator requires lessbandwidth (and therefore, lower spectrum licensing costs) toserve its subscribers with a given median rate. However, thisassumes that demand for network services is fixed, then itactually varies according to the quality of service, and it alsoignores competition between service providers. The early workon mmWave resource sharing also does not address the case ofasymmetric service providers, where the large service provider

3

contributes more resources to the partnership, but then offersits subscribers the same quality of service as the small serviceprovider.

Some of the literature on cellular networks addresses eco-nomic or regulatory aspects of resource sharing. For exam-ple, [29] models the tradeoff associated with competition reg-ulation and resource sharing in the context of the planned evo-lution of cellular networks. A coalition game described in [30]suggests that resource sharing and cooperation can sometimesimprove individual cellular service providers’ payoff. A ten-year case study on cellular service providers in Sweden [31]lists incentives, obstacles, and key drivers for cooperation,and a similar investigation of cellular infrastructure sharingin emerging markets is described in [32]. However, none ofthese address resource sharing in mmWave networks which,as mentioned above, are fundamentally different from previouscellular networks in ways that can affect the decision to shareresources or not. An early economic perspective on mmWavenetworks (although not on resource sharing) in [33] suggeststhat the limited coverage range of mmWave-based 5G systemsis a key challenge for its cost efficiency. Resource sharingcould potentially be a way to address this challenge, butthe economic implications of resource sharing in mmWavenetworks have not been studied yet.

C. Paper Organization

The rest of this paper is organized as follows. We beginwith a brief introduction to the economic framework used inthis paper, in Section II. In Section III, we describe the systemmodel and simulation results showing the benefit of resourcesharing in mmWave networks with respect to fifth percentilerate. Section IV describes mmWave service as a network good(in the economic sense), and uses simulation results to quantifythe network externalities associated with increasing networksize. We build on results from Section III and Section IV inSection V to show how demand for mmWave network servicesevolves as a service provider increases its network size, andwe compare the likelihood of market entry with and without“open” resources such as unlicensed spectrum or an opendeployment of neutral small cells. In Section VI, we describe aduopoly game involving two vertically differentiated mmWavenetwork service providers, and compare their profits with andwithout resource sharing, for simultaneous market entry andsequential market entry. Finally, in Section VII, we concludewith a discussion of the implications of this work and areasof further research.

II. ECONOMIC FOUNDATIONS

We briefly summarize here the economic framework usedin the rest of this paper. We define a network good and showhow its demand is fundamentally different from demand fornon-network goods, give equilbria and conditions for reachingcritical mass in a market for a network good, explain theconcept of compatibility, and describe a model of verticallydifferentiated network good. For a more detailed overview ofthis area of economics, see [13].

A. Network goods

In economics, a network good or service [13] is a productfor which the utility that a consumer gains from the productvaries with the number of other consumers of the product(the size of the network). This effect on utility - which iscalled the network externality or the network effect - may bedirect or indirect. The classic example of a direct networkeffect is the telephone network, which is more valuable whenthe service has more subscribers. The classic example of anindirect effect is the hardware-software model, e.g. a consumerwho purchases an Android smartphone will benefit if otherconsumers also purchase Android smartphones, because thiswill incentivize the development of new and varied applica-tions for the Android platform. The network externality mayalso be negative, for example, if an Internet service providerbecomes oversubscribed, its subscribers will suffer from thecongestion externality.

1.0

0.8

0.6

0.4

0.2

n=0

n' n''

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

n (network size, i.e. demand)

p (

pri

ce)

Fig. 1. An example of demand curves for a series of expected network sizes(dashed lines, each labeled with expected network size ne), and the fulfilledexpectations demand curve p(n;n) (solid line), which is the collection ofpoints where the demand curve for an expected network of size ne intersectsthe vertical line at n = ne. Three equilibria for a perfectly competitive marketwith marginal cost c = 0.1 are labeled in white text on a dark background,one at n = 0 and two at the intersesctions of p(n;n) and p = c (dottedline).

A fundamental difference between a network good and onewith no network effects is the behavior of the demand curve,which describes the relationship between two key quantities:consumer demand for a good, n (or equivalently, the numberof units of the good that are sold i.e., network size), and theprice of the good, p.

Consider a set of consumers in a market for a non-networkgood. The total demand for the good, n, is normalized so thatn = 1 when all consumers purchase the good, and n = 0 whenno consumers purchase the good. A consumers’ willingnessto pay for the good is ω, with ω varying among the set ofconsumers up to ω (different consumers are willing to paydifferent prices for an identical good). A consumer of type ωis indifferent between purchasing the good or not when pricep = ω. For p > ω, none of the consumers will purchase thegood (n = 0), because it is too expensive even for consumerswith the highest value of ω. At p = 0, all of the consumers

4

will purchase the good (n = 1). The demand curve, whichindicates what portion of the consumers will purchase the goodat a given price, has a negative slope for most kinds of non-network goods, because the quantity demanded n (typicallyshown on the horizontal axis) increases as the price of the goodp (typically shown on the vertical axis) decreases. Conversely,to increase demand for a typical non-network good, a producerof the good must reduce its price. The dashed line labeled“1.0” in Fig. 1 shows a sample demand curve when ω isdistributed uniformly on the interval [0, 1].

Now let us consider a network good. We still assumeheterogenous consumers, with type ω up to ω, but a consumerof type ω has willingness to pay ωh(n), where n is the networksize, and h(n) is a network externalities function indicatinghow consumer utility scales with n. For a positive networkexternality, h(n) increases with n, and for a negative networkexternality h(n) decreases with n. A consumer purchasing thegood at price p gains utility u(ω, n, p) = ωh(n)− p.

Under these circumstances, a consumers’ decision to pur-chase the good or not depends on how many units of the goodthey expect will be sold, i.e., the expected network size ne.A consumer of type ω is indifferent between purchasing thegood or not when p = ωh(ne). We can draw a demand curvefor any expected network size ne. For p > ωh(ne), none ofthe consumers will purchase the good (n = 0). For p = 0,all will purchase the good (n = 1). At the point where thedemand curve intersects the vertical line at n = ne, i.e., whenthe demand for the good at a given price equals the expectednetwork size, we say that consumers’ expectations are fulfilled.

We can construct a series of such demand curves p(n;ne)for different values of ne. Each curve gives the willingnessto pay of the nth consumer when the expected size of thenetwork is ne. The collection of points at n = ne, wherethe actual size of the network and consumers’ expectationsregarding the size of the network are the same, then make upthe fulfilled expectations demand curve, p(n;n). Fig. 1 showsthe fulfilled expectations demand curve p(n;n) and demandcurves for selected values of ne when h(n) = n and ω isdistributed uniformly on the interval [0, 1].

This fulfilled expectations demand curve gives the size ofthe network that could be supported at equilibrium for a givenprice, in the same way that the demand curve for a typicalnon-network good defines the demand that can be supportedat a given price. We notice two key differences between thebehavior of the demand curve for a network good and a non-network good.

First, the demand for a network good depends on theconsumers’ expected utility, which in turn depends on theexpected network size ne. However, the expected networksize depends on consumer demand, creating a self-fulfillingexpectation. When many consumers expect the good to beunpopular, they will not purchase the good, and the networksize will be small.

Second, we note that although a traditional demand curvealways slopes down, the fulfilled expectations demand curvefirst increases with n, then decreases. That is, for goods witha positive network externality, p(n;n) first increases withn due to the network externality, but eventually begins to

slope downward, as it becomes increasingly difficult to findcustomers who have not yet purchased the good, but havea high enough willingness to pay for it. For a traditionalgood, to gain market share a producer has to reduce the price.For a network good, a producer can sometimes demand ahigher price as more units of the good are sold, because theconsumers’ utility increases with the number of units sold.

B. Equilbria and critical mass

An important feature of a fulfilled expectations demandcurve is how it relates to the size of a network at equilibrium,and particularly, how it relates to the critical mass of thenetwork: the smallest network size that can be sustained inequilibrium [14].

In a monopoly, the producer of the good has no competitionto drive down the price. The fulfilled expectations demandcurve p(n;n) defines the price that the producer can charge tosustain demand of size n, earning total revenue of np(n;n).The producer will choose the network size that maximizes itsprofits π(n, p, c) = n(p(n;n) − c), where c is the marginalcost, i.e., the cost to the producer of providing one unit of thegood [14].

In a perfectly competitive market, with many producers of-fering goods that are perfect substitutes, competing producerswill drive down the price until it is equal to marginal cost, i.e.p(n;n) = c, yielding three possible equilibria when c is lessthan the maximum of p(n;n):

1) one stable equilibrium at n = 0 (representing a zero sizenetwork),

2) an unstable equilibrium for a network of size n′ at thefirst (smaller n) intersection of the fulfilled expectationsdemand curve p(n;n) with the horizontal line p = c,and

3) a stable equilibrium for a network of size n′′ at the sec-ond (larger n) intersection of p(n;n) and the horizontalp = c.

These are illustrated in Fig. 1. When the line p = c intersectsp(n;n) once, at its maximum value, there is a stable equilib-rium at that point and one at n = 0. When c is greater thanthe maximum value of p(n;n), the producers would only beable to sell the good at a loss, so the only equilibrium will beat n = 0, and no producer will offer the good [14].

It is shown in [14] that under perfect competition, the criticalmass is equal to the network size n0 at which p(n;n) ismaximized. At this point, competing pressures on demandare perfectly balanced. For network sizes between zero andn′, there is “downward pressure” toward the first equilibriumat n = 0, since there are not enough consumers willing topay for the good at the lowest price at which the produceris prepared to offer it. When n′ < n < n′′, there are moreconsumers willing to pay price c, and the service increasesin value as more units are sold, exerting “upward pressure”on the demand toward the equilibrium at n′′. For n > n′′,there is again “downward pressure” on the demand toward n′′

because producers are trying to sell the good to the part of thepopulation with a low willingness to pay.

5

Because of these pressures on demand, n′′ has a strongstability property, and n′ is highly unstable. A producerentering a new market is interested in selling enough units at asmall network size for the network to grow to at least n′, sincebeyond that “tipping point” the upward pressure on demandhelps the network reach its non-zero stable equilibrium. Theslope of the fulfilled expectations demand curve p(n;n) forsmall network sizes is very important, since it describes howeasy it is for the network size to reach critical mass. Whenthis slope is large, then for a given value of c, n′ occurs at asmaller network size, making it easier to reach critical massand from there, the stable equilibrium at n′′.

C. Compatibility

In the previous section, we model a consumer’s willingnessto pay for good i as ωh(ni), where h(ni) is the networkexternalities function and ni is the number of consumers whohave purchased the good. In this model, h(ni) is not affectedby the number of units sold of any other good. Now weconsider a market where producers may choose to make theirgoods compatible [15], [16]. When two network goods arecompatible, then the total network effect for a consumer ofeither good is based on the sum network size of both goods,so the network externalities function for good i is evaluatedusing the total network size for all the goods: h(

∑j∈I nj),

where I is a set of firms producing compatible goods andi ∈ I .

A firm producing a network good has conflicting incentivesfor and against compatibility:• Positive network externalities: A firm that makes its

product compatible increases its value to consumers,since the argument to h(·) is greater.

• Market power: A firm that chooses to make its product in-compatible reduces the value of its competitors’ product,so it avoids losing market share, and can charge higherprices.

Compatibility is often used to model a firm’s choice to use aproprietary technical standard or a common industry standard.For example, the developers of a word processing applicationmight choose to use a proprietary file format so that allconsumers who need to open these files must purchase theirsoftware, or they might choose to use an open standard so thattheir users can share the files produced with their software withusers of other word processors.

D. Vertical differentiation

In our previous model, when two network goods are com-patible, consumers prefer them equally, since the value of thenetwork externalities function is the same for both. This canshift demand from one producer to another. A producer maytry to disinguish itself from competitors by improving thevalue of its good in other ways (not by increasing n). Forexample, consumers of Android-based smartphones benefitfrom the network effects due to consumers of all Android-compatible smartphones. However, a firm that produces An-droid phones can distinguish itself in the market by selling

handsets with better hardware specifications than its competi-tors’.

In Section VI of this paper, where we consider serviceproviders that are already well established in the market, weuse a model of consumer utility described in [34] whichincludes vertical differentiation. In this model, a consumer’swillingness to pay for good i is ωqi + qih(

∑j∈I nj), where

I is a set of firms producing compatible goods, i ∈ I , andqi is a scaling factor that represents aspects of the good’squality that are not a function of the network size. Firms thatproduce compatible goods can distinguish themselves fromone another by choosing different quality levels. However, afirm that chooses to produce a higher-quality good also hashigher marginal costs. Where c was a constant marginal costin previous sections, now we scale marginal cost according tothe quality level, so that the cost to the producer of producingone unit of good i at quality level qi is qi, and its profits areπi(qi, ni, pi) = nipi − qini.

E. Application to mmWave network service

Given this economic framework, we are interested in mod-eling mmWave network service as a network good, to betterunderstand:• What kind of network effects apply to mmWave network

service? What is the relative benefit to a consumer ofsubscribing to a large service provider? (Section IV)

• What is the behavior of the fulfilled expectations demandcurve for mmWave network service? How difficult is itfor a service provider just entering the market to reachcritical mass? Do open resources help a service providerreach critical mass? (Section V)

• Under what conditions will mmWave network serviceproviders want to share resources? Is it desirable for aregulator to enforce resource sharing? (Section VI)

To answer these questions, we must connect the economicmodels described in this section to an accurate technical modelof mmWave cellular systems, described in Section III.

III. TECHNICAL BENEFITS OF RESOURCE SHARING

In this section, we describe the system model of themmWave network used in the rest of the paper. We outlinethe technical benefits of resource sharing between identicalservice providers, confirming some of the results of [23], [25]–[27]. We also describe the sharing gains achieved by serviceproviders that are asymmetric with respect to number ofsubscribers, spectrum holdings, and base station deployments.The simulations of this section will be used in Section IV andSection V to devise an economic model for mmWave resourcesharing.

A. mmWave System Model

We consider a system with multiple mmWave networkservice providers (NSPs) operating in the 73 GHz band. Aservice provider i ∈ {1, . . . , I} has bandwidth Wi, a setof base stations (BSs) distributed in the network area usinga homogeneous Poisson Point Process (hPPP) with intensity

6

λBi , and a set of user equipment (UEs) whose locations aremodeled by an independent hPPP with intensity λUi .

Both BSs and UEs use antenna arrays for directional beam-forming. For the sake of tractability, we approximate the actualarray patterns using a simplified pattern as in [24], [35]. LetG(φ) denote the simplified antenna directivity pattern depictedin Fig. 2, where M is the main lobe power gain, m is the backlobe gain and θ is the beamwidth of the main lobe. In general,m and M are proportional to the number of antennas in thearray and M/m depends on the type of the array. Furthermore,θ is inversely proportional to the number of antennas, i.e., thegreater the number of antennas, the more beam directionality.We let GB(φ) (which is parameterized by MB , mB , andθB) be the antenna pattern of the BS, and GU (φ) (which isparameterized by MU , mU , and θU ) be the antenna patternof the UE.

✞✒�✁

✂

M

m m

✌✄ ☎✆✝

✟

✠

✡

m M☛

Fig. 2. Simplified antenna pattern with main lobe M , back lobe m andbeamwidth θ.

✞☛�

✁✂

✞✂✄

✁☛

☎✆

✝✟

Fig. 3. Intercell interference from base station Bm received at the UE Ujk ,

which is associated with neighboring base station Bk . φD is the departureangle from Bm and φA is the arrival angle at Uj

k .

We consider a time-slotted downlink of a mmWave cellularsystem. The channel model we use includes path loss, shad-owing, outage, and small scale fading. For path loss, shadow-ing, and outage, line of sight (LOS), and NLOS probabilitydistributions, we use models adopted from [36]. We assumeRayleigh block fading. Finally, the data rate is modeled as

R = (1− α)W log2

(1 + β

PGU (0)GB(0)H

NfN0W + I

), (1)

where α and β (which are specified in Section III-C) areoverhead and loss factors, respectively, and are introduced tofit a specific physical layer to the Shannon capacity curve.Furthermore, P is the BS transmit power, H is the channelpower gain derived from the model discussed above, andNf , N0, W and I are UE noise figure, noise power spectraldensity, bandwidth, and interference power, respectively. Weassume perfect beam alignment between BS and UE within a

cell, therefore the antenna power gain (link directionality) isGU (0)GB(0) = MUMB . The SINR of a UE is defined asPGU (0)GB(0)HNfN0W+I .Let Bk denote the BS of cell k ∈ {1, . . . ,K}, where K

is the total number of cells. Also, let U jk denote the UE j ∈{1, . . . , Nk} of cell k, where Nk is the total number of theUEs in cell k. We do not introduce any intercell coordination tomanage interference. The power of intercell interference frombase station Bm received at UE U jk depends on the beamwidthθ, the arrival angle φA at U jm, and the departure angle φD fromBk, as shown in Fig. 3. In the scenario illustrated we haveI = PGU (φA)G

B(φD)Hk,jm , where Hk,j

m here is the channelgain between Bm and U jk . While the probability of strongintercell interference is low due to the beam directionality,interference still exists. There are four main factors affectingintercell interference:• Frequency: The interference signal is weaker at high

frequencies due to path loss [23].• Antenna Pattern: Antenna pattern parameters, especially

the beamwidth (θ in Fig. 2), directly affect the intercellinterference. A larger beamwidth results in stronger in-tercell interference [23], [24].

• UE and BS density: In mmWave networks, strongintercell interference occurs infrequently in sparse net-works [24]. This is also true for conventional cellularnetworks (like LTE), but with the greater beam direction-ality used in mmWave networks, the network becomesinterference limited for much higher UE and BS densitiesthan in conventional networks. However, for ultra-densedeployments, intercell interference can be dominant evenwith high beam directionalities if there is no coordina-tion [24], [25].

• Bandwidth: When the network bandwidth is large, noisepower becomes the dominant factor and interference isless important [36].

We consider two kinds of intercell interference. Intra-NSPinterference comes from transmissions of neighboring BSs toUEs of the same NSP. Inter-NSP interference comes fromtransmissions of neighboring BSs to UEs of a different NSPon the same frequency, and occurs only when the serviceproviders use spectrum on a non-exclusive basis.

To model interference, we consider four kinds of resourceuse:

1) Exclusive Spectrum, Exclusive BSs (No Sharing):Each NSP works independently on its own part of thespectrum, with its own BSs. Obviously, there is no inter-NSP interference. Each UE associates with the closestBS belonging to its own NSP.

2) Exclusive Spectrum, Non-Exclusive BSs (BS SharingOnly): A UE associates with the closest BS, regardlessof which NSP it belongs to. As a result, the averagedistance between a UE and its serving BS is reduced,which leads to a higher received signal power. However,this also increases the intra-NSP interference powerrelative to the no sharing case, because the distancebetween the interferer BSs and a UE shrinks along withcell radius. There is no inter-NSP interference.

7

3) Non-Exclusive Spectrum, Exclusive BSs (SpectrumSharing Only): NSPs use spectrum on a non-exclusivebasis. Besides for the increase in noise power due tolarger bandwidth, inter-NSP interference also occurs inaddition to intra-NSP interference, and these togetherdecrease the SINR of UEs compared to the previouscases, but may still increase the rate.

4) Non-Exclusive Spectrum, Non-Exclusive BSs (FullSharing): NSPs use both spectrum and BSs on a non-exclusive basis. This case is a combination of Case 2and Case 3.

B. Scheduling

Early work on resource sharing in mmWave networks [23],[25]–[27] has focused on signal propagation and interferenceeffects in networks with shared resources. To approximatedata rate at a UE, these papers divide the link capacity asdetermined by the UE’s average SINR by the total numberof UEs in the cell. In a realistic network with opportunisticscheduling, however, a UE may achieve a higher data ratethan its average SINR would suggest, because it is scheduledwith higher priority in time slots when its SINR is high.This scheduling gain increases with the number of UEs.For an economic analysis we need to accurately model howconsumers’ utility scales with all aspects of network size,including the number of subscribers, so our model mustinclude this scheduling gain.

We adopt a modified scheduler based on the multicelltemporal fair opportunistic scheduler proposed in [37]. We useonly the first stage (UE nomination stage) of this scheduler,since there is no coordination among the BSs. We expectthe difference from the two-stage scheduler with coordina-tion to be neglible, since intercell interference is limitedby the directional nature of the transmissions in mmWavenetworks [27]. Thus each BS runs the scheduler and selectsa UE independently, without considering intercell interferenceand making scheduling decisions based only on the signal tonoise ratio of the UEs in each time slot.

C. Performance Evaluation

In order to establish benefits of resouce sharing froma technical perspective, we present simulation results of ammWave network with two NSPs (i ∈ {1, 2}) operating inthe 73 GHz band. We fix BS transmit power (P ) and lossfactor (β) as 30 dBm and 0.5, respectively, as in [36]. Table Ishows the specific parameters we use in the simulations. Weassume two different cases: symmetric NSPs and asymmetricNSPs. In the symmetric case, NSPs are identical in terms ofnetwork resource (BS density, bandwidth) and UE density.In the asymmetric case, one of the NSPs has more networkresources and UE density than the other. Parameters such asUE and BS densities (λUi , λBi ), and bandwidth (Wi) will bespecified for each NSP separately.

1) Symmetric NSPs: Fig. 4 shows the cumulative distribu-tion function (CDF) of the UE rate, for symmetric NSPs, eachwith 500 MHz of 73 GHz spectrum licensed for exclusive use,50 BSs, and 250 UEs in a single square kilometer. All UEs

TABLE INETWORK PARAMETERS

Parameter Value

Frequency 73 GHzTotal bandwidth (W1 +W2) 1 GHzTotal BS density (λB1 +λB2 ) 100 BSs/km2

Total BS density (λU1 +λU2 ) 500 UEs/km2

BS transmit power P 30 dBmBS antenna model (MB ,mB ,θB) (20 dB, -10 dB, 5◦)UE antenna model (MU ,mU ,θU ) (10 dB, -10 dB, 30◦)Rate model (α, β) (0.2, 0.5)UE noise figure Nf 7 dBNoise PSD N0 -174 dBm/HzSimulation duration T 105 slots

0.00

0.25

0.50

0.75

1.00

10−2

10−1

100

101

Data rate (Gbps)

Em

pir

ica

l C

DF

BS Sharing Only

Full Sharing

No Sharing

Spectrum Sharing Only

Fig. 4. There is a positive effect on UE rate when symmetric NSPs pooltheir BS and spectrum resources.

benefit the most from when the NSPs pool both spectrum andBSs. However, both spectrum sharing alone and BS sharingalone improve UE rate relative to the case where no resourcesare shared. BS sharing has a greater effect on rate thanspectrum sharing for UEs outside the coverage range or witha poor signal quality. For UEs in outage, BS sharing improvescoverage probability. For UEs with a low SNR, there is littlebenefit to adding bandwidth because they are in a power-limited regime. UEs with a good signal quality (high SNR)are bandwidth-limited and benefit more from spectrum sharingthan from BS sharing. At this BS density (100 BSs total persquare kilometer), the effects of interference are neglible dueto the directional nature of the transmissions, so there is nonegative effect due to spectrum sharing without coordination.

When two NSPs pool their BS and spectrum resources,they offer UEs a higher data rate. This is consistent withthe results described in [23], [25]–[27]. However, the earlywork on mmWave resource sharing in [23], [25]–[27] doesnot consider NSPs, which we address next.

2) Asymmetric NSPs: Fig. 5 shows the CDF of UE ratefor consumers of two asymmetric NSPs operating in the samegeographic area. The larger NSP has 70% of resources andsubscribers: 700 MHz of 73 GHz spectrum licensed for ex-clusive use, 70 BSs, and 350 UEs in a single square kilometer.The smaller NSP has 30% of resources and subscribers: 300MHz of spectrum, 30 BSs, and 150 UEs.

When there is no resource sharing, the larger NSP gainsmarket power by offering its subscribers a higher rate than

8

Small service provider (30%) Large service provider (70%)

0.00

0.25

0.50

0.75

1.00

10−2

10−1

100

101

10−2

10−1

100

101

Data rate (Gbps)

Em

pir

ical C

DF

BS Sharing Only

Full Sharing

No Sharing

Spectrum Sharing Only

Fig. 5. When asymmetric NSPs share spectrum and BS resources, they provide a higher quality of service, but are no longer able to differentiate themselveson the data rates they offer.

competing services. With fully shared resources, however,their subscribers’ rate distributions are identical and the ser-vices are perfect substitutes. Sharing increases the value ofthe service that both NSPs offer, but eliminates the ability ofthe larger NSP to distinguish itself in the market by offeringhigher rates.

With only spectrum sharing or only BS sharing, the largerNSP retains some ability to differentiate itself in the marketby offering higher rates. However, even under these circum-stances, the small NSP enjoys greater relative gains than thelarge NSP from sharing of any kind (compared to no sharing),while the large NSP contributes more resources. The smallNSP especially benefits from full sharing or BS sharing byincreasing the coverage probability of UEs that were in outage.This gives it an extra competitive edge in the market amongconsumers who consider stability and coverage probability asa primary factor in choosing their NSP.

IV. MMWAVE SERVICE AS A NETWORK GOOD

In the simulations of Section III, we assumed a fixed numberof UEs and resources. Now we model mmWave networkservice as a network good (as explained in Section II), withvarying demand and resources. Subscribers benefit from anindirect positive network externality: a large wireless serviceprovider with more subscribers will build a denser deploymentof BSs, and purchase more spectrum. (Given large availablebandwidth at mmWave frequencies, we expect it will befeasible for NSPs to acquire more spectrum at will.) Wedescribe the impact of increasing network size on consumers’data rates, using the simulation described in Section III.

The network size n of a mmWave network is defineddifferently depending on the scenario:• No open resources: In this scenario, an NSP scales its

spectrum licenses and BSs according to the number ofsubscribers it has. Network size, n, is the normalizeddemand for the service, but is also a scaling factor onthe BS density (λB) and bandwidth (W ) of the NSP.

• Open BS deployment: In this scenario, there is a pre-existing deployment of neutral small cells, operated bya coalition of service providers or by a third party (as

suggested in [22]). These cells have an open associationpolicy, and will serve UEs of any NSP. Network size, n,refers to demand for the service and is also a scalingfactor on the bandwidth of the NSP, W , but the BSdensity of the NSP is constant and equal to the size ofthe “open” deployment (λBmax) for all values of n.

• Open spectrum: In this scenario, spectrum is unlicensedand may be used by any NSP. Here, n refers to demandfor the service and is also a scaling factor on the BSdensity of the NSP, λB . However, the bandwidth ofthe NSP is constant and equal to the full unlicensedbandwidth Wmax for all values of n.

Note that the use of “open” resources is not the same as sharingresources acquired by individual NSPs. Open resources arefixed in size and available to all NSPs. Shared resourcesare also available to participating NSPs, but their size variesaccording to the network size of the NSPs. Table II enumeratesthe BS density and bandwidth used by NSP i in all threescenarios, when there is no sharing between individual NSPsand when resources are shared among NSPs in I , with i ∈ I .Resources that are used by NSP i on a non-exclusive basis arein bold font.

TABLE IISCALING RESOURCES WITH NETWORK SIZE

No open re-sources

Open BS de-ployment

Openspectrum

No sharing λBi niλBmax λB

max niλBmax

Wi niWmax niWmax Wmax

Sharing λBi∑∑∑

j∈I njλBmax λB

max∑∑∑

j∈I njλBmax

Wi∑∑∑

j∈I njWmax∑∑∑

j∈I njWmax Wmax

Depending on the scenario, the total network externalityh(n) is the sum of the network effects associated with up tothree aspects of n. We separately quantify the contribution toh(n) of each:

1) BS density: We simulate a network of increasing BSdensity, with bandwidth constant at 1 GHz and ratio ofUEs to BSs constant at 5 UEs per BS. To separatelyquantify the effect of interference, we also compute

9

●● ●

●

●

●

●

●●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●●

●

● ●

●

0.0

0.2

0.4

0.6

0 100 200 300

Base stations per square km

5th

perc

entile

rate

(G

pbs)

Outage

NLOS to LOS

LOS, increasing signal power

● Rate

Rate without interference

Fig. 6. Effect of increasing BS density on fifth percentile rate as bandwidth isheld constant at 1 GHz and UE density per BS is held constant at 5 UEs/BS.Error bars show 95% bootstrap confidence intervals.

● ● ● ● ● ● ● ● ● ●0.0

0.2

0.4

0.6

250 500 750 1000

Bandwidth (MHz)

5th

perc

entile

rate

(G

pbs)

Base station density(per square km)

● 20

50

100

150

200

250

Fig. 7. Effect of increasing bandwidth on fifth percentile rate for differentBS densities, with UE density per BS held constant at 5 UEs/BS. Error barsshow 95% bootstrap confidence intervals.

the UE rate based on SNR (neglecting the effect ofinterference) and compare this to the actual UE rate.These results are shown in Fig. 6.

2) Bandwidth: We simulate a network of increasing band-width, while keeping BS density constant at a moderatevalue of 100 BSs per square kilometer, and UE densityconstant at 500 UEs per square kilometer. These resultsare shown in Fig. 7.

3) UE density: We simulate a network of increasing UEdensity, while keeping bandwidth constant at 1 GHz andBS density constant at 100 BSs per square kilometer.To quantify the scheduling gain, we also simulate thisnetwork with a round robin scheduler, and compare thisto UE rate with a partially opportunistic scheduler. Theseresults are shown in Fig. 8.

We assume that consumers decide whether or not to sub-scribe to a mmWave network based on its fifth percentilerates. This is supported by research on human behavior,which suggests that service reliability is rated more highlythan overall connection quality in perceived quality of mobilevalue-added services [38]. We take fifth percentile rate as aproxy for service reliability.

From Fig. 6, Fig. 7, and Fig. 8, we find that the fifthpercentile data rate of UEs is affected by network size as

●

●

●

●

●

●●

● ●

●●

● ●

● ●●

● ●●

0.25

0.50

0.75

1.00

1.25

250 500 750 1000User density per square km, 100 BS

5th

perc

entil

e ra

te (

Gpb

s)

● Opportunistic

Round Robin

Fig. 8. Effect of increasing UE density per BS as bandwidth is held constantat 1 GHz and BS density is held constant at 100 BSs per square kilometer.Error bars show 95% bootstrap confidence intervals.

follows:

1) BS density: (Fig. 6) As BS density increases, UEstransition between three regions, marked by the verticallines in Fig. 6. In the outage region on the left, thedeployment of mmWave BSs is very sparse, and UEsare likely to be outside the coverage area. In the middleregion, increasing BS density improves the probabilityof having a LOS link, and so the rate grows very quicklywith BS density until a point at which virtually all linksare LOS. In the third region, there is a smaller marginalbenefit associated with higher BS density due to increas-ing SNR. When the deployment of BSs is extremelydense, there may be a small negative interference effectthat partially mitigates the positive effect of increasingBS density, consistent with [25].

2) Bandwidth: (Fig. 7) For UEs with a moderate or highSNR, link capacity scales linearly with bandwidth, sothese will benefit from subscribing to a large serviceprovider with more mmWave spectrum holdings. How-ever, UEs with a very weak signal power (e.g., in anetwork where the density of BSs is very low) arepower-limited and do not benefit much from increasedbandwidth.

3) UE density: (Fig. 8) Given a fixed number of BSs,increasing the total number of UEs can overburden thenetwork and create a negative congestion externality.Specifically, when there are N UEs in the cell, eachUE is allocated approximately 1

N th of frequency-timeresources. However, because the scheduling of UEsis partially opportunistic, a UE is more likely to bescheduled in high-SINR time periods, and this effectincreases with the number of UEs in the cell as the“competition” to be scheduled becomes more intense.Thus with an increasing number of UEs in a cell, anindividual UE is scheduled in fewer time slots, but hasa higher average data rate in the time slots in which itis scheduled.

In general, the benefit to a UE of increasing any of thethree elements of network size discussed above depends on theUE’s signal quality. In sparse deployments, UEs are outside the

10

● ● ● ●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

0.0

0.1

0.2

0.3

0.4

0.5

0.00 0.25 0.50 0.75 1.00

n

5th

perc

entile

rate

(G

pbs)

● No Open Resources

Open BS Deployment

Open Spectrum

Fig. 9. Effect of increasing network size on fifth percentile UE rate. Errorbars show 95% bootstrap confidence intervals.

coverage area or at its edge, and derive little or no benefit fromincreasing bandwidth alone. Under moderate SNR conditions,UEs benefit both from increasing BS density and increasingbandwidth. In a very dense network, virtually all UEs have aLOS link and high SNR, and have little to gain from increasingBS density, but benefit from increasing bandwidth.

V. DEMAND FOR MMWAVE SERVICES, WITH ANDWITHOUT OPEN RESOURCES

Having quantified the technical effects on fifth percentilerate of increasing mmWave network size in Section IV, wefocus on how demand for wireless service, price an NSPcan charge, and NSP’s revenue, depend on network size. Weare especially interested in the evolution of demand at smallnetwork sizes, when an NSP first begins to offer mmWaveservices, and whether the network will reach critical mass. Toaddress this, we model the willingness to pay of consumers asa function of n (using h(n)), then construct a curve of fulfilledexpectations demand p(n;n) that shows how demand, price,and revenue scale with n. We assume again that consumersdecide to subscribe or not based on fifth percentile rates,and consider the three scenarios (no open resources, open BSdeployment, and open spectrum) in the “No sharing” row inTable II.

Fig. 9 shows the simulated fifth percentile rate for UEs ina mmWave network, from which we derive h(n) empirically.In the open BS deployment scenario, 100 BSs are availableregardless of n; in the open spectrum scenario, bandwidth is1 GHz at all values of n. Otherwise, bandwidth, BS density,and the number of UEs in the network scale with n up to1 GHz, 100 BSs, and 500 UEs, respectively, at n = 1. Wenote that the behavior of the fifth percentile rate is roughlypiecewise linear in n, with breakpoints when we transitionbetween the SNR regions of Section IV.

Next, we construct a fulfilled expectations demand curve foreach of the three scenarios, using the values in Fig. 9 for thenetwork externalities function h(n). We assume consumers areheterogeneous in their willingness to pay for service, and theparameter ω, which denotes the consumer type, is uniformlydistributed over [0, 1]. A consumer of type ω subscribing to

the network will gain surplus u(ω, n, p) = ωh(n)− p from anetwork of size n at price p.

The fulfilled expectations demand curve in Fig. 10 isconstructed as described in Section II. We also show therevenue of the NSP, np(n;n). We observe in Fig. 10 theinitially upward-sloping fulfilled expectations demand curvep(n;n) that is a feature of network goods.

The slope of this curve at small network sizes is worthy ofextra attention. As described in Section II, this determines howeasily the network will reach critical mass in a perfectly com-petitive market where competing networks are homogeneousin every way except for size.• When there are no open resources, the slope of p(n;n)

is small for 0 ≤ n ≤ 0.262, suggesting that the marginalbenefit of increasing network size is very small whenthe network size is small. Under these conditions, it isdifficult for a network to reach its tipping point.

• When there is an open BS deployment, there is robustdemand and strong marginal network externalities evenat very small network sizes. Assuming a pre-existing BSdeployment, is relatively easy to reach the tipping pointunder these circumstances.

• With open spectrum, the positive slope of p(n;n) is largefor moderate network sizes, but again there is a very smallpositive slope when n is small (n ≤ 0.235).

We observe that there is a strong marginal network externalityat small network sizes when there is an open BS deployment,and NSPs can grow their network by incrementally addingspectrum holdings and subscribers. This suggests that basedpurely on the ability of a small NSP to generate revenue(not considering startup costs) and to reach its tipping point,an open BS deployment could ease the barrier to entry forcellular network providers who are considering extendingtheir networks to include mmWave service, encouraging newservice offerings. Furthermore, we note that open BSs helpencourage new service offerings more than open (shared orunlicensed) spectrum would. However, the open BS scenariorelies on a third party having invested in BSs, potentially aheadof demand if there are no existing mmWave NSPs in themarket yet and the BSs are only useful for mmWave service.The open spectrum approach relies only on regulators havingreleased the spectrum for unlicensed use in cellular systems.

VI. RESOURCE SHARING IN A COMPETITIVE MARKET

Now we turn our attention to mmWave NSPs that are al-ready established, and are mainly concerned with maximizingtheir profits in a competitive market, rather than strugglingto reach critical mass. We model the NSPs’ decision toshare mmWave network resources or not as a compatibilityproblem (introduced in Section II), where mmWave NSPsare considered compatible if their subscribers can connect toany of the set of NSPs’ BSs, and use a bandwidth equal totheir pooled spectrum holdings. We previously addressed thetechnical benefits of this in Section III. We consider all of thecombinations of “shared” and “open” resources in Table II.

In many locations in the United States and around the world,the market for cellular service is effectively a duopoly. We

11

● ● ● ● ●●

●

●● ● ● ● ● ●

● ●●

●

●

● ●0.2

0.4

0.6

0.8

●

●

●●

● ● ●● ●

● ●

●● ●

●●

●●

●● ●

0.2

0.40.60.8

● ● ● ●●

●

●

●● ● ●

●● ●

●●

●

●

●

● ●0.2

0.4

0.6

0.8

No Open Resources Open BS Deployment Open Spectrum

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0n

pFulfilled ExpectationsDemand: p(n;n)Revenue: np(n;n)Demand at given n e

Fig. 10. The fulfilled expectations demand curve p(n;n) and NSP revenue np(n;n) for a mmWave network deployment, where the value of the networkexternalities function for a network of size n, h(n), is computed by simulation as the fifth percentile rate of UEs in a network of that size (in Fig. 9). Weconsider three scenarios: one where the size of the network, n, is proportional to the number of BSs it has deployed and its spectrum resources (left), onewhere 100 BSs are open to all networks and n is proportional to the network’s spectrum resources (middle), and one where 1 GHz of open spectrum is usedby all networks and n is proportional to the number of BSs the network has deployed (right).

therefore consider a market with two vertically differentiatedNSPs, and the three-stage game with complete informationdescribed in [34]:

1) NSPs i ∈ {1, 2} simultaneously choose inherent qualityqi from the interval [0, q].

2) NSPs i ∈ {1, 2} simultaneously set price pi.3) Each consumer chooses to subscribe to one of the NSPs

i ∈ {1, 2} or to neither.The quality qi is the inherent quality (with maximum

feasible value q). It refers to aspects of service unrelated tothe size of the mmWave network, such as the quality of voicecalls, the quality of the legacy data network, customer service,and the availability of desirable handsets. These measures ofquality are increasingly important once the service is wellestablished in the market, and has a sufficiently large networkthat network size is no longer the single main criterion bywhich consumers decide which NSP to subscribe to.

An NSP’s marginal costs are increasing in qi, with costfunction c(qi, ni) = qini, and so each NSP i ∈ {1, 2} seeksto maximize its profits

πi(qi, ni, pi) = nipi − qini (2)

Consumers evaluate competing services in terms of thedifference in their inherent qualities as well as their networkexternalities. We have heterogeneous consumers parameterizedby ω, with ω distributed uniformly from [0, ω]. The surplus ofa consumer of type ω is given by

u(ω, qi, ni, pi) =

{ωqi + µqini − pi if subscribes to i0 if no subscription

(3)with i ∈ {1, 2} and 0 ≤ µ < min[1, ω/2], where µ is explainedin the next paragraph. If the NSPs share their mmWavenetwork resources, then ni =

∑i∈{1,2} ni, otherwise ni = ni.

The fifth percentile rate in a mmWave network is piecewiselinear in the network size (Fig. 9). Here we consider onlymoderate- to large-sized networks, where the curves in Fig. 9are linear. Thus the network externalities function h(ni) islinear in ni. The scaling factor µ determines the intensity ofthe network externality, i.e., h(ni) = µni, and is empirically

derived from slopes of the lines in Fig. 9. We consider threescenarios, each with established NSPs:• No open resources: In this scenario we use µ = 0.7,

corresponding to the slope of the “no open resources”line in Figure 9 for n ≥ 0.25.

• Open BS deployment: There is an open BS deploymentserving all NSPs. We use µ = 0.25, corresponsing to theslope of the “open BS deployment” line in Fig. 9 forn ≥ 0.35.

• Open spectrum: Spectrum is unlicensed and used by allNSPs. We use µ = 0.4, corresponsing to the slope of the“open spectrum” line in Fig. 9 for n ≥ 0.45.

By their choice of quality level, the NSPs segment themarket into a low-end group (small-ω type) and a high-endgroup (large-ω type). Without loss of generality, we say thatNSP 1 chooses a higher quality than NSP 2, i.e., q1 > q2, andsubscribers of NSP 1 belong to the large-ω group. We definetwo marginal consumers: the consumer of type ω is indifferentbetween choosing no subscription and subscribing to NSP 2,and the consumer of type ω is indifferent between subscribingto NSP 1 and subscribing to NSP 2.

Then the utility of the marginal consumer of type ω satisfies

ωq1 + µq1n1 − p1 = ωq2 + µq2n2 − p2 (4)

and the utility of the marginal consumer of type ω satisfies

ωq2 + µq2n2 − p2 = 0 (5)

Also, the marginal consumer of type ω defines the marketshare of the high-end service

n1 =ω − ωω

(6)

and the marginal consumers together define the market shareof the low-end service

n2 =ω − ωω

(7)

We can solve (4), (5), (6), and (7) for n1, n2, ω, and ω, andthus determine the decisions of the consumers and the marketshare of each NSP given pi, qi, i ∈ {1, 2}.

12

Given pi and qi i ∈ {1, 2}, it is shown in [34] that if theratio of quality levels satisfies

q1q2>

(ω2

(ω − µ)(ω − 2µ)

)(8)

then there is a unique Nash equilibrium in which both NSPsset prices higher than their marginal costs. Furthermore, if thesolution to (4), (5), (6), and (7) satisfies

0 < ω < ω < ω (9)

then both NSPs have market share greater than zero. Whenboth (8) and (9) hold, then there is a unique Nash equilibriumin which both NSPs earn non-zero profit. We restrict ourattention to these circumstances, since these are of primaryinterest to us.

If (8) and (9) hold and the NSPs do not share resources,then according to [34] their equilibrium prices p∗1,NS , p

∗2,NS

are as follows:

p∗1,NS = q1

[1+

(ω − 1)[2q1(ω − µ)2 − q2ω(2ω − µ)]4q1(ω − µ)2 − q2ω2

]> q1

(10)

p∗2,NS = q2

[1 +

(ω − 1)[q1(ω − µ)(ω − 2µ)− q2ω2]

4q1(ω − µ)2 − q2ω2

]> q2

(11)and their equilibrium quality levels q∗1,NS , q

∗2,NS are:

q∗1,NS = q (12)

q∗2,NS =q(ω−µ)2

[11ω−10µ−

√3(3ω2+28ωµ−20µ2)

]2ω2(7ω−5µ) < q (13)

The profits of the high-end NSP always increase with q1,so it will use q. The low-end NSP sets q2 to balance twocompeting effects: at high values of q2 the low-end NSP has agreater market share, but is also more similar to q1, whichincreases price competition and drives the price of servicedown.

Note that reducing the parameter q increases price competi-tion, since the difference in quality levels between NSPs willbe small and so consumers’ decisions will be more sensitive toprice. Similarly, reducing ω increases price competition, sincethis decreases the dispersion of consumers’ willingness to payand the market is less segmented.

If the NSPs share resources, then per [34] their equilibriumprices p∗1,S , p

∗2,S are:

p∗1,S = q1

[1 +

2ω(ω − 1)(q1 − q2)(4ω − 3µ)q1 − ωq2

]> q1 (14)

p∗2,S = q2

[1 +

ω(ω − 1)(q1 − q2)(4ω − 3µ)q1 − ωq2

]> q2 (15)

and their equilibrium quality levels q∗1,S , q∗2,S are:

q∗1,S = q (16)

q∗2,S =q(4ω − 3µ)

7ω − 6µ< q (17)

For the sake of comparison, we are also interested in theprofits of a monopoly NSP. When there is only one NSP, themarginal consumer is defined by

ωq1 + µq1n1 − p1 = 0 (18)

and the market share of the NSP is

n1 =ω − ωω

(19)

At equilibrium, the monopoly NSP will choose price

p∗1,M =q1(ω − 1)

2(20)

and quality levelq∗1,M = q (21)

Fig. 11 shows the profits of each NSP in various circum-stances, as ω (and dispersion of consumers’ willingness topay) increases. First, we note that the low-end NSP alwaysprefers to share resources. Since it captures the low end of themarket (the consumers who are less willing to pay for inherentquality), the network effects are especially important to thisNSP. In the duopoly market, the high-end NSP paradoxicallyprefers to share resources when the intensity of the networkeffect is small (as when there are open resources), because itscompetitor gains less of an advantage from the larger (shared)network size. (This is consistent with the results describedin [16]). When the intensity of the network effect is large(as it is when there are no open resources) then the high-endNSP prefers resource sharing only when the market is highlysegmented and there is little price competition (i.e., for largeω).

Fig. 12 shows the market share of each NSP and total marketcoverage (i.e., share of consumers who subscribe to eitherNSP) under the same set of circumstances. When ω (and thedispersion of consumers’ willingness to pay) is small, sharingoffers the best overall market coverage. When ω is large, thebest market coverage is achieved by not sharing resources. Thevalue of ω at which the benefit of market segmentation beginsto dominate the benefit of network effects is greater when theintensity of the network effect is high (large µ).

VII. CONCLUSIONS

In this paper, we have connected economic models of thestrategic decision making of cellular network service providersand subscribers, to detailed simulations of mmWave networks,with and without resource sharing. While we have confirmedthe benefits of resource sharing from a purely technical view(without considering the effect on demand), with the economicanalysis we have illustrated that resource sharing is not alwaysthe preferred strategy of service providers, and some kindsof resource sharing may be preferred over others. We haveshown that “open” deployments of neutral small cells make iteasier for networks to reach critical mass, encouraging marketentry more than “open” spectrum would. Furthermore, we haveshown that the leading service provider in a duopoly marketprefers to share resources only when sharing gains are smallor the market is highly segmented. Our technical simulationsof asymmetric service providers have hinted at this, with

13

No Open Resources (0.7) Open BS Deployment (0.25) Open Spectrum (0.4)

0

1

2

3

2 4 6 8 10 2 4 6 8 10 2 4 6 8 10

ω

Pro

fit

Monopoly

No Sharing

Sharing

Monopoly

High−end provider

Low−end provider

Fig. 11. Profit of each NSP for different cases of the intensity of the network effect, µ. The dotted line shows the profits of each NSP if they choose not toshare their respective resources (though they may still use any available “open” resources). The dashed line shows their profits if they share resources. Thesolid line shows the profits in a monopoly market, for comparison. The horizontal axis indicates the dispersion of the consumers’ valuation of the services.(q = 1.5)

No Open Resources (0.7) Open BS Deployment (0.25) Open Spectrum (0.4)

0.00

0.25

0.50

0.75

1.00

2 4 6 8 10 2 4 6 8 10 2 4 6 8 10

ω

Mark

et S

hare

Monopoly

No Sharing

Sharing

Monopoly

High−end provider

Low−end provider

Total

Fig. 12. Market share of each NSP and total market coverage for different cases of the intensity of the network effect, µ. The dotted line shows the marketshare of each service provider and their total market coverage if they choose not to share their respective resources (though they may still use any available“open” resources). The dashed line shows their market shares and total market coverage if they share resources. The solid line shows the market share of asingle firm in a monopoly market, for comparison. The horizontal axis indicates the dispersion of the consumers’ valuation of the services. (q = 1.5)

greater gains for the smaller service provider than the marketleader. However with a purely technical approach, one wouldconclude that resource sharing is always beneficial (albeit lessbeneficial for the market leader), while the economic analysiswith consideration of price and demand in addition to technicalgains has suggested a different conclusion.

We briefly discuss here some assumptions of our approach.Our results are predicated on an assumed indirect networkeffect benefitting consumers subscribing to a large serviceprovider. That is, we assume that the resources held by aservice provider in a given market scale together with the num-ber of subscribers it serves. Practically, building out physicalinfrastructure and licensing spectrum requires a tremendouscapital investment. A service provider is unlikely to buildout a very large network, at great cost, when it has fewsubscribers and so a limited revenue stream. For this reason,we consider it justified to tie the level of investment in thenetwork - and thus, the size of the network resources - to thenumber of subscribers. Another assumption is that consumersare homogeneous in their preference for one firm or theother, given their overall valuation of network service, i.e.,that consumers with the same ω will make the same choicebetween service providers, given their price, network size, andinherent quality. Actually, consumers and are not identicalin their valuations of competing services. However, despitethis common simplifying assumption, the general economicframework we have applied in this paper has been empirically

validated in a variety of other industries with network effects.The work in this paper suggests several interesting avenues

for further research. We would like to extend this modelto include the investment costs associated with deploying anew mmWave network, which we expect to be substantiallydifferent from traditional cellular networks given the uniquephysical characteristics of the mmWave bands. We would alsolike to investigate scheduling strategies that divide sharedresources among mmWave service providers in ways thatincrease the benefit of resource sharing for both serviceproviders and consumers.

REFERENCES

[1] Federal Communications Commission, “In the matter of use of spectrumbands above 24 GHz for mobile radio services,” Notice of Inquiry inGN Docket No. 14-177, FCC 14-154, Nov. 2014.

[2] T-Mobile USA, Inc., “Comments of T-Mobile USA, Inc.” GN DocketNo. 14-177, Jan. 2015.

[3] Verizon and Verizon Wireless, “Comments of Verizon,” GN Docket No.14-177, Jan. 2015.

[4] Straight Path Communications Inc., “Reply comments of Straight PathCommunications Inc.” GN Docket No. 14-177, Feb. 2015.

[5] Qualcomm Incorporated, “Comments of Qualcomm Incorporated,” GNDocket No. 14-177, Jan. 2015.

[6] CTIA - The Wireless Association, “Comments of CTIA - The WirelessAssociation,” GN Docket No. 14-177, Jan. 2015.

[7] Nokia, “Comments of Nokia (d/b/a Nokia Solutions and Networks USLLC),” GN Docket No. 14-177, Jan. 2015.

[8] AT&T Services, Inc., “Reply comments of AT&T,” GN Docket No. 14-177, Feb. 2015.

[9] National Cable & Telecommunications Association, “Comments ofNational Cable & Telecommunications Association,” GN Docket No.14-177, Jan. 2015.

14

[10] Google Inc., “Comments of Google Inc.” GN Docket No. 14-177, Jan.2015.

[11] Federal Communications Commission, “In the matter of use of spectrumbands above 24 GHz for mobile radio services,” Notice of ProposedRulemaking in FCC 15-138, Oct. 2015.

[12] T. S. Rappaport, R. W. Heath Jr., R. C. Daniels, and J. N. Murdock,Millimeter Wave Wireless Communications. Pearson Education, 2014.

[13] N. Economides, “The economics of networks,” International journal ofindustrial organization, vol. 14, no. 6, pp. 673–699, 1996.

[14] N. Economides and C. P. Himmelberg, “Critical mass and network sizewith application to the US fax market,” NYU Stern School of BusinessEC-95-11, 1995.

[15] M. L. Katz and C. Shapiro, “Network externalities, competition, andcompatibility,” The American economic review, vol. 75, no. 3, pp. 424–440, 1985.

[16] N. Economides and F. Flyer, “Equilibrium coalition structures in marketsfor network goods,” Annales d’Economie et de Statistique, pp. 361–380,1998.

[17] J. S. Panchal, R. Yates, and M. M. Buddhikot, “Mobile network resourcesharing options: Performance comparisons,” IEEE Trans. on WirelessCommunication, vol. 12, no. 9, pp. 4470–4482, 2013.

[18] S. Hua, P. Liu, and S. S. Panwar, “The urge to merge: When cellularservice providers pool capacity,” in IEEE ICC 2012, 2012, pp. 5020–5025.

[19] J. Kibilda, P. Di Francesco, F. Malandrino, and L. A. DaSilva, “Infras-tructure and spectrum sharing trade-offs in mobile networks,” in IEEEDySPAN 2015, 2015, pp. 348–357.

[20] E. A. Jorswieck, L. Badia, T. Fahldieck, E. Karipidis, and J. Luo, “Spec-trum sharing improves the network efficiency for cellular operators,”IEEE Communications Magazine, vol. 52, no. 3, pp. 129–136, 2014.

[21] J. Luo, J. Eichinger, Z. Zhao, and E. Schulz, “Multi-carrier waveformbased flexible inter-operator spectrum sharing for 5G systems,” in IEEEDYSPAN 2014, 2014, pp. 449–457.

[22] J.-C. Guey, P.-K. Liao, Y.-S. Chen, A. Hsu, C.-H. Hwang, and G. Lin,“On 5G radio access architecture and technology [industry perspec-tives],” IEEE Wireless Communications, vol. 22, no. 5, pp. 2–5, 2015.

[23] A. K. Gupta, J. G. Andrews, and R. W. Heath Jr., “On the feasibility ofsharing spectrum licenses in mmwave cellular systems,” arXiv preprintarXiv:1512.01290, 2015.

[24] T. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wavecellular networks,” IEEE Trans. on Wireless Communications, vol. 14,no. 2, pp. 1100–1114, 2015.

[25] F. Boccardi, H. Shokri-Ghadikolaei, G. Fodor, E. Erkip, C. Fischione,M. Kountouris, P. Popovski et al., “Spectrum pooling in mmwavenetworks: Opportunities, challenges, and enablers,” arXiv preprintarXiv:1603.01080, 2016.

[26] M. Rebato, M. Mezzavilla, S. Rangan, and M. Zorzi, “Resource sharingin 5G mmwave cellular networks,” arXiv preprint arXiv:1603.02651,2016.

[27] ——, “The potential of resource sharing in 5G millimeter-wave bands,”arXiv preprint arXiv:1602.07732, 2016.

[28] G. Li, T. Irnich, and C. Shi, “Coordination context-based spectrumsharing for 5G millimeter-wave networks,” in CROWNCOM 2014, 2014,pp. 32–38.

[29] P. D. Francesco, F. Malandrino, T. K. Forde, and L. A. DaSilva,“A sharing- and competition-aware framework for cellular networkevolution planning,” IEEE Trans. on Cognitive Communications andNetworking, vol. 1, no. 2, pp. 230–243, June 2015.

[30] C. Singh, S. Sarkar, A. Aram, and A. Kumar, “Cooperative profit sharingin coalition-based resource allocation in wireless networks,” IEEE/ACMTrans. Netw., vol. 20, no. 1, pp. 69–83, Feb. 2012.

[31] J. Markendahl and B. G. Mölleryd, “On co-opetition between mobilenetwork operators: Why and how competitors cooperate,” in 19th ITSBiennial Conference 2012. ITS, 2012.

[32] D.-E. Meddour, T. Rasheed, and Y. Gourhant, “On the role of infras-tructure sharing for mobile network operators in emerging markets,”Comput. Netw., vol. 55, no. 7, pp. 1576–1591, May 2011.

[33] V. Nikolikj and T. Janevski, “State-of-the-art business performanceevaluation of the advanced wireless heterogeneous networks to bedeployed for the "TERA age",” Wirel. Pers. Commun., vol. 84, no. 3,pp. 2241–2270, Oct. 2015.

[34] P. Baake and A. Boom, “Vertical product differentiation, network exter-nalities, and compatibility decisions,” International Journal of IndustrialOrganization, vol. 19, no. 1, pp. 267–284, 2001.

[35] A. M. Hunter, J. G. Andrews, and S. Weber, “Transmission capacityof ad hoc networks with spatial diversity,” IEEE Trans. on WirelessCommunications, vol. 7, no. 12, pp. 5058–5071, 2008.

[36] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rap-paport, and E. Erkip, “Millimeter wave channel modeling and cellularcapacity evaluation,” IEEE Journal on Sel. Areas in Communications,vol. 32, no. 6, pp. 1164–1179, 2014.

[37] S. Shahsavari and N. Akar, “A two-level temporal fair scheduler formulti-cell wireless networks,” IEEE Wireless Communications Letters,vol. 4, no. 3, pp. 269–272, June 2015.

[38] Y.-F. Kuo, C.-M. Wu, and W.-J. Deng, “The relationships among servicequality, perceived value, customer satisfaction, and post-purchase inten-tion in mobile value-added services,” Computers in Human Behavior,vol. 25, no. 4, pp. 887 – 896, 2009.