On the Feasibility of Sharing Spectrum Licenses in mmWave Cellular Systems
11 On the Feasibility of Sharing SpectrumLicenses in mmWave Cellular Systems
Abhishek K. Gupta, Jeffrey G. Andrews, Robert W. Heath, Jr.
Abstract
The highly directional and adaptive antennas used in mmWave communication open up the pos-sibility of uncoordinated sharing of spectrum licenses between commercial cellular operators. Thereare several advantages to sharing including a reduction in license costs and an increase in spectrumutilization. In this paper, we establish the theoretical feasibility of spectrum license sharing amongmmWave cellular operators. We consider a heterogeneous multi-operator system containing multipleindependent cellular networks, each owned by an operator. We then compute the SINR and ratedistribution for downlink mobile users of each network. Using the analysis, we compare systems withfully shared licenses and exclusive licenses for different access rules and explore the trade-offs betweensystem performance and spectrum cost. We show that sharing spectrum licenses increases the per-userrate when antennas have narrow beams and is also favored when there is a low density of users. Wealso consider a multi-operator system where BSs of all the networks are co-located to show that thesimultaneous sharing of spectrum and infrastructure is also feasible. We show that all networks canshare licenses with less bandwidth and still achieve the same per-user median rate as if they each hadan exclusive license to spectrum with more bandwidth.
I. I
NTRODUCTION
Due to scarcity of spectrum at conventional cellular frequencies (CCF), the use of higherfrequencies such as mmWave has been proposed for 5G cellular networks [2]–[4]. The mmWavecellular systems are expected to consist of multiple networks, each deployed by an independent
A. K. Gupta ( [email protected] ), J. G. Andrews ( [email protected] ) and R. W. Heath Jr.( [email protected] ) are with Wireless Networking and Communications Group, The University of Texas at Austin, Austin,TX 78712 USA. This work was supported by the National Science Foundation under Grant 1514275.A part of this paper is presented at ITA Workshop, Feb. 2016 in San Diego, CA, USA [1]. a r X i v : . [ c s . I T ] M a y cellular operator with possibly closed access only to its customers. It is worth noting that commu-nication at mmWave frequencies has non-trivial differences when compared to communicationat CCF. For example, the typical use of many antennas in mmWave systems results in highlydirectional communication [5], [6], and under some circumstances, it is noise limited [7]–[10].In general, mmWave communication causes less interference to neighboring BSs operating inthe same frequency bands compared to communication at frequencies below 6 GHz [2], [11].This leads to possibility of a new kind of sharing which we call as spectrum license sharing :independent cellular operators who own licenses for separate frequency bands agree to share thecomplete rights of operation in each other’s bands without any explicit coordination. Sharinglicenses (if possible), will allow all networks to use the full spectrum simultaneously withoutimpacting the individual achieved rates and help networks to reduce their expenses by sharing thelicense costs. Such uncoordinated sharing of licenses was not possible in conventional cellularfrequencies due to high interference caused by serving BSs which renders the channel unusableto any nearby BS of other networks operating in the same frequency band. A. Background and Related Work
In conventional cellular licensing, commercial operators buy exclusive licenses where theyhave exclusive and complete control over a band of spectrum. For mmWave bands, there are noregulatory rules for cellular services yet, although there are many incumbent services includingfixed services, satellite to earth communications, military, and research activities and unlicensedoperations. It has been reported that the spectrum remains underutilized [12] due to this exclusivelicensing at CCF which is expected to be even worse for mmWave bands. There has beensignificant work related to cognitive radios to help fill the gaps in the underutilized spectrum byletting secondary users make use of the spectrum band via sensing based access control [13]–[17]for CCF. In [18]–[21], various aspects and performance of spectrum sharing were studied in acellular setting. Most related to this paper, in [20], dynamic spectrum sharing between differentoperators was shown to achieve reasonable sensing performance in 3GPP LTE-A systems withcarrier aggregation. Various cognitive license sharing schemes such as licensed shared access(LSA) and authorized shared access (ASA) were proposed [22], [23] which allow more than oneentity to use the spectrum. In the presence of incumbent services, the above mentioned techniques[22], [23] would authorize a cellular system to transmit, only when the incumbent services are idle [24]–[26]. Implementation would require some kind of sensing or central coordination,which may waste important resources resulting in underutilization of spectrum. Further, it wasshown in [27], [28] that spectrum gaps will be rare for ultra-dense deployment of small cells in amulti-tier network and cognitive sensing may not give the desired gain. Another way to resolvethe transmission conflicts between multiple licensees (or entities in case of unlicensed spectrum)is by the use of a central database which keeps track of transmission of each licensee [29]. Thisreduces the requirement of continuous listening/sensing of the spectrum by each licensee/entitybut creates significant feedback/transmission overhead and possibly delay due to the centraldatabase.In recent work, stochastic geometry has emerged as a tractable approach to model and analyzevarious wireless systems. For example, the performance of a single operator mmWave systemhad been investigated in prior work using tools from stochastic geometry [11], [30]–[32]. In [30],a stochastic geometry framework for analysis of mmWave network was proposed. In [11], [31],a blocking model for mmWave communication was proposed to distinguish between line-of-sight (LOS) or non-LOS (NLOS) transmission links and performance metrics such as coverageprobability and per-user rate were derived using stochastic geometry. In [32], a stochastic geom-etry framework was presented for mmWave network with backhaul and co-existent microwavenetwork. In [33], it was shown that even a very dense mmWave network tends to be noise limitedfor certain choices of parameters. A major limitation of the prior work on mmWave systems[11], [30]–[33] is the assumption of a single operator. To study the impact of an active BS of anoperator on neighboring ones of another operators, a more general framework containing multipleoperators is needed. For CCF, a heterogeneous system (HetNet) consisting of multiple tiers ofthe same operator was studied in [34] where all the tiers were operating in the same frequencyband. Similarly for mmWave, a numerical framework was presented for a mmWave HetNetwith generalized fading in [35]. In [35], only single operator was considered and BSs wereassumed to have open access to all the users, while as envisioned in this paper, commercialproviders are expected to have closed access to its customers or users. Cognitive networkshave also been studied using tools from stochastic geometry for CCF. For example, in [36],a stochastic geometry model was presented for spectrum sharing to characterize the systemperformance in terms of transmit capacity. In [37], a cognitive carrier sensing protocol wasproposed and analyzed for a network consisting of multiple primary and secondary users and spectrum access probabilities and transmission capacity were computed. There is limited workto analyze the impact of spectrum sharing among operators at CCF using stochastic geometry.For example, in [38], the impact of infrastructure and CCF spectrum sharing was studied usingstochastic geometry. As discussed above, communication at mmWave frequencies has differentcharacteristics and modeling considerations when compared to CCF. Therefore, these results needto be reevaluated for mmWave systems and a new mathematical model is required for possiblelicense sharing in mmWave systems. In the above mentioned prior work [11], [30]–[38], theimpact of inter-operator interference in mmWave systems and the feasibility of license sharingare not studied, which is the main focus here.
B. Contributions
We establish the feasibility of uncoordinated sharing of spectrum licenses in a multi-operatormmWave system. We model a multi-operator mmWave system where every operator owns aspectrum license of fixed bandwidth with a provision to share the complete rights over its licensedspectrum with other networks. Next, we compute the performance of such system in terms ofsignal-to-interference-and-noise (SINR) and rate coverage probability. A main conclusion of ourwork is that it is feasible to share spectrum licenses among multiple mmWave operators withoutrequiring any coordination among them. The main contributions of this paper can be categorizedbroadly as follows.
Modeling a multi-operator cellular system:
We model a multi-operator mmWave systemusing a two-level architecture, where the system consists of multiple heterogeneous networks,each owned by an independent mmWave cellular operator. Each operator owns a license for afrequency band and its network consists of its own BSs and users which are independent ofother networks. This model provides a general framework for access and license sharing byintroducing the notion of access and sharing groups. The operators can form multiple accessgroups where users of each member of an access group can connect to BSs of any member ofthe same access group. Similarly, the operators can form sharing groups where each member ofa sharing group can use the spectrum of all members of the same sharing group.
Establishing the feasibility of uncoordinated sharing of spectrum licenses:
We derive thecombined mean user-load on BSs of each network. We derive expressions for SINR and ratecoverage of a typical user of an operator using the tools from stochastic geometry. Then, we compare systems with shared licenses and exclusive licenses and show that spectrum licensesharing achieves higher performance in terms of per user rate. We investigate the effect ofantenna beamwidth on the feasibility of license sharing and show that spectrum license sharingis more favorable as communication becomes more directional. Finally, we quantify the benefitof license sharing in term of license cost reduction and show that networks can save significantamount of money if they share their licenses.
Modeling co-located operators:
To include the impact of correlation among deploymentsof operators, we also consider a co-located deployment where site locations are defined by asingle PPP and each operator has one BS at each site. We show that spectrum sharing is feasibleeven in this case. This important result indicates that multiple operators can also share siteinfrastructure (such as macrocell towers, as is common practice today) while also sharing theirspectrum licenses. A practical deployment with some co-location and some unique sites will liebetween these two extremes of independent and totally co-located BS locations.The rest of the paper is organized as follows. Section II explains the system model andenumerates few special combinations of accesses and license sharing schemes. In Section III,the expressions for SINR and rate coverage probability are derived. In Section IV, we comparethe aforementioned cases and Section V discusses the impact of partial loading where some BSsare turned off due to lack of users associated to them. Section VI presents numerical results andderives main insights of the paper. We finally conclude in Section VII.II. S
YSTEM M ODEL
We consider a system consisting of M different cellular operators which coexist in a particularmmWave band. Each operators owns a network Φ m consisting of its own BSs and users. Thelocations of BSs are modeled using a Poisson Point Process (PPP) with intensity λ m and thelocations of users are modeled as independent PPP with intensity λ u m . The PPP assumption canbe justified by the fact that nearly any BS distribution in 2D results in a small fixed SINR shiftrelative to the PPP [39], [40]. The locations of BSs of multiple operators have been modeled bysuperposition of independent PPPs in the past work [38]. Different point processes such as theLog-Gaussian Cox process can also be used to model the locations of a multi-operator systemas discussed in [41]. For tractability we have assumed independence among operators; but, toaddress the impact of correlation of locations among the operators, we also consider another case with full spatial correlation among operators where all operators are co-located with theirlocations defined by a single PPP. One can think of real multi-operator deployments as lyingbetween these two corner cases of complete independence and co-location. The BSs of eachnetwork can transmit with power P m . We denote the total spectrum by B and suppose that the m th operator owns a license for an orthogonal spectrum of B m bandwidth which it can sharewith others. We consider a typical user UE at the origin without loss of generality. Let usassume this is a user of the n th operator. A. Channel Model
Let us consider a link between this typical user and a BS of network m located at x distancefrom the origin. This link can be LOS or NLOS link which we denote by the variable link type s , which can take values s = L (for LOS) or s = N (for NLOS). We assume that the probabilityof a link being LOS is dependent on x and independent of types of other links and is given by p ( x ) = exp( − βx ) [11], [42]. The analysis can be extended to other blocking models. The pathloss from the BS to the user is modeled as (cid:96) s ( x ) = C s ( x ) − α s where α s is the pathloss exponentand C s is the gain for s type links. Conditioned on this typical user, the BSs of the eachnetwork m can be categorized into LOS or NLOS based on the type of their link to this typicaluser. Therefore, the BS PPP of network m can be seen as superposition of the following twoindependent (non-homogeneous) BS PPPs because of the independent thinning theorem [43],1. Φ m, L with density λ m, L ( x ) = λ m p ( x ) containing all the BSs with LOS link to UE , and2. Φ m, N with density λ m, N ( x ) = λ m (1 − p ( x )) containing all the BSs with NLOS link to UE .Note that this results in total M classes (known as tiers) of BSs where each tier is denotedby { m, s } . Here m and s represent the index of the network and the link type, respectively.The average number of BSs of tier { m, s } in area A is given by Λ m,s ( A ) = (cid:82) A λ m,s ( (cid:107) x (cid:107) )d x .Therefore, the average number of LOS and NLOS BSs in a ball B ( r ) with radius r for network m are given by Λ m, L ( B ( r )) = 2 π (cid:90) r λ m p ( x ) x d x = 2 π (cid:90) r λ m e − βx x d x = πλ m β γ (2 , βr )Λ m, N ( B ( r )) = 2 π (cid:90) r λ m (1 − p ( x )) x d x = πλ m (cid:18) r − β γ (2 , βr ) (cid:19) . Let us denote the j th BS of network m as BS mj . Hence, the effective channel between BS mj and the user UE is given as h mj P m (cid:96) s mj ( x mj ) where s mj denotes the link type between the user TABLE IS
UMMARY OF N OTATION
Notation Description Φ m , λ m PPP consisting of the locations of BSs of operator m , BS density of operator mP m , B m , W m Transmit power of BSs of operator m , licensed bandwidth of operator m , bandwidth available tooperator m via sharing of licenses Φ u m , λ u m PPP modeling locations of users of operator m , the density of this PPP L , N Possible values of link type: L denotes LOS, N denotes NLOS { m, p } Notation representing the tier having all BSs of link type p of operator m Φ mp , λ mp The PPP modeling the locations of BSs of the tier { m, p } , the density of this PPP G , G , θ b BS antenna parameters: maximum gain, minimum gain and half beamwidth C p , α p Path-loss model parameters: path-loss gain and path-loss exponent of any link of type p ∈ { L , N } p ( r ) , β p ( r ) is the probability of being LOS for a link of distance r , β is the blocking parameter. UE A typical user at origin for which analysis is performed. n The operator which UE belongs to S n The set consisting of all operators which a user of operator n has access to BS mj , x mj , x mj j th BS of the network m , its location (here x mj ∈ Φ m ), its distance from the origin s mj Fading faced by the link between BS mj and the user, the type of this link k, i The operator associated with UE , the index of the serving BS of this operator s, x Type of the link between the serving BS and the user, its distance from the origin Q k The sharing group of operator k which is also the set of all operators interfering to the operator k . m Indices representing an operator or a network, in particular a member of set Q k D ksmp ( x ) Exclusion radius for BSs of tier { m, p } when UE is associated with BS of tier { k, s } located at xA nk Association probability of a user of network n to be associated with BS of operator k P c ks Probability of SINR coverage of UE when associated with tier { k, s } N um The mean number of users associated to a BS of network mκ m ( z ) Probability of a BS of network m having z number of associated users R ki , R c k Instantaneous rate and rate coverage of UE when associated with network k R c ( ρ ) Rate coverage of UE , i.e. P [ R k ≥ ρ ] and BS mj and h mj is an exponential random variable denoting Rayleigh fading. We observe andshow in the numerical section that considering a more general fading model such as Nakagamidoes not provide any additional design insights, but it does complicate the analysis significantly.Therefore we will consider only Rayleigh fading for our analysis. We do not consider shadowingseparately as it is mostly covered by the blocking model. We show in the numerical section thatincluding shadowing does not change any of the observed trends. We assume that BSs of every network are equipped with a steerable antenna having radiationpattern given as [11] G ( θ ) = G | θ | < θ b G otherwise . Here G (cid:29) G and θ b denotes half beamwidth. The angle between the BS BS mj antennaand direction pointing to the user UE is denoted by θ mj . We have assumed the same antennapattern for all networks to avoid complicating the expressions unnecessarily. The analysis can beextended to a system where each network has a different transmit antenna pattern. We assumethat the user is equipped with a single omni-directional antenna. Although users will also havedirectional antennas, it will be analytically equivalent to aggregating the transmitter and receivergains at BS antennas. Considering the UE antenna gain and the BS antenna gain separately doesnot change the observed trends and hence, is left for future work. Note that we consider singlestream operation in this work. More advanced mmWave cellular systems may employ massiveMIMO [44], [45] or multi-stream MIMO using hybrid beamforming [4]. Generalizing to theseother architectures is a topic of future work. B. Access and License Sharing Model
We assume that a user of operator n can be associated with any BS from a particular set ofoperators denoted by access set S n . Two special cases of access are open and closed. In an openaccess system, a user can connect to any operator and therefore S n = { , , · · · M } . In a closedaccess system, a user can connect only to the operator it belongs to, and therefore S n = { n } .We assume that license sharing is performed by forming mutually exclusive groups, known as sharing groups . All the operators in each group share the whole spectrum license such that eachoperator within a group has equal bandwidth available to it. The effective bandwidth available toeach operator after sharing is denoted by W m . For example, in a system of 5 operators, supposethat operators 1, 2 and 3 form a group and operators 4 and 5 are in second group. Hence, afterlicense sharing, 1, 2 and 3 will have access to the the aggregate band of total B + B + B bandwidth, i.e. W = W = W = B + B + B . Similarly operators 4 and 5 will have access tothe aggregate band of bandwidth W = W = B + B . We denote the sharing group containingthe k th operator by Q k . The user UE experiences interference from all networks operating in the spectrum of associated operator k . The set of the interfering BS is equal to the sharing groupcontaining the k th operator which is Q k . Note that the aggregate spectrum of a sharing group canbe fully used by all members. Hence, for a particular network, the set of interfering networksremains the same for its complete available spectrum band. One example of license sharing isa system with fully shared licenses, in which there is only one sharing group containing all theoperators and all of them can use the whole frequency band. Here, the available bandwidth W k to each operator is B and the set of interfering networks is Q k = { , , · · · M } . For the casewhen all operators have exclusive licenses, there are M sharing groups each containing onlyone operator which indicates that no operator shares its license. Hence, the bandwidth availableto the operator k is W k = B k and the set of networks interfering to it is Q k = { k } .Now, the effective received power from a BS BS mj at user UE is given as P mj = P j h mj (cid:96) s mj ( x mj ) G ( θ mj ) . (1)Hence, the average received power from BS mj at UE without the antenna gain is given by P avg mj = P m (cid:96) s mj ( x mj ) . (2)We assume the maximum average received power based association in which any user associateswith the BS providing highest P avg mj among all the networks it has access to ( i.e. access set).Let us denote the operator the user UE associates with by k and the index of the serving BSby i . Since the serving BS aligns its antenna with the user, the angle θ ki between the servingBS antenna and user direction is o and the effective received power of this BS is given as P ki = P k h ki (cid:96) s ki ( x ki ) G (0) = P k h ki (cid:96) s ki ( x ki ) G . For each interfering BS BS mj where m ∈ Q k and ( m, j ) (cid:54) = ( k, i ) , the angle θ mj is assumed to be uniformly distributed between − π and π .Now, the SINR at the typical user UE that is associated with the i th BS of the operator k is given as SINR ki = P k h ki (cid:96) s ki ( x ki ) G σ k + I (3)where I is the interference from all BSs of networks in set Q k and is given by I = (cid:88) m ∈ Q k (cid:88) j ∈ Φ m P m h mj (cid:96) s mj ( x mj ) G ( θ mj ) = (cid:88) m ∈ Q k (cid:88) p ∈{ L , N } (cid:88) x mj ∈ Φ m,p \{ x ki } P m h mj (cid:96) p ( x mj ) G ( θ mj ) . (4)The noise power for operator k is given by σ k = N W k where N is the noise power density.Since σ k is dependent on the allocated bandwidth, it varies accordingly with association. III. SINR
AND R ATE C OVERAGE P ROBABILITY
One metric that can be used to compare systems is the SINR coverage probability. It is definedas the probability that the SINR at the user from its associated BS is above a threshold T P c ( T ) = P [ SINR > T ] , (5)and is equivalently the CCDF (complementary cumulative distribution function) of the SINR.In this section, we will first investigate the association of a typical user of n th network to a BSand then compute the coverage probability for this user. A. Association Criterion and Probability
Recall that a user of the n th operator can be associated with a BS of any operator from theset S n . Let E ki denote the event that the typical user is associated with the BS BS ki ( i.e. the i th BS of operator k ). Let us denote the distance of this BS by x = x ki and type by s = s ki for compactness. The event E ki is equivalent to the event that no other BS has higher P avg atthe user. This event can be further written as combination of following two events: (i) the eventthat no other BS of operator k has higher P avg at the user, and (ii) that no BS of any otheraccessible operator m has higher P avg at the user: E ki = { P avg ki > P avg kj ∀ i (cid:54) = j } ∩ { P avg ki > P avg mj ∀ m ∈ S n \ { k }} . (6)Substituting (2) in (6), E ki = (cid:26) C s ki ( x ki ) α ski > C s kj ( x kj ) α skj ∀ i (cid:54) = j (cid:27) (cid:92) (cid:26) C s ki P k ( x ki ) α ski > C s mj P m ( x mj ) α smj ∀ m ∈ S n \ { k } (cid:27) . The above condition can also be further split over LOS and NLOS tiers of each network, thenit can be expressed as an equivalent condition over locations of all BSs as follows: E ki = { x kj > x ∀ s kj = s, i (cid:54) = j } (cid:92) (cid:40) x kj > (cid:18) C s (cid:48) C s (cid:19) αs (cid:48) x αsαs (cid:48) ∀ s kj (cid:54) = s, i (cid:54) = j (cid:41)(cid:92) (cid:40) x mj > (cid:18) P m P k (cid:19) αs x ∀ s mj = s, m ∈ S n \ { k } (cid:41)(cid:92) (cid:40) x mj > (cid:18) P m C s (cid:48) P k C s (cid:19) αs (cid:48) x αsαs (cid:48) ∀ s mj (cid:54) = s, m ∈ S n \ { k } (cid:41) where s (cid:48) denotes the complement of the link type s . In other words, if s = L , then s (cid:48) = N and if s = N , then s (cid:48) = L . The first condition restricts all BSs of operator k with link type s (same as type of serving BS) to be located outside a 2D ball. The second term is for all BSs of operator k and link type s (cid:48) . Similarly the third and fourth terms are for BSs of all other accessible operatorswith link type s and s (cid:48) , respectively.As seen from these conditions, the average received power based association rule effectivelycreates exclusion regions around the user for BSs of each network in S n . Let us denote theexclusion radius for the tier { m, p } by D ksmp ( x ) . For example, the exclusion region for all LOSBSs of operator m when the user is associated with a NLOS BS of operator k is given by D k N m L ( x ) = (cid:18) P m P k C L C N (cid:19) α L x α N α L . (7)Note that for BSs of the networks that are not in set S n , there are no exclusion regions, i.e. D ksmp ( x ) = 0 ∀ m / ∈ S n . This exclusion region denotes the region where interfering BSs cannotbe located and hence, affects the sum interference.The probability that all BSs of the tier { m, p } are outside the exclusion radius d is given by thevoid probability of the PPP Φ mp which is µ m,p ( d ) = exp( − Λ m,p ( B ( d ))) . Since the PPPs of thetiers are mutually independent, the probability that BSs of the tiers other than { k, i } are locatedoutside the exclusion region, can be calculated by multiplying the individual void probabilitiesof each tier: f ok,s ( x ) = µ k,s (cid:48) (cid:0) D ksks (cid:48) ( x ) (cid:1) (cid:89) m ∈ S n \{ k } µ m,s (cid:0) D ksms ( x ) (cid:1) µ m,s (cid:48) (cid:0) D ksms (cid:48) ( x ) (cid:1) . (8)Therefore, the probability density function of the distance x to this associated BS is given as f k,s ( x ) = 2 πλ k,s x µ k,s ( x ) µ k,s (cid:48) (cid:0) D ksks (cid:48) ( x ) (cid:1) (cid:89) m ∈ S n \{ k } µ m,s (cid:0) D ksms ( x ) (cid:1) µ m,s (cid:48) (cid:0) D ksms (cid:48) ( x ) (cid:1) . (9)The probability that a user of operator n is associated with a BS of operator k can be computedby summation over both LOS and NLOS tiers: A nk = (cid:90) ∞ ( f k, L ( x ) + f k, N ( x )) d x. (10)Let P c k L and P c k N denote the coverage probabilities for the typical user which is associatedwith a LOS and NLOS BS of operator k , respectively. They can be computed by integrating theCCDF of SINR from serving BS over pdf of distance x from serving BS as follows: P c ks ( T ) = (cid:90) ∞ P [ SINR ks ( x ) > T ] f k, L ( x )d x = (cid:90) ∞ P (cid:2) P k h ks C s G (0) > T ( I + σ k ) x α s (cid:3) f k,s ( x )d x. (11) Since h ks ∼ exp(1) , the probability in (11) can be replaced as P c ks ( T ) = (cid:90) ∞ E (cid:20) exp (cid:18) − T σ k x α s C s G P k − T Ix α s C s G P k (cid:19)(cid:21) f k,s ( x )d x = (cid:90) ∞ exp (cid:18) − T N W k x α s C L G P k (cid:19) L I (cid:18) T x α s C s G P k (cid:19) f k, L ( x )d x (12)where L I ( t ) denotes the Laplace transform of the interference I caused by BSs of all networksin set Q k and is defined as L I ( t ) = E (cid:2) e − tI (cid:3) .Since the association with different tiers are disjoint events, the SINR coverage probability ofthe typical user can be computed by summing these individual tier coverage probabilities overall accessible tiers: P c ( T ) = (cid:88) k ∈ S n P c k ( T ) = (cid:88) k ∈ S n [P c k L ( T ) + P c k N ( T )] , (13)where P c k ( T ) is the sum of coverage probabilities of both tiers of the operator k . To proceedfurther, we need to first characterize the interference I for which we will compute its Laplacetransform defined as L I ( t ) = E (cid:2) e − tI (cid:3) . B. Interference Characterization
If the user of operator n is associated with operator k , it experiences interference from allnetworks operating in spectrum W k . Recall that all these interfering networks form set Q k .Hence, the total interference is given by (4). Due to mutual independence of the tiers, its Laplacetransform can written as product of the following terms: L I ( t ) = E (cid:2) e − tI (cid:3) = (cid:89) m ∈ Q k L I m ( t ) = (cid:89) m ∈ Q k ( L I m L ( t ) L I m N ( t )) (14)where L I m ( t ) refers to the interference caused by network m and L I m L ( t ) and L I m N ( t ) denotethe Laplace transforms of LOS and NLOS interference from network m which are given in thefollowing Lemma. Lemma 1.
The Laplace transforms of the interference from LOS and NLOS BSs of network m to a user of operator n which is associated with a type s BS of operator k in a multi-operatorsystem are given as L I m L ( t ) = exp (cid:0) − λ m [ θ b F L ( β, α L , tG P m C L , D ksm L ( x )) + ( π − θ b ) F L ( β, α L , tG P m C L , D ksm L ( x ))] (cid:1) L I m N ( t ) = exp (cid:0) − λ m [ θ b F N ( β, α N , tG P m C N , D ksm N ( x )) + ( π − θ b ) F N ( β, α N , tG P m C N , D ksm N ( x ))] (cid:1) where F L ( b, a, A, x ) = (cid:90) ∞ x e − by Ay − a Ay − a y d y, and F N ( b, a, A, x ) = (cid:90) ∞ x (1 − e − by ) Ay − a Ay − a y d y. Proof:
See Appendix ANote that the term containing G and θ b denotes the interference from the aligned BSs whoseantennas are directed towards the considered user while the term containing G and ( π − θ b ) represents the interference from the unaligned BSs. Since G (cid:29) G , the interference fromthe aligned BS is significantly larger than the interference from the unaligned BS and hence,dominates the Laplace transform expression. Therefore, the value of θ b plays a significant rolein characterizing the interference and determining the benefits of spectrum license sharing.Now, we provide the final expression for SINR coverage probability. Theorem 1.
The SINR coverage probability of a typical user of operator n in a multi-operatormmWave cellular system is given as P c ( T ) = (cid:88) k ∈ S n (cid:88) s ∈{ L , N } (cid:90) ∞ (cid:89) m ∈ Q k L I m L (cid:18) T x α s C s G P k (cid:19) L I m N (cid:18) T x α s C s G P k (cid:19) exp (cid:18) − N W k T x α s C s G P k (cid:19) f k,s ( x )d x (15) where L I mp ( t ) is computed in Lemma 1 and f k,s ( x ) is given in (9) .Proof: Substituting the value of L I ( t ) from Lemma 1 in (13), we get the result.In (15), the first summation is over all networks which the user of operator n can connect to,weighted by the association probability. This weighting is included inside the term f k,s ( x ) .Since due to complexity of the above expressions, it is difficult to derive direct insights, wealso consider a simple case to simplify the expressions. Corollary 1.1.
Consider a mmWave system with n identical operators with closed access andfully shared licenses. Assuming that the LOS and NLOS channel have same pathloss parameterswith α L = α N = 4 and the side lobe gain G = 0 , the probability of coverage for the interferencelimited scenario is given as P c ( T ) = 11 + θ b π T (cid:16) n π − arctan( T − ) (cid:17) . (16)It can be observed from the (16) that the probability of coverage decreases monotonicallywith the number of operators n . C. Rate Coverage
While the SINR shows the serving link quality, the per-user rate represents the data bitsreceived per second per user and is one of the main goal for using mmWave bands. In thissection, we derive the downlink rate coverage which is defined as the probability of the rate ofa typical user being greater than the threshold ρ , R( ρ ) = P [Rate > ρ ] . (17)Let us assume that O k denotes the time-frequency resources allocated to each user associatedwith the ‘tagged’ BS of operator k . Therefore, the instantaneous rate of the considered typicaluser is given as R ki = O k log (1 + SINR ki ) . The value of O k depends upon the number of users( N u k ), equivalently the load, served by the tagged BS. The load N u k is a random variable due tothe randomly sized coverage areas of each BS and random number of users in the coverage areas.As shown in [32], [46] approximating this load with its respective mean does not compromisethe accuracy of results. Since user distribution of each network is assumed to be PPP, the averagenumber of users associated with the tagged BS of network k associated with the typical usercan be modeled similarly to [32], [46]: N u k = 1 + 1 .
28 1 λ k (cid:88) m : k ∈ S m λ u m A mk . (18)Note that the summation is over all the networks whose users can connect to the network k andthe sum denotes the combined density of associated users from each network.Now, we assume that the scheduler at the tagged BS gives /N u k fraction of resources to eachof the N u k users. This assumption can be justified as most schedulers such as round robin orproportional fair give approximately /N u k fraction of resources to each user on average. Usingthe mean load approximation, the instantaneous rate of a typical user of operator n which isassociated with i th BS of operator k is given as R ki = W k N u k log (1 + SINR ki ) . (19)Let R c k ( ρ ) denote the rate coverage probability when user is associated with operator k . Thenthe total rate coverage will be equal to the sum of R c k ( ρ ) ’s over all accessible networks: R c ( ρ ) = (cid:88) k ∈ S n R c k ( ρ ) . (20) R c k ( ρ ) can be derived in terms of SINR coverage probability as follows: R c k ( ρ ) = P [ R ki > ρ ] = P [ W k /N u k log (1 + SINR ki ) > ρ ]= P (cid:20) SINR ki > ρ N u kWk − (cid:21) = P c k (cid:0) ρN u k /W k − (cid:1) . Now, the rate coverage is given as R c ( ρ ) = (cid:88) k ∈ S n P c k (cid:0) ρN u k /W k − (cid:1) . (21)We, now, turn back to the simple case considered in Corollary 1.1 to simplify the expressionsand provide further insights. Corollary 1.2.
Consider a mmWave system with n identical operators with closed access andfully shared licenses of bandwidth ¯ B each as considered in Corollary 1.1. The rate coverage forthe interference limited scenario is given as R c ( ρ ) = 11 + θ b π (2 ρ (cid:48) /n − (cid:16) n π − arctan((2 ρ (cid:48) /n − − ) (cid:17) (22)where ρ (cid:48) = ρN u / ¯ B .The denominator in (22) behaves differently with respect to n for different regimes of ρ (cid:48) . Forsmall ρ , rate coverage decreases with respect to n making exclusive licenses more beneficial. Forlarge ρ , the rate coverage increases with respect to n which favors sharing spectrum licenses. Letus compare the case of n = 1 and n = 2 . For θ b = π , the license sharing becomes more beneficialthan exclusive licenses when ρ (cid:48) > . which is equivalent to R c = 0 . indicating that atmaximum, only 20% users would be benefiting if the licenses are shared. This effectively meansthat spectrum sharing is not beneficial from the operator’s perspective. Whereas for θ b = 10 o , theswitch occurs at R c = 0 . indicating that 83% users are benefiting from spectrum sharing andmaking sharing of licenses favorable from the operator’s perspective. Therefore, the directionalityof antennas ( θ b ) plays a crucial role in determining the least value of R c where license sharingstarts becoming more beneficial than exclusive licenses. We later show similar trends with targetrate and beamwidth using simulations. IV. P
ERFORMANCE C OMPARISON
We use the preceding mathematical framework to compare the benefits of spectrum licensesharing. We enumerate three specific cases (or systems) considering different combinations ofaccesses and license sharing schemes. Also see Fig. 1 for a visual explanation.
System 1: Closed Access and Exclusive Licenses:
In System 1, each user must associate withonly its own network and each operator can use its own spectrum only. This case is equivalentto a set of M single operator systems which has been studied in prior work [11]. This systemserves as a baseline case to evaluate benefits of sharing. The SINR coverage probability of UE isgiven by (15) with k = n , S n = { n } . Recall that in this system, the spectrum accessible to eachoperator is its own licensed spectrum only, i.e. W n = B n , Q n = { n } . System 2: Open Access and Full Spectrum License Sharing:
In System 2, each user canbe associated with any network and the spectrum license is shared between all operators. In thiscase, the probability of association of a user with any BS of the operator k is independent ofwhich operator this user belongs to and is given as A nk = A k = A k, L + A k, N = (cid:90) ∞ ( f k, L ( x ) + f k, N ( x )) d x. The SINR coverage probability of UE is given by (15) with S n = { , , · · · M } . The averageload to the tagged BS of operator k is given as N u k = 1+1 . (cid:80) m λ u m A k λ k . The spectrum accessibleto each operator W k is the complete band B and Q n = { , · · · M } . Since users can be servedby BS of any operator, it requires full coordination, sharing of control channels and technologysharing among operators. The quality of service to a user will be the same regardless of theoperator it belongs to. This will limit the technological advantage of an operator over otheroperators and hence, operators may not want to open up their networks to each other. Therefore,System 2 is likely not a practical system but instead serves as an upper bound to the two othermore practical systems. Note that if all M networks are identical with respect to every parameter,then this system is equivalent to a single operator system with the aggregate BS and UE density.In this case, from a user association perspective, there is no discrimination based on the networkwhich a particular BS belongs to. Also all networks transmit in the same band. So the userseffectively see a single network with aggregate BS density of all the networks. Similarly fromthe operator’s perspective, users of all the networks look the same due to open access. Hence, System 1 System 2 System 3 System 4 Accessible links
Interfering links
Spectrum: Licensed Available B B B B B B B B Network 2 BS Network 1 BS User of network 1
Fig. 1. Illustration describing the differences between the four systems. For a typical user of operator 1, the figure shows allaccessible networks this user can connect to, all interfering networks, and the available spectrum after license sharing. the users of different operators can be replaced by users of a single operator with the aggregateUE density.
System 3: Closed Access and Full Spectrum License Sharing:
In System 3, each usermust associate with its own network but the whole spectrum is shared between all the operators.This case does not require any transmission coordination among networks or common controlchannels, nor does it require sharing of infrastructure or back-haul resources. This system isclose to the practical implementation where subscribers must connect to their respective serviceproviders only. The SINR coverage probability of the considered typical user of operator n isgiven by (15) with k = n , S n = { n } . For this system, the spectrum accessible to each operatoris B and Q n = { , · · · M } . Along with the above three systems, we consider one additional system System 4 as definedbelow. This system will help us understand if independent operators can still share BS towerinfrastructure while sharing the spectrum licenses. As mentioned before, there may be correlationamong BSs locations of different operators with possible co-location of BSs, therefore it isimportant to understand how gain from license sharing will be affected when the correlation ispresent.
System 4: Co-located BSs with Closed Access and Full Spectrum License Sharing:
System4 has closed access and fully shared licenses where the respective BSs of all the operators areall co-located. The system model for this case remains the same as the previous three systemsexcept for the following two differences: 1) The BS locations are modeled by a single PPP
Φ = { x j } with intensity λ and 2) for a typical user, the BSs of all the operators located atthe same location are either all LOS or all NLOS. We first briefly show the computation theprobability of SINR coverage of this system. Consider a typical user of operator n . The BS PPP Φ can be divided into two independent PPPs, Φ L and Φ N with intensity λ L ( x ) = λp ( x ) and λ N ( x ) = λ (1 − p ( x )) . The probability density function of the distance x to the associated BS ofoperator n is given as f s ( x ) = 2 λ s πx exp ( − Λ s ( B ( x ))) exp ( − Λ s (cid:48) ( B ( D ss (cid:48) ( x )))) (23)where the exclusion radius D sp ( x ) is the same for BSs of all the operators and given as D sp ( x ) = (cid:16) C p C s (cid:17) αp x αsαp . The interference I at the user from BSs of all the operators is given as I = x − α s C s (cid:88) m ∈ Q n \{ n } P m h mi G ( θ mi ) + (cid:88) p =L , N (cid:88) x j ∈ Φ p \{ x i } x − α p j C p (cid:88) m ∈ Q n P m h mj G ( θ mj ) . (24)The following Lemma characterizes the Laplace transform of the interference in (24) in theco-located BSs case. Lemma 2.
The Laplace transform of interference to a typical user of operator n with closedaccess which is associated to a BS of type s in a multi-operator system with co-located BSs isgiven as L I ( t ) = (cid:89) m ∈ Q n \{ k } (cid:18) θ b /π tx − α s C s P m G + ( π − θ b ) /π tx − α s C s P m G (cid:19) × (cid:89) p =L , N exp (cid:32) − πλ (cid:90) ∞ D sp ( x ) p ( y ) (cid:32) − (cid:89) m ∈ Q n (cid:18) θ b /π ty − α p C p P m G + ( π − θ b ) /π ty − α p C p P m G (cid:19)(cid:33) y d y (cid:33) Proof:
See Appendix B.Similar to previous subsections, the coverage probability of the considered typical user isgiven as P c ( T ) = (cid:88) s =L , N (cid:90) ∞ exp (cid:18) − T N W n x α s C s G (0) P n (cid:19) L I (cid:18) T x α s C s G (0) P n (cid:19) f s ( x )d x (25)where L I ( t ) is given in Lemma 2 and f s ( x ) is given in (23). Since full sharing of license is as-sumed for this system, the spectrum accessible to each operator is B and and Q n = { , · · · M } .Hence, similar to System 3, the average load is given as N u n = 1 + 1 . λ u n λ n .V. P ARTIAL L OADING OF T HE N ETWORK
In the previous section, we assumed that all the BSs have at least one user associated andthey are all transmitting. Such an assumption is justified when the user density is very high incomparison to the BS density resulting in many associated users per BS and negligible probabilityof any BS being inactive. For very dense networks, however, this assumption will break downand many BSs will not be occupied at all times. An interesting case to consider is where thesystem is not fully loaded and there are some BSs having no (active) users to associate with. Insuch a case, interference will reduce which should favor license sharing. From [46], the numberof users associated with a BS of network m for a closed access system can be approximated asthe following distribution: κ m ( N u m ) = 3 . . Γ( n + 1) Γ( n + 3 . .
5) ( η m ) n (3 . η m ) − n − . (26)where η m denotes the ratio between density of associated users and BS density for network m .The κ m ( N u m ) approximation has been shown to closely match the load distribution in simulations[32], [46]. For the multi-network system, η m is computed as η m = (cid:88) q : m ∈ S q λ u q A qm λ m , (27)where the sum is over all the networks whose users can connect to BSs of network m . Themultiplication of association probability of a user of network q and user density of network q gives the density of the network q ’s associated users for network m .The probability that a BS is off is equal to the probability that a BS has no user associated toit which is equal to κ m (0) . Therefore for Systems 1, 2 and 3, the interfering BSs can be obtained by independent thinning of original BS PPP with probability − κ m (0) . The coverage probabilityof the Systems 1, 2 and 3 are given by Theorem 1 with λ m substituted by λ (cid:48) m = λ m (1 − κ m (0)) .For System 4, the sum interference I at the user from BSs of all the networks in partialloading case is given as I = x − α s C s (cid:88) m ∈ Q n \{ n } P m h im G ( θ im ) δ mi + (cid:88) p =L , N (cid:88) x j ∈ Φ p \{ x i } x − α p j C p (cid:88) m ∈ Q n P m h mj G ( θ mj ) δ mj (28)where δ mj is an indicator which is 1 when BS BS mj is on. Hence, the Laplace transform ofinterference in partial loading can be computed as (see Appendix C for proof) = L I ( t ) = (cid:89) m ∈ Q k \{ k } (cid:32) κ (0) + (1 − κ (0)) (cid:88) j =1 , a j /π tx − α s C s P m G j (cid:33) × (cid:89) p =L , N exp (cid:32) − λ (cid:90) ∞ D sp ( x ) p ( y ) (cid:32) − (cid:89) m ∈ Q k (cid:32) κ (0) + (1 − κ (0)) (cid:88) j =1 , a j y − α p tC p P m G j (cid:33) (cid:33) y d y (cid:33) (29)where a = θ b , a = π − θ b . The coverage probability of this system is given by (25) with L I ( t ) given in (29) and f s ( x ) given in (23).VI. N UMERICAL R ESULTS
In this section, we provide results numerically computed from the analytic expressions derivedin previous sections. We compare the four aforementioned systems to provide insights and discussthe impact of license sharing. For these numerical results, we consider a system consisting oftwo cellular operators with identical parameters, both operating in mmWave band. Each operatorowns a network of BSs with density of / km which is equivalent to average cell radius of 103m and have users with density of / km . We consider the exponential blockage model i.e. p ( x ) = exp( − βx ) with β = 0 . which has an average LOS region of 144 m. The transmitpower is assumed to be 26dBm. For most of the results, the operating frequency is 28GHz forwhich pathloss exponents for LOS and NLOS are α L = 2 , α N = 4 and the corresponding gainsare C L = − dB , C N = − dB. The total system bandwidth is 200 MHz. We assume that eachoperator owns a license to 100 MHz. Recall that for System 1, each operator can use only itsown spectrum. In Systems 2, 3 and 4, both operators share each other’s spectrum licenses andtherefore, get 200MHz of spectrum. −40 −30 −20 −10 0 10 20 30 400.10.20.30.40.50.60.70.80.9 SINR Threshold T (dB) P r ob a b ili t y o f C o ve r a g e P c SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co−located BSsSimsTheory
Fig. 2. Probability of SINR coverage in a two-network mmWave system with BS antenna half beamwidth θ b = 10 o for differentcases. Line-curves denote values from the analysis and markers denote respective values from simulation. Validation of analysis and SINR coverage trends:
Fig. 2 compares the probability ofcoverage for these systems and validates our analysis with simulation. The typical user in System2 has high SINR coverage due to its open access. The closed access in System 3 allows BSs ofanother networks to be located closer than the serving BS and may lead to large interference.Therefore the user in System 3 has low SINR coverage. System 1 has the same closed accessas System 3, but the spectrum is not shared. Therefore the user faces no interference fromother networks and hence, the SINR coverage is greater than System 3. In comparing System1 and System 2, we observe different behaviors for different SINR ranges. Recall that System2 is similar to System 1, but with double BS and MS density and double bandwidth. Hence,a serving BS is relatively closer in System 2 from System 1 by a factor of √ . Now, for thehigh SINR region (which is mainly due to LOS serving links), this increases the received powerof a serving BS by a factor of √ α L = 2 . Since the noise power also increases by a factorof 2 due to increase in bandwidth, it effectively cancels the increase in the power caused byincreased proximity of serving BS. As far as interference is considered, since doubling densityincreases the probability of interferers to be LOS, interference increases significantly in System2. Therefore SINR coverage is higher in System 1 than System 2. For the low SINR region(which is mostly due to NLOS serving links), the received power of a serving BS increasesby a factor of √ α N = 4 and the probability of serving link turning to LOS is also increased due to increased proximity of serving BS. Therefore System 2 has higher SINR coverage thanSystem 1 in the low SINR region. System 4 has similar values and trends as System 3 whencompared to other systems. When we compare System 3 and System 4, we see a tradeoff dueto co-location of BSs. Since BSs are co-located in System 4, there cannot be any BS of otheroperators closer than the serving BS from the typical user. But in System 3, there can be aninterfering BS closer to the user than the serving BS due to the independence assumption. Thiscauses System 4 to perform better than System 3 for cell edge users where serving signal poweris already low. But the same co-location argument also guarantees that System 4 will alwayshave M − interfering BSs of the other operators at the same location as serving BS, whichis not the case in System 3. Therefore, System 3 performs even better for the users with highserving signal power. Hence, we can observe that SINR coverage of System 3 is better for highvalues of SINR thresholds, while System 4 performs better for low SINR thresholds. Sharing licenses achieves higher rate coverage:
Fig. 3 compares the probability of ratecoverage for four systems which incorporates the effect of load and bandwidth. Since eachnetwork has a large bandwidth and large SINR coverage in System 2, its rate coverage is thehighest among all systems. Such a system, however, as mentioned earlier, may not be practicaland mainly serves as a benchmark for practical systems. A more interesting comparison isbetween System 1 and 3 (or 4). Here we can see that even though System 1 has higher SINRcoverage than System 3 (and 4), the latter achieves higher median rate, due to the extra bandwidthgained from spectrum license sharing. In particular, System 3 and 4 have respectively 25% and32% higher median rates than System 1.
Validation of the model with realistic scenario:
To validate our PPP assumption and toshow that it is reasonable, we also present a simulation result where the BSs are deployed ina square grid and the BS antenna pattern is parabolic, as specified by the 3GPP standard [47].In System 2 and 3, both operators have their own square grid deployments which is shiftedfrom each other by a random amount in each realization of the simulation. We also considerlog-normal shadowing ( σ LOS = 5 . dB and σ NLOS = 7 . dB). Fig. 4 compares the probability ofrate coverage for four systems with these modifications. We observe similar trends for spectrumlicense sharing which justifies our assumptions regarding deployment and shadowing. Fig. 5shows the probability of rate coverage for four systems with Nakagami fading (with parameter ) instead of Rayleigh. It can be seen that the trends are similar to those of Rayleigh distribution Rate Threshold ρ ( × R a t e C o ve r a g e R c SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 3. Rate coverage in a two-network mmWave system with Rayleigh fading and BS antenna half beamwidth θ b = 10 o fordifferent cases. Systems 3 and 4 with shared license perform better than System 1 with exclusive licenses. Rate Threshold ρ ( × R a t e C o ve r a g e R c SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 4. Rate coverage in a two-network mmWave system with grid BS deployment and 3GPP antenna pattern [47] for differentcases. The trends for this case are similar to Fig. 3 which validates our assumptions regarding the system model. as we claimed in Section IIA.
Impact of beamwidth on median rate:
Fig. 6 compares the median rate of the four systemsfor various values of beamwidth. It can be seen that above a certain threshold for the beamwidth,it becomes more beneficial to have exclusive license, due to high interference. As the beamwidthdecreases, license sharing becomes more beneficial. For the given parameters, the threshold isat about o . Since mmWave has typical beamwidth less than o , sharing should increase the Rate Threshold ρ ( × R a t e C o ve r a g e R c SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 5. Rate coverage in a two-network mmWave system with Nakagami fading (with parameter 10) and BS antenna halfbeamwidth θ b = 10 o for different cases. When compared to Rayleigh fading (Fig. 3), the insights are similar which justifiesthe Rayleigh fading assumption for analysis Half Beamwidth θ b (degrees)0 10 20 30 40 50 60 M e d i a n R a t e ( M bp s ) SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 6. Median rate versus BS antenna beamwidth in two-network mmWave system under different cases for Rayleigh fading.Systems with sharing of license outperforms System 1 with no shared license for moderate and low values of antenna beamwidth. achievable rate.
Partial loading favors sharing:
Fig. 7 compares the probability of rate coverage for foursystems under partial loading with user density of / km . It can be observed that due to reducedinterference, System 3 (and 4) has even higher gain than System 1. In particular, System 3 has40% higher median rate than System 1 in partial loading case compared to only 25% gain in Rate Threshold ρ ( × R a t e C o ve r a g e R c SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 7. Rate coverage in a two-network mmWave system under partial loading with Rayleigh fading for different cases. Partialloading favors spectrum license sharing.
Half Beamwidth θ b (degrees)0 10 20 30 40 50 60 R e qu i r e d B a nd w i d t h B k ( M H z ) SYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 8. Required bandwidth for each network (with sharing of licenses) to achieve the same median rate achieved by the networkwith no sharing of spectrum with each network having 100MHz spectrum license. Sharing can reduce the license cost by morethan 25% the previous case when user density was / km . Sharing reduces spectrum cost significantly:
Now, we compare the following two cases. Inthe first case, each network owns a 100 MHz bandwidth exclusive license. This case is the sameas System 1. In the second case, the networks share licenses completely and choose to buy justenough spectrum to achieve the same median rate as in the first case. Fig. 8 shows this requiredspectral bandwidth for each network. With a o beamwidth antenna, each network only needs Number of sharing networks |Q n |0 2 4 6 8 10 R a t e ( i n M bp s ) Median rate75 th percentile rate25 th percentile rate Fig. 9. Rate versus number of sharing networks in a mmWave cellular system with 10 networks. A trade-off between increasingthe available bandwidth and increasing interference is observed. to buy 75 MHz of bandwidth which would save 25% of the license cost assuming linear pricingof the spectrum.
Optimal cardinality of sharing groups depends on the target rate:
Now, we consider asystem with 10 operators with 50MHz bandwidth each and closed access. Fig. 9 shows variationof the per-user rate for different percentiles with respect to cardinality | Q n | of sharing groupwhich is equal to the number of operators sharing licenses with network n . We can see that the75 th percentile rate increases with | Q n | while the 25 th percentile rate decreases. For the medianrate, we see an increase up to | Q n | = 3 and then the median rate decreases. This trade-off isdue to the fact that as more operators share their licenses, the total available bandwidth and thesum interference both increase. It can be observed that depending on the target performance, theoptimal number of networks that should share their licenses varies. Impact of asymmetry among operators:
We now consider the case when both operators arenot identical. Fig. 10 shows the variation of median rate of the first and the second operatorswith respect to BS density of the first operator in a two-operator system with fixed user densityof 200/km . The BS density of first operator is fixed at 30/km . The gain from license sharingincreases as the BS density increases for the operator with higher density and decreases for theother operator. We can observe that both operators can simultaneously gain from license sharingonly when both operators have similar BS densities which is between 15 to 45/km here. First Operator's BS density (/km )
20 40 60 80 M e d i a n R a t e ( i n M bp s ) First Operator
SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharing
First Operator's BS density (/km )
20 40 60 80050100150200
Second Operator
SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharing
Fig. 10. Variation of median rate with BS density of first operator in a two operator system. The BS density of second operatoris fixed at 30/km . The user densities of both operators are same and fixed. Both operators gain from license sharing only whenfirst operator density between 15 to 45/km . Spectrum license sharing is more beneficial for the operator with higher density. Rate Threshold ρ ( × R a t e C o ve r a g e R c SYS1: Closed, No sharingSYS2: Open, Full sharingSYS3: Closed, Full sharingSYS4: Co-located BSs
Fig. 11. Rate coverage in a two-network mmWave system at 73 GHz frequency with Rayleigh fading for different cases. Similartrends for spectrum sharing are observed for the 28 GHz and 73 GHz bands.
Results for 73GHz band:
We also consider mmWave communication at 73GHz with 1GHz bandwidth. The near-field path-loss gains are decreased by a factor of
10 log 10(73 / =8 . dB when compared with 28GHz for both LOS and NLOS. Fig. 11 compares the rate coveragefor four systems for 73GHz and shows similar trends as 28 GHz. Due to reduced interference,license sharing achieves slightly higher gain (28%) compared to 28 GHz case ( i.e. VII. C
ONCLUSIONS
We have modeled a two-level architecture of a mmWave multi-operator system and derivedthe SINR and per-user rate distribution. We show that license sharing among operators improvessystem performance by increasing per-user rate. We conclude that it is economical for operatorsto share their spectrum licenses without increasing any overhead. We show that narrow beamsplay a key role in determining the feasibility of spectrum sharing. Since an increasing numberof networks increases both the sum interference and bandwidth, the optimal cardinality of thesharing group will depend on the target rate.This work would seem to have numerous extensions. First, it is worth investigating howmulti-antenna techniques such as multiplexing including hybrid beamforming affect the insightsabout license sharing. Second, it is interesting to understand how other essential infrastructureincluding backhaul can be shared among networks reducing the cost further, especially in thecase of co-located BSs. A
PPENDIX AP ROOF OF L EMMA m at UE is given as I m L = (cid:88) x mj ∈ Φ (cid:48) m, L h mj (cid:107) x mj (cid:107) − α L P m C L G ( θ mj ) where Φ (cid:48) m, L = Φ m, L ∩ ¯ B (cid:0) , D ksm L ( x ) (cid:1) and ¯ B (0 , r ) denotes the compliment of a ball of radius r located at origin. This is due to the fact that all interfering BSs are located outside the radius D ksm L ( x ) . The Laplace transform of I m L is given as L I m L ( t ) = E exp − t (cid:88) x mj ∈ Φ (cid:48) m L h mj (cid:107) x mj (cid:107) − α L P m C L G ( θ mj ) . Now, using the PGFL of PPP [43], the Laplace transform can be written as L I m L ( t ) = exp (cid:32) − πλ m (cid:90) ∞ D ksm L ( x ) p ( y ) (cid:16) − E (cid:104) e − thG ( θ ) P m C L y − α L (cid:105)(cid:17) y d y (cid:33) . Now, using the moment generating function (MGF) of exponentially distributed h and pdf ofuniformly distributed θ , we get L I m L ( t ) = exp (cid:32) − λ m (cid:90) ∞ D ksm L p ( y ) (cid:18) π − (cid:90) π dθ tG ( θ ) P m C L y − α L (cid:19) y d y (cid:33) . Now, integrating with θ and then using the definition of function F L ( · ) , we get L I m L ( t ) = exp (cid:32) − λ m (cid:90) ∞ D ksm L (cid:90) π tG ( θ ) P m C L y − α L tG ( θ ) P m C L y − α L d θe − βy y d y (cid:33) = exp (cid:32) − λ m (cid:90) ∞ D ksm L (cid:18) θ b tG P m C L y − α L tG P m C L y − α L + ( π − θ b ) tG P m C L y − α L tG P m C L y − α L (cid:19) e − βy y d y (cid:33) = exp (cid:0) − λ m (cid:2) θ b F L (cid:0) β, α L , tG P m C L , D ksm L ( x ) (cid:1) + ( π − θ b ) F L (cid:0) β, α L , tG P m C L , D ksm L ( x ) (cid:1)(cid:3)(cid:1) . The Laplace transform of the interference from the NLOS BSs can be computed similarly.A
PPENDIX BP ROOF OF L EMMA L I ( t ) = E h,θ (cid:104) e − tx − αs C s (cid:80) m ∈ Qk \{ k } P m h mi G ( θ mi ) (cid:105) (cid:89) p =L , N E Φ p ,h,θ (cid:20) e − t (cid:80) x j ∈ Φ p \{ x i } x − αpj C p (cid:80) m ∈ Qk P m h mj G ( θ mj ) (cid:21) . Now, using the PGFL of PPP and independence of h mi ’s and θ mi ’s, L I ( t ) can be written as L I ( t ) = (cid:89) m ∈ Q k \{ k } E h,θ (cid:104) e − tx − αs C s P m h mi G ( θ mi ) (cid:105)(cid:89) p =L , N exp (cid:32) − πλ (cid:90) ∞ D sp ( x ) p ( y ) E h,θ (cid:104) − e − ty − αp C p (cid:80) m ∈ Qk P m h m G ( θ m ) (cid:105) y d y (cid:33) . Now, using the MGF of exponentially distributed h mi ’s in first product term and the independenceof h m ’s in the second product term, we get L I ( t ) = (cid:89) m ∈ Q k \{ k } E θ (cid:20)
11 + tx − α s C s P m G ( θ mi ) (cid:21)(cid:89) p =L , N exp (cid:32) − πλ (cid:90) ∞ D sp ( x ) p ( y ) E θ (cid:34) − (cid:89) m ∈ Q k E h (cid:104) e − ty − αp C p P m h m G ( θ m ) (cid:105)(cid:35) y d y (cid:33) . Now, using the MGF of exponentially distributed h m ’s, we can write E h (cid:2) e − ty αp C p P m h m G ( θ m ) (cid:3) = 1 / (cid:0) ty − α p C p P m G ( θ m ) (cid:1) . Since G ( θ m ) ’s are discrete random variables with P [ G ( θ m ) = G ] = θ b /π and P [ G ( θ m ) = G ] =1 − θ b /π , L I ( t ) can be further written as L I ( t ) = (cid:89) m ∈ Q k \{ k } (cid:18) θ b /π tx − α s C s P m G + ( π − θ b ) /π tx − α s C s P m G (cid:19) × (cid:89) p =L , N exp (cid:32) − πλ (cid:90) ∞ D sp ( x ) p ( y ) (cid:32) − (cid:89) m ∈ Q k (cid:18) θ b /π ty − α p C p P m G + ( π − θ b ) /π ty − α p C p P m G (cid:19)(cid:33) y d y (cid:33) which proves the Lemma. A PPENDIX CP ROOF OF THE L APLACE T RANSFORM FOR C O - LOCATED BS UNDER P ARTIAL L OADING
The Laplace transform of interference in partial loading case (28) is given as L I ( t ) = E h,θ (cid:104) e − tx − αs C s (cid:80) m ∈ Qk \{ k } P m h mi G ( θ mi ) δ mi (cid:105) (cid:89) p =L , N E Φ p ,h,θ (cid:20) e − t (cid:80) x j ∈ Φ p \{ x i } x − αpj C p (cid:80) m ∈ Qk P m h mj G ( θ mj ) δ mi (cid:21) . Since δ mi ’s are Bernoulli random variables with P [ δ mi = 0] = κ (0) , L I ( t ) can be written as L I ( t ) = (cid:89) m ∈ Q k \{ k } E h,θ (cid:104) κ (0) + (1 − κ (0)) e − tx − αs C s P m h mi G ( θ mi ) (cid:105)(cid:89) p =L , N exp (cid:32) − πλ (cid:90) ∞ D sp ( x ) p ( y ) E h,θ,δ (cid:104) − e − ty − αp C p (cid:80) m ∈ Qk P m h m G ( θ m ) δ m (cid:105) y d y (cid:33) = (cid:89) m ∈ Q k \{ k } E θ (cid:20) κ (0) + (1 − κ (0)) 11 + tx − α s C s P m G ( θ mi ) (cid:21)(cid:89) p =L , N exp (cid:32) − πλ (cid:90) ∞ D sp ( x ) p ( y ) E θ (cid:34) − (cid:89) m ∈ Q k (cid:0) κ (0) + (1 − κ (0)) E h (cid:2) e − ty αp C p P m h m G ( θ m ) (cid:3)(cid:1)(cid:35) y d y (cid:33) which can be simplified further following the same steps as Appendix B to get (29).R EFERENCES [1] A. K. Gupta, J. G. Andrews, and R. W. Heath Jr, “Can operators simply share millimeter wave spectrum licenses?” in
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