On the Feasibility of Sharing Spectrum. Licenses in mmwave Cellular Systems

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1 On the Feasibility of Sharing Spectrum 1 Licenses in mmwave Cellular Systems Abhishek K. Gupta, Jeffrey G. Andrews, Robert W. Heath, Jr. arxiv: v1 [cs.it] 4 Dec 215 Abstract The highly directional and adaptive antennas used in mmwave communication open up the possibility of uncoordinated sharing of spectrum licenses between commercial cellular providers. There are several advantages to sharing including a reduction in license costs and an increase in spectrum utilization. In this paper, we establish the theoretical feasibility of spectrum license sharing among mmwave cellular providers. We consider a heterogeneous multi-network system containing multiple independent cellular networks. We then compute the SINR and rate distribution for downlink mobile users of each network. Using the analysis, we compare systems with fully shared licenses and exclusive licenses for different access rules and explore the trade-offs between system performance and spectrum cost. We show that sharing spectrum licenses increases the per-user rate when antennas have narrow beams and is also favored when there is a low density of users. We also consider a multi-network system where BSs of all the networks are co-located to show that the simultaneous sharing of spectrum and infrastructure is also feasible. We show that all networks can share licenses with less bandwidth and still achieve the same per-user median rate as if they each had an exclusive license to spectrum with more bandwidth. I. INTRODUCTION Due to scarcity of spectrum at conventional cellular frequencies, the use of higher frequencies such as mmwave has been proposed for 5G cellular networks [1] [3]. In a cellular setup, mmwave systems are expected to consist of multiple mmwave providers networks. Each of these networks may be deployed by an independent cellular provider with possibly closed access only A. K. Gupta g.kr.abhishek@utexas.edu), J. G. Andrews jandrews@ece.utexas.edu) and R. W. Heath Jr. rheath@utexas.edu) are with Wireless Networking and Communications Group, The University of Texas at Austin, Austin, TX USA. This work was supported by the National Science Foundation under Grant

2 2 to its customers or users. Therefore the effect of an active network on their adjacent networks need to be investigated to efficiently design co-existence of these independently owned networks. It is worth noting that communication at mmwave frequencies has non-trivial differences when compared to communication at conventional frequencies. For example, the typical use of many antennas in mmwave systems results in highly directional communication [4], [5] and under some circumstances, it is noise limited [6] [9]. In general, mmwave communication causes less interference to neighboring BSs operating in the same frequency bands compared to communication at frequencies below 6 GHz [1], [1]. This leads to possibility of new kind of sharing of the spectrum licenses between independent cellular networks where there is no need of any coordination between these providers in terms of transmission, interference or user association. Such uncoordinated sharing of licenses was not possible in conventional cellular frequencies due to high interference caused by serving BSs which renders the channel unusable to any nearby BS of other networks operating in the same frequency band. A. Background and Related Work There are two important lines of prior research that are related to our work: spectrum sharing and mmwave cellular analysis using stochastic geometry. It is important to understand how the spectrum licenses should be shared among service providers in mmwave communication systems to reduce the license costs and increase utilization of spectrum. It has been reported that the spectrum remains underutilized [11] due to exclusive access to operators and services. There has been significant work related to cognitive radios to help fill the gaps in the underutilized spectrum by letting secondary users make use of the spectrum band via sensing based access control [12] [16]. In [17] [2], various aspects and performance of spectrum sharing were studied in a cellular setting. Most related to this paper, in [19], dynamic spectrum sharing between different operators was shown to achieve reasonable sensing performance in 3GPP LTE-A systems with carrier aggregation. Various cognitive license sharing schemes such as licensed shared access LSA) and authorized shared access ASA) were proposed [21], [22] and found to allow more than one entity to use the spectrum. As there are incumbent services, the above mentioned techniques [21], [22] would authorize a cellular system to transmit, only when the incumbent services are idle [23] [25]. Implementation would require some kind of sensing or central coordination, which may waste important resources

3 3 in sensing/coordination resulting in underutilization of spectrum. Another way to resolve the 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 [26]. This reduces the requirement of continuous listening/sensing of the spectrum by each licensee/entity but creates significant feedback/transmission overhead due to the central database. Sharing licenses if possible), will allow all networks to use the full spectrum simultaneously without impacting the individual achieved rates and help networks to reduce their expenses by sharing the license costs. In [27], the optimality of sharing was shown to highly depend on density of BSs in a multi-tier network. Further, it was shown in [28], [29] that spectrum gaps will be rare for ultra-dense deployment of small cells in a multi-tier network and cognitive sensing may not give the desired gain. Therefore the possibility of an uncoordinated sharing of spectrum licenses is worth exploring and a mathematical model is required to derive insights to properly implement license sharing in mmwave systems. In recent work, stochastic geometry has emerged as an analytical tool to model and analyze various similar wireless communication systems. For example, the performance of a single operator network) mmwave system had been investigated in prior work using tools from stochastic geometry [1], [3] [32]. In [3], a stochastic geometry framework for analysis of mmwave network was proposed. In [1], [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 coverage probability and rate were derived using stochastic geometry. In [32], a stochastic geometry framework was presented for mmwave network with backhaul and co-existent microwave network. In [33], it was shown that even a very dense mmwave network tends to be noise limited for certain choices of parameters. A major limitation of the prior work on mmwave systems [1], [3] [33] is the assumption of a single operator. To study the impact of independent operators networks) on adjacent ones, a more general framework containing multiple networks is needed. For conventional frequencies, a heterogeneous system HetNet) consisting of multiple tiers of the same network was studied in [34] where all the tiers were operating in the same frequency band. Similarly for mmwave, a numerical framework was presented for a mmwave HetNet with generalized fading in [35]. In [35], BSs were assumed to have open access to all the users, while as indicated above, commercial provider networks are expected to have closed access to its customers or users. Cognitive networks have also been studied using tools from stochastic geometry for conventional

4 4 frequencies. For example, in [36], stochastic geometry model was presented for spectrum sharing at conventional frequencies to characterize the system performance in terms of transmit capacity. In [37], a cognitive carrier sensing protocol was proposed and analyzed for a network consisting of multiple primary and secondary users and spectrum access probabilities and transmission capacity were computed. In the above mentioned prior work [1], [3] [37], the impact of interoperator interference for mmwave systems and feasibility of license sharing are not studied which is the main focus of this paper. B. Contributions In this paper, we establish the feasibility of uncoordinated sharing of spectrum licenses in a multi-operator mmwave system. We model a multi-network mmwave system where every network operator) owns a spectrum license of fixed bandwidth with a provision to share the complete rights over its licensed spectrum with other networks. Next, we compute the performance of such system in terms of signal-to-interference-and-noise SINR) and rate coverage probability. A main conclusion of our work is that it is feasible to share spectrum licenses among multiple mmwave operators without requiring any coordination among them. The main contributions of this paper can be categorized broadly as follows. Modeling multi-network using two-level architecture: We model a multi-network mmwave system using a two-level architecture. In the proposed architecture, the system consists of independent heterogeneous networks each operating in mmwave bands. Each network is itself modeled as a set of multiple tiers e.g., LOS and NLOS tiers). The need of modeling the network as a two-level architecture is due to the fact that the networks are owned by independent cellular providers while all the tiers in a network are deployed by the same provider and therefore, the infrastructure issues such as customer billing, service and performance requirements, market share) are very different at these two levels. The proposed two level architecture can effectively handle such operational issues between network providers and hence can be used in modeling various recently deployed systems such as system with coexisting microwave and mmwave networks of multiple operators in future work. Establishing the feasibility of uncoordinated sharing of spectrum licenses: We derive the combined mean user-load on BSs of each network. We derive expressions for SINR and rate coverage of a typical user of a network using the tools from stochastic geometry. Then we analyze

5 5 different combinations of accesses e.g. open and closed) and license sharing schemes e.g. with full shared licenses and exclusive licenses) and show that spectrum license sharing achieves higher performance in terms of per user rate. We also consider the case where BSs of different networks are co-located to show that multiple networks can still share BS infrastructure while sharing the spectrum licenses. We investigate the effect of antenna beamwidth on the feasibility of license sharing and show that spectrum licenses sharing is more favorable as communication becomes more directional. Finally, we quantify the benefit of license sharing in term of license cost reduction and show that networks can save significantly if they share their licenses. The rest of the paper is organized as follows. Section II explains the system model and enumerates few special combinations of accesses and sharing schemes. In Section III, the expressions for SINR and rate coverage probability are derived. In Section IV, we compare the aforementioned cases and Section V discusses the impact of partial loading where some BSs are turned off due to lack of users associated to them. Section VI presents numerical results and derives main insights of the paper. We finally conclude in Section VII. II. SYSTEM MODEL We consider a system consisting of M different cellular providers networks) which coexist in a particular mmwave band. Each network Φ m consists of BSs whose locations are modeled using a Poisson Point Process PPP) with intensity λ m and users whose locations are distributed as independent PPP with intensity λ u m. The BSs of each network can transmit with power P m. We denote the total spectrum by B and suppose that the m th network owns a license for an orthogonal spectrum of B m bandwidth which it can share with others. Slivnyak s theorem states that the properties observed by a typical point of the point process Φ u are the same as those observed by the origin in the point process Φ u [38]. Therefore we consider a typical user UE at origin without loss of generality. Let us index the network it belongs to by n. Now consider a link between this user and a BS of network m located at distance x. 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 probability of a link being LOS is dependent on x and independent of types of other links and is given by px) = exp βx) [1], [39]. The analysis can be extended to other blocking models. The path loss from the BS to user is modelled as l s x) = C s x ) αs where α s is the pathloss exponent

6 6 and C s is the gain for s type links. A. Two-level Architecture From the independent thinning theorem [38], the BS PPP of network m can be divided into two independent non-homogeneous) BS PPPs: a PPP containing all the BSs with LOS link to the user UE, Φ m,l and a PPP containing all the BSs with NLOS link to the user UE Φ m,n. They have intensity λ m,l x) = λ m px) and λ m,n x) = λ m 1 px)), respectively. Note that this results in total 2M classes known as tiers) of BSs where each tier is denoted by {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) = A λ m,s x)dx. 1) Therefore, the average number of LOS and NLOS BSs in a ball Br) with radius r for network m are given by Λ m,l Br)) = 2π Λ m,n Br)) = 2π r r r λ m px)xdx = 2π λ m e βx 2 xdx = πλ m γ2, βr) β2 λ m 1 px))xdx = πλ m r 2 2 β 2 γ2, βr) ). Let us denote the j th BS of network m as B mj. Hence the effective channel between BS B mj and the user UE is given as h mj P m l smj x mj ) where s mj denotes the link type between the user and BS B mj and h mj is a exponential random variable denoting Rayeigh fading. We observe and show in the numerical section that considering a more general fading model such as Nakagami does 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 shadowing separately as it is mostly covered by the blocking model. We show in the numerical section that including shadowing does not change any of the observed trends. We assume that BSs of every network are equipped with a steerable antenna having radiation pattern given as [1] Gθ) = G 1 G 2 θ < θ b otherwise.

7 7 TABLE I SUMMARY OF NOTATION Notation Φ m, λ m P m, B m, W m Φ u m, λ u m Description PPP consisting of the locations of BSs of network m, BS density of network m Transmit power of BSs of network m, licensed bandwidth of network m, bandwidth available to network m via sharing of licenses PPP modeling locations of users of network m, user density of this PPP L, N Possible values of link type: L denotes LOS, N denotes NLOS {m, p} Φ mp, λ mp G 1, G 2, θ b Notation representing the tier having all BSs of link type p of network m PPP modeling the locations of BSs of the tier {m, p}, density of this PPP BS antenna parameters: maximum gain, minimum gain and half beamwidth C p and α p Path-loss model parameters: path-loss gain and path-loss exponent of any link of type p {L, N} pr), β n S n pr) is the probability of being LOS for a link of distance r, β is the blocking parameter. The network which the considered user belongs to The set consisting of all networks which the considered user has access to B mj j th BS of network m here j Φ m) x mj, h mj, s mj Location of the BS B mj, fading faced by the link between this BS and the user, the type of this link k, i The network associated with the considered user, the index of the serving BS of this network s, x Type of the link between the serving BS and the user, location of this BS Q k m The set consisting of networks which are interfering to the network k Indices representing a network, in particular a member of set Q k D ks mpx) Exclusion radius for BSs of tier {m, p} when the considered user is associated with BS of tier {k, s} A n k located at x Association probability of a user of network n to be associated with BS of network k P c ks Probability of coverage of the considered user when associated with tier {k, s} N u m κ mz) R ki, R c k The mean number of users associated to a BS of network m Probability of a BS of network m having z number of associated users Instantaneous rate and rate coverage of the considered user when associated with network k R c ρ) Rate coverage of the considered user, i.e. P [R k ρ] Here G 1 G 2 and θ b denotes half beamwidth. The angle between the BS B mj antenna and direction pointing to the user UE is denoted by θ mj. We have assumed the same antenna pattern for all networks to avoid complicating the expressions unnecessarily. The analysis can be extended for a system where each network has a different transmit antenna pattern. We assume that the user is equipped with a single omni-directional antenna. Although users will also have directional antennas, it will be analytically equivalent to aggregating the transmitter and receiver

8 8 gains at BS antennas. Considering the UE antenna gain and the BS antenna gain separately does not change the observed trends and hence is left for the future work. Note that we consider single stream operation in this work. More advanced mmwave cellular systems may employ massive MIMO [4], [41] or multi-stream MIMO using hybrid beamforming [3]. Generalizing to these other architectures is a topic of future work. B. Access and Spectrum Sharing Model We assume that a user of operator n can be associated with any BS from a particular set of operators denoted by access set S n. Two special cases of access are open and closed. In an open access system, a user can connect to any system and therefore S n = {1, 2, M}. In a closed access system, a user can connect only to the network it belongs to, and therefore S n = {n}. We assume that license sharing is performed by forming mutually exclusive groups. All the networks in each group share the whole spectrum license such that each network within a group has equal bandwidth available to it. The effective bandwidth available to each network after sharing is denoted by W m. For example, in a system of 5 networks, suppose that networks 1, 2 and 3 form a group and networks 4 and 5 are in second group. Hence after license sharing, 1, 2 and 3 will have access to the total B 1 +B 2 +B 3 bandwidth band i.e. W 1 = W 2 = W 3 = B 1 +B 2 +B 3. Similarly networks 4 and 5 will have access to the sum band of bandwidth W 4 = W 5 = B 4 +B 5. The user UE experiences interference from all networks operating in the spectrum of associated network k. We denote this interfering set by Q k which is the same as the sharing group containing k th network. Note that the spectrum is shared by every member in each sharing group. Therefore, for a particular network, the set of interfering networks remains the same for its complete available spectrum band. Two extreme examples of license sharing are exclusive license and fully shared license. In the exclusive license scheme, each network can use only its own license. Therefore the bandwidth available to each network is W n = B n and the interfering networks set is Q n = {n}. In full sharing, all networks can use whole frequency band. Therefore the available bandwidth W n to each network is B and the interfering network set is Q n = {1, 2, M}. Now, the effective received power from a BS B mj at user UE is given as P mj = P j h mj l smj x mj )Gθ mj ). 2)

9 9 Hence, the average received power from B mj at UE without the antenna gain is given by P avg mj = P m l smj x mj ). 3) We assume the maximum average received power based association in which any user associates with the BS providing highest P avg mj among all the networks it has access to i.e. access set). Let us denote the network the user UE associates with by k and the index of the serving BS by i. Since the serving BS aligns its antenna with the user so that angle θ ki between the serving BS antenna and user direction is o and the effective received power of this BS is given as P ki = P k h ki l ski x ki )G) = P k h ki l ski x ki )G 1. For each interfering BS B mj for m Q k, the angle θ mj is assumed to be uniformly distributed between π and π. Now, the SINR at the typical user UE of network n at origin and associated with the i th BS of network k is given as SINR ki = P kh ki l ski x ki )G 1 σ 2 k + I 4) where I is the interference from all BSs of networks in set Q k and is given by I = P m h mj l smj x mj )Gθ mj ) = m Q k j Φ m m Q k p {L,N} j Φ m,p P m h mj l p x mj )Gθ mj ). 5) The noise power for network m is given by σ 2 m = N W m where N is the noise power density. Since σ 2 m is dependent on the allocated bandwidth, it varies accordingly with association. III. SINR AND RATE COVERAGE PROBABILITY One metric that can be used to compare systems is the SINR coverage probability. It is defined as the probability that the SINR at the user from its associated BS is above a threshold T P c T ) = P [SINR > T ], 6) 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 BS and then compute the coverage probability for this user.

10 1 A. Association Criterion and Probability Recall that a user of the n th network can be associated with any network from the set S n. Let E ki denote the event that the user is associated with the BS B ki i.e. the i th BS of network 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 at the user. This event can be further written as combination of following two events: i) the event that no other BS of network k has higher P avg at the user, and ii) that no BS of any other accessible network m has higher P avg at the user: E ki = {P avg ki > P avg kj i j} {P avg ki > P avg mj m S n \ {k}}. 7) Substituting 3) in 7), E ki = { Csik x ki ) αs ik > C s kj x kj ) αs kj } { Csik i j P k x ki ) αs ik > C s mj P m x mj ) αs mj } m S n \ {k}. The above condition can also be further split over LOS and NLOS tiers of each network, then it can be expressed as an equivalent condition over locations of all BSs as follows: E ki = {x kj > x s kj = s, i j} { x kj > { x mj > Pm P k { x mj > Pm C s Cs C s ) 1 α s ) } 1 αs x smj = s, m S n \ {k} P k C s ) 1 α s x αs α s x αs α s s mj s, m S n \ {k} s kj s, i j where s denotes the complement of the link type s. In other words, if s = L, then s = N and if s = N, then s = L. The first condition restricts all BSs of network 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 network k and link type s. Similarly the third and fourth terms are for BSs of all other accessible networks with link type s and s, respectively. As seen from these conditions, the average received power based association rule effectively creates exclusion regions around the user for BSs of each network in S n. Let us denote the exclusion radius of a tier {m, p} by D ks mpx). For example, the exclusion region of all the LOS BSs of network m when the user is associated with a NLOS BS of network k is given by ) 1 DmLx) kn Pm C α L L α N = x αl. 9) P k C N } } 8)

11 11 Note that for the BSs of the networks which are not in set S n, there are no exclusion regions, i.e. D ks mpx) = m / S n. This exclusion region denotes the region where interfering BSs cannot be located and hence, affects the sum interference. The probability that all BSs of tier {m, p} are outside the exclusion radius d is given by the void probability of the PPP Φ mp which is µ m,p d) = exp Λ m,p Bd))). Since the PPPs of the tiers are mutually independent, the probability that the BSs of tiers other than {k, i} are located outside the exclusion is given by multiplication of individual void probabilities of each tier: fk,sx) o = µ k,s D ks ks x)) µ m,s D ks msx) ) µ m,s D ks ms x)). 1) m S n\{k} Therefore, the probability density function of the distance x to this associated BS is given as f k,s x) = 2πλ k,s x µ k,s x) µ k,s D ks ks x)) µ m,s D ks msx) ) µ m,s D ks ms x)). 11) m S n\{k} The probability that a user of network n is associated with a BS of network k can be computed by summation over both LOS and NLOS tiers: A n k = f k,l x) + f k,n x)) dx. 12) Let P c kl and Pc kn denote the probabilities of coverage for the typical user which is associated with a LOS and NLOS BS of network k, respectively. They can be computed by integrating the CCDF of SINR from serving BS over pdf of distance x from serving BS as follows: P c kl = = P [SINR kl x) > T ] f k,l x)dx P [ P k h kl C L G) > T I + σ 2 k)x α L] fk,l x)dx. 13) Since h kl exp1), the probability in 13) can be replaced as [ P c kl = E exp T σ2 k xα L T IxαL C L G 1 P k = exp T N W k x α L C L G 1 P k C L G 1 P k ) T x α L L I C L G 1 P k )] f k,l x)dx 14) ) f k,l x)dx 15) where L I t) denotes the Laplace Transform of the interference I caused by BSs of all networks in set Q k and is defined as L I t) = E [ e ti]. Similarly for NLOS, P c kn = exp T N ) W k x α N T x α N ) L I f k,n x)dx. 16) C N G 1 P k C N G 1 P k

12 12 Since the association with different tiers are disjoint events, the SINR coverage probability of the typical user can be computed by summing these individual tier coverage probabilities over all accessible tiers: P c T ) = P c kt ), 17) k S n where P c k T ) is the sum probability of coverage over both tiers of network k and is defined as P c kt ) = P c klt ) + P c knt ). 18) To proceed further, we need to first characterize the interference I for which we will compute the Laplace Transform of interference given as L I t) = E [ e ti]. B. Interference Characterization If the user of network n is associated with network k, it experiences interference from all networks operating in spectrum W k. Recall that all these interfering networks form set Q k. Hence the total interference is given as I = m Q k p {L,N} j Φ m,p h mj x mj αp P m C p Gθ mj ) 19) where the first sum is over all interfering networks, the second sum is over both the LOS and NLOS tiers, and the third sum is over all the BSs of the tier. Due to mutual independence of the tiers, its Laplace transform can written as product of the following terms: L I t) = E [ e ti] = m Q k L Im t) = m Q k L ImL t)l ImN t)) 2) where L Im t) refers to the interference caused by network m and L ImL t) and L ImN t) denote the Laplace transforms of LOS and NLOS interference from network m which are given in the following Lemma. Lemma 1. The Laplace transforms of the interference from LOS and NLOS BSs of network m to a user of network n which is associated with s type BS of network k in a multi-network system are given as L ImL t) = exp 2λ m [θ b F L β, α L, tg 1 P m C L, D ks mlx)) + π θ b )F L β, α L, tg 2 P m C L, D ks mlx))] ) L ImN t) = exp 2λ m [θ b F N β, α N, tg 1 P m C N, D ks mnx)) + π θ b )F N β, α N, tg 2 P m C N, D ks mnx))] )

13 13 where F L b, a, A, x) = Proof: See Appendix A x e by Ay a 1 + Ay a ydy, and F Nb, a, A, x) = x Ay a 1 e by ) ydy. 1 + Ay a Note that the term containing G 1 and θ b denotes the interference from aligned BSs whose antennas are directed towards the considered user while the term containing G 2 and π θ b ) represents the interference from the unaligned BSs. Since G 1 G 2, the interference from aligned BS is significantly larger than the interference from the unaligned BS and hence dominates the Laplace Transform expression. Therefore the value of half beamwidth θ b plays a significant role in characterizing the interference and deciding 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 network n in a multi-network system is given as P c = k S n s {L,N} m Q k L ImL T r α s C s G 1 P k ) T r α s ) L ImN exp C s G 1 P k where L Imp t) is computed in Lemma 1 and f k,s x) is given as 11). NT rαs C s G 1 P k Proof: Substituting the value of L I t) from Lemma 1 in 17), we get the result. ) f k,s x)dx In 21), the first summation is over all networks which the user of network n can connect to, weighted by the association probability. This weighing is included inside the term f k,s x). 21) C. Rate Coverage Rate coverage is another important metric for comparing performance. While the SINR shows the serving link quality, the rate represents the data bits received per second per user and hence is more realistic indicator of the system performance. In this section, we derive the downlink rate coverage which is defined as the probability of the rate of a typical user being greater than the threshold ρ Rρ) = P [Rate > ρ]. 22) Let us assume that O k denote the time-frequency resources allocated to each user associated with the tagged BS of network k. Therefore the instantaneous rate of the considered typical

14 14 user 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 to the randomly sized coverage areas of each BS and random number of users in the coverage areas. As shown in [32], [42] approximating this load with its respective mean does not compromise the accuracy of results. Since the user distribution of each network is assumed to be PPP, the average number of users associated with the tagged BS of network k associated with the typical user can be modeled similarly to [32], [42]: Nk u = λ u λ ma m k. 23) k m:k S m Note that the summation is over all the networks whose users can connect to the network k and the sum denotes the combined density of associated users from each network. Now we assume that the scheduler at the tagged BS gives 1/N u k of the N u k fraction of resources to each users. This assumption can be justified as most schedulers such as round robin or proportional fair scheduler gives approximately 1/N u k fraction of resources to each user. Using the mean load approximation, the instantaneous rate of a typical user of network n which is associated with network k is given as R ki = W k N u k log 1 + SINR ki ). 24) Let R c k ρ) denote the rate coverage probability when user is associated with network k. Then the total rate coverage will be equal to sum of R c k ρ) s over all accessible networks: R c ρ) = k S n R c kρ). 25) Now R c k ρ) can be derived in terms of SINR coverage probability as follows: R c kρ) = P [R ki > ρ] = P [W k /Nk u log 1 + SINR ki ) > ρ] [ = P SINR ki > 2 ρ N u ] k W k 1 = P c k 2 ρnk u/w k 1 ). Therefore the rate coverage is given as R c ρ) = k S n P c k 2 ρn u k /W k 1 ). 26)

15 15 IV. PERFORMANCE COMPARISON We use our mathematical framework to compare the benefits of spectrum licensing. We enumerate three specific cases or systems) considering different combinations of accesses and license sharing schemes. Also see Fig. 1. System 1: Closed Access and Exclusive Licenses: In System 1, each user must associate with only its own network and each network can use its own spectrum only. This case is equivalent to a set of M single network systems which has been studied in prior work [1]. This system serves as a baseline case to evaluate benefits of sharing. The SINR coverage probability of such a user is given by 21) with k = n, S n = {n}. Recall that in this system, the spectrum accessible to each network is its own licensed spectrum only i.e. W n = B n. System 2: Open Access and Full Spectrum License Sharing: In System 2, each user can be associated with any network and the spectrum license is shared between all networks. In this case, the probability of association with BS of network k is independent of which network the considered user belongs to and is given as A n k = A k = A k,l + A k,n = f k,l x) + f k,n x)) dx. The SINR coverage probability of the user is given by 21) with S n = {1, 2, M}. The average load to the tagged BS of network k is given as Nk u = m λu A k m λ k. The spectrum accessible to each network W k is the complete band B. Since users can connect to any network, it requires full coordination between the networks including sharing of control channel and other resources. Therefore such system serves as an upper bound to the other two more practical systems. Note that if all M networks are identical with respect to every parameter, then this system is equivalent to a single network system with the aggregate BS and UE density. In this case, from a user association perspective, there is no discrimination based on the network which a particular BS belongs to. Also all networks transmit in the same band. So the users effectively see a single network with aggregate BS density of all the networks. Similarly from the network perspective, users of all the networks looks the same due to open access. Hence, the users of different networks can be replaced by users of a single network with the aggregate UE density. System 3: Closed Access and Full Spectrum License Sharing: In System 3, each user must associate with its own network but whole spectrum is shared between all the networks.

16 16 System 1 System 2 System 3 System 4 Accessible links Interfering links Spectrum: B 1 B 2 B 1 B 2 B 1 B 2 B 1 B 2 Licensed Available Network 1 BS Network 2 BS User of network 1 Fig. 1. Illustration describing the differences between the four systems. For a typical user of network 1, the figure shows all accessible networks, interfering networks and available spectrum after sharing. This case does not require any transmission coordination among networks or common control channel, nor does it require sharing of infrastructure or back-haul resources. This system is close to the practical implementation where subscribers must connect to their respective service providers only. The SINR coverage probability of the considered typical user of network n is given by 21) with k = n, S n = {1, 2, M}. For this system, the spectrum accessible to each network is B. Along with the above three systems, we consider the one additional system System 4 as defined below. This system will help us understand if independent networks can still share BS infrastructure while sharing the spectrum licenses. System 4: Co-located BSs with Closed Access and Full Spectrum License Sharing: System

17 17 4 has closed access and complete spectrum sharing license where the respective BSs of all the networks are all co-located. The system model for this case remains the same as the previous three systems except 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 networks located at the same location are either all LOS or all NLOS. We first briefly show the computation the probability of SINR coverage of this system. Consider a typical user of network n. Recall from our assumption that for a user, BSs of all the networks located at the same location are either all LOS or all NLOS. Hence the BS PPP Φ can be divided in to two independent PPP, Φ L and Φ N with intensity λ L x) = λpx) and λ N x) = λ1 px)). The probability density function of the distance x of the associated BS of network n is given as f s x) = 2λ s πx exp Λ s Bx))) exp Λ s B Ds s x)))) 27) where the exclusion radius Dpx) s is the same for BSs of all the networks and given as ) 1 Dpx) s Cp αp αs = x αp. 28) The interference I at the user from BSs of all the networks is given as I = x αs C s m Q n\{n} P m h mi Gθ mi ) + C s p=l,n j Φ p\{i} x αp j C p m Q n P m h mj Gθ mj ). 29) The following Lemma characterizes the Laplace Transform of the interference in 29) in the co-located BSs case. Lemma 2. The Laplace Transform of interference to a typical user of network n with closed access which is associated to a BS of type s in a multi-network system with co-located BSs is given as L I t) = p=l,n ) θ b /π π θ b )/π tx αs C s P m G tx αs C s P m G 2 exp 2πλ py) 1 ) ) θ b /π π θ b )/π + ydy) 1 + ty αp C p P m G ty αp C p P m G 2 m Q n\{k} D s px) Proof: See Appendix B. m Q n

18 18 Similar to previous subsections, the coverage probability of the considered typical user is given as P c T ) = s=l,n exp T N ) W n x αs L I C s G)P n T x α s C s G)P n ) f s x)dx 3) where L I t) is given in Lemma 2 and f s x) is given in 27). Since full sharing of license is assumed for this system, the spectrum accessible to each network is B. Hence, similar to System 3, the average load is given as N u n = λ u n 1 λ n. V. PARTIAL LOADING OF THE NETWORK In the previous section, we assumed that all the BSs have at least one user associated and they are all transmitting. Such an assumption is justified when the user density is very high in comparison to the BS density resulting in many associated users per BS and negligible probability of any BS being inactive. For very dense networks, however, this assumption will break down and many BSs will not be occupied at all times. An interesting case to consider is where the system is not fully loaded and there are some BSs having no active) users to associate with. In such a case, interference will reduce which should favor license sharing. From [42], the number of users associated with a BS of network m for a closed access system can be approximated as the following distribution: κ m N u m) = Γn + 1) Γn + 3.5) η m ) n η m ) n ) Γ3.5) 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 match the load distribution in simulation closely [32], [42]. For the multi-network system, η m is computed as η m = A q m, 32) λ m q:m S q λ u q where the sum is over all the networks whose users can connect to BSs of network m. The multiplication 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 each BS is off is equal to probability that that BS has no user associated to it which is equal to κ m ). Therefore for Systems 1, 2 and 3, the interfering BSs can be obtained by independent thinning of original BS PPP with probability 1 κ m ). The coverage probability of the Systems 1, 2 and 3 are given by Theorem 1 with λ m substituted with λ m = λ m 1 κ m )).

19 19 For System 4, the sum interference I at the user from BSs of all the networks in partial loading case is given as I = x αs C s P m h im Gθ im )δ mi + m Q n\{n} p=l,n j Φ p\{i} x αp j C p m Q n P m h mj Gθ mj )δ mj 33) where δ mj is an indicator which is 1 when BS B mj is on. Hence the Laplace Transform of interference in partial loading can be computed as see Appendix C for proof) L I t) = κ) + 1 κ)) ) a j /π 1 + tx αs C m Q k \{k} j=1,2 s P m G j exp 2λ py) 1 κ) + 1 κ)) )) ) a j ydy Dp sx) 1 + y αp tc p P m G j p=l,n m Q k where a 1 = θ b, a 2 = π θ b. The coverage probability of this system is given by 3) with L I t) given in 34) and f s x) given in 27). j=1,2 34) VI. NUMERICAL RESULTS In this section, we provide results numerically computed from the analytic expressions derived in previous sections. We compare the four aforementioned systems to provide insights and discuss the impact of license sharing. For these numerical results, we consider a system consisting of two networks with identical parameters. Here each network represents a cellular provider operating in mmwave frequency with BS intensity 3/km 2 which is equivalent to average cell radius of 13 m. Each network has user density of 2/km 2. For blockage, we have considered the exponential model i.e. px) = exp βx) with β =.7 which has an average LOS region of 144 m. The transmit power is assumed to be 3dBm. For most of the results, the operating frequency is 28GHz for which pathloss exponents for LOS and NLOS are α L = 2, α N = 4 and the corresponding gains are C L = 6dB, C N = 7dB. The total system bandwidth is 2 MHz. We assume that each network owns a license for 1 MHz. Recall that for System 1, each network can use only its own spectrum. In Systems 2, 3 and 4, both networks share each other s spectrum licenses and therefore get 2MHz of spectrum. Validation of analysis and SINR coverage trends: Fig. 2 compares the probability of coverage for these systems and validates our analysis with simulation. The typical user in System 2 has high SINR coverage due to its open access. The closed access in System 3 allows BSs of

20 2 Probability of Coverage P c Theory Sims SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs SINR Threshold T db) Fig. 2. Probability of SINR coverage in a two-network mmwave system with BS antenna half beamwidth θ b = 1 o for different cases. Line-curves denote values from the analysis and markers denote respective values from simulation. another 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 access as System 3, but the spectrum is not shared. Therefore the user faces no interference from other networks and hence the SINR coverage is greater than System 3. In comparing System 1 and System 2, we observe different behaviors for different SINR ranges. Recall that System 2 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 2. Now for the high SINR region which is mainly due to LOS serving links), this increases the received power of a serving BS by a factor of 2 α L = 2. Since the noise power also increases by a factor of 2 due to increase in bandwidth, it effectively cancels the increase in the power caused by increased proximity of serving BS. As far as interference is considered, since doubling density increases the probability of interferers to be LOS, interference increases significantly in System 2. 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 increases by a factor of 2 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 than System 1 in the low SINR region. System 4 has similar values and trends as System 3 when compared to other systems. Due to co-location of BSs, System 4 always guarantees that for a typical user, there is no other BS of other networks closer to the associated BS of its network

21 SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Rate Coverage R c Rate Threshold ρ 1Mbps) Fig. 3. Rate coverage in a two-network mmwave system with Rayleigh fading and BS antenna half beamwidth θ b = 1 o for different cases. Systems 3 and 4 with shared license perform better than System 1 with exclusive licenses. while in System 3, there can be other interfering BS closer to the user than the serving BS. But due to the same reason, it also guarantees that there is always K 1 interfering BSs of the other networks at the same location in System 4, while in System 3, that is not the case. Therefore we see a trade-off between System 3 and System 4 where for high values of SINR thresholds, SINR of System 3 is better, while System 4 performs better for low SINR thresholds. Sharing licenses achieves higher rate coverage: Fig. 3 compares the probability of rate coverage for four systems which incorporates the effect of load and bandwidth. Since each network has a large bandwidth and large SINR coverage in System 2, its rate coverage is the highest among all systems. Such a system, however, requires full coordination, thus, it is not very practical and mainly serves as a benchmark for practical systems. A more interesting comparison is between System 1 and 3 or 4). Here we can see that even though System 1 has higher SINR coverage than System 3 and 4), the latter achieves higher median rate, due to the extra bandwidth gained from spectrum license sharing. In particular, System 3 and 4 have respectively 25% and 32% higher median rates than System 1. Validation of the model with realistic scenario: To validate our PPP assumption and to show that it is reasonable, we also present a simulation result where the BSs are deployed in a square grid and the BS antenna pattern is parabolic, as specified by the 3GPP standard [43]. We also consider log-normal shadowing σ LOS = 5.2dB and σ NLOS = 7.6dB). Fig. 4 compares

22 SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Rate Coverage R c Rate Threshold ρ 1Mbps) Fig. 4. Rate coverage in a two-network mmwave system with grid BS deployment and 3GPP antenna pattern [43] for different cases. The trends for this case are similar to Fig. 3 which validates our assumptions regarding the system model. 1.8 SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Rate Coverage R c Fig Rate Threshold ρ 1Mbps) Rate coverage in a two-network mmwave system with Nakagami fading with parameter 1) and BS antenna half beamwidth θ b = 1 o for different cases. When compared to Rayleigh fading Fig. 3), the insights are similar which justifies the Rayleigh fading assumption for analysis the probability of rate coverage for four systems with these modifications. We observe similar trends for spectrum license sharing which justifies our assumptions regarding deployment and shadowing. Fig. 5 shows the probability of rate coverage for four systems with Nakagami fading with parameter 1) instead of Rayleigh. It can be seen that the trends are similar to those of Rayleigh distribution as we claimed in Section IIA. Impact of beamwidth on mediate rate: Fig. 6 compares the median rate of the four

23 23 Median Rate Mbps) SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Half Beamwidth θ b degrees) 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. Rate Coverage R c SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Rate Threshold ρ 1Mbps) Fig. 7. Rate coverage in a two-network mmwave system under partial loading with Rayleigh fading for different cases. Partial loading favors spectrum license sharing. systems for various values of beamwidth. It can be seen that above a certain threshold for the beamwidth, it becomes optimal to have exclusive license, due to high interference. As the beamwidth decreases, license sharing becomes optimal. For the given parameters, the threshold is at about 25 o. Since mmwave has typical beamwidth less than 15 o, sharing should increase the achievable rate. Partial loading favors sharing: Fig. 7 compares the probability of rate coverage for four systems under partial loading with user density of 3/km 2. It can be observed that due to reduced

24 24 Required Bandwidth B k MHz) 15 1 SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Half Beamwidth θ b degrees) Fig. 8. Required bandwidth for each network with sharing of licenses) to achieve the same median rate achieved by the network with no sharing of spectrum with each network having 1MHz spectrum license. Sharing can reduce the license cost by more than 25% interference, System 3 and 4) has even higher gain than System 1. In particular, System 3 has 4% higher median rate than System 1 in partial loading case compared to only 25% gain in the previous case when user density was 2/km 2. Sharing reduces spectrum cost significantly: Now we compare the following two cases. In the first case, each network owns a 1 MHz bandwidth exclusive license. This case is the same as System 1. In the second case, the networks share licenses completely and choose to buy just enough spectrum to achieve the same median rate as in the first case. Fig. 8 shows this required spectral bandwidth for each network. With a 1 o beamwidth antenna, each network only needs to buy 75 MHz of bandwidth which would save 25% of the license cost assuming linear pricing of the spectrum. Optimal cardinality of sharing groups depends on the target rate: Now we consider a system with 1 operators with 5MHz bandwidth each and closed access. Fig. 9 shows variation of the per-user rate for different percentiles with respect to cardinality S n of spectrum group which is equal to the number of operators sharing licenses with network n. We can see that the 75 th percentile rate increases with S n while the 25 th percentile rate decreases. For the median rate, we see an increase up to S n =3 and then the median rate deceases. This trade-off is due to the fact that as more operators share their licenses, the total available bandwidth and the sum interference both increase. It can be observed that depending on the target performance, the

25 Median rate 75 th percentile rate 25 th percentile rate Rate in Mbps) Number of sharing networks S n Fig. 9. Rate versus number of sharing networks in a mmwave cellular system with 1 networks. A trade-off between increasing the available bandwidth and increasing interference is observed. optimal number of networks that should share their licenses varies SYS1: Closed, No sharing SYS2: Open, Full sharing SYS3: Closed, Full sharing SYS4: Co located BSs Rate Coverage R c Rate Threshold ρ 1Mbps) Fig. 1. Rate coverage in a two-network mmwave system at 73 GHz frequency with Rayleigh fading for different cases. Similar trends 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 1 GHz bandwidth. The near-field path-loss gains are decreased by a factor of 1 log 173/28) 2 = 8.32dB when compared with 28GHz for both LOS and NLOS. Fig. 1 compares the rate coverage for 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. 25%).