Modeling and Analyzing the Coexistence of Wi-Fi and LTE in Unlicensed Spectrum
Yingzhe Li, Francois Baccelli, Jeffrey G. Andrews, Thomas D. Novlan, Jianzhong Charlie Zhang
aa r X i v : . [ c s . I T ] F e b Modeling and Analyzing the Coexistence ofWi-Fi and LTE in Unlicensed Spectrum
Yingzhe Li, Franc¸ois Baccelli, Jeffrey G. Andrews, Thomas D. Novlan,Jianzhong Charlie Zhang
Abstract
We leverage stochastic geometry to characterize key performance metrics for neighboring Wi-Fiand LTE networks in unlicensed spectrum. Our analysis focuses on a single unlicensed frequency band,where the locations for the Wi-Fi access points (APs) and LTE eNodeBs (eNBs) are modeled as twoindependent homogeneous Poisson point processes. Three LTE coexistence mechanisms are investigated:(1) LTE with continuous transmission and no protocol modifications; (2) LTE with discontinuoustransmission; and (3) LTE with listen-before-talk (LBT) and random back-off (BO). For each scenario,we derive the medium access probability (MAP), the signal-to-interference-plus-noise ratio (SINR)coverage probability, the density of successful transmissions (DST), and the rate coverage probabilityfor both Wi-Fi and LTE. Compared to the baseline scenario where one Wi-Fi network coexists withan additional Wi-Fi network, our results show that Wi-Fi performance is severely degraded when LTEtransmits continuously. However, LTE is able to improve the DST and rate coverage probability ofWi-Fi while maintaining acceptable data rate performance when it adopts one or more of the followingcoexistence features: a shorter transmission duty cycle, lower channel access priority, or more sensitiveclear channel assessment (CCA) thresholds.
I. I
NTRODUCTION
As is well-established, licensed spectrum below 6 GHz is scarce and extremely expensive.Given that there is over 400 MHz of generally lightly used unlicensed spectrum in the 5 GHzband – e.g. in the USA, the U-NII bands from 5.15-5.35 GHz and 5.47-5.825 GHz [2] –
Y. Li, F. Baccelli and J. G. Andrews are with the Wireless Networking and Communications Group (WNCG), The University ofTexas at Austin (email: [email protected], [email protected], [email protected]). T. Novlan and J. Zhangare with Samsung Research America-Dallas (email: [email protected], [email protected]). Part of this paper waspresented at IEEE Globecom 2015, th International Workshop on Heterogeneous and Small Cell Networks (HetSNets) [1]. extending LTE’s carrier aggregation capabilities to be able to opportunistically use such spectrumis an interesting proposition [3]–[5]. Such an approach utilizes an anchor primary carrier in LTEoperator’s licensed spectrum holdings to provide control signaling and data, and a secondarycarrier in the unlicensed spectrum that when available, offers a significant boost in data rate.However, IEEE 802.11/Wi-Fi is an important incumbent system in these bands. Thus, a keydesign objective for LTE is to not only obey existing regulations for unlicensed spectrum, but alsoto achieve fair coexistence with Wi-Fi. In this paper, we propose a theoretical framework basedon stochastic geometry [6]–[10] to analyze the coexistence issues that arise in such scenario.
A. Related Work and Motivation
LTE is a centrally-scheduled system which was designed for exclusive usage of licensedspectrum. In contrast, Wi-Fi is built on distributed carrier sense multiple access with collisionavoidance (CSMA/CA), where the carrier sensing mechanism allows transmissions only when thechannel is sensed as idle. This distinctive medium access control (MAC) layer can potentially leadto very poor Wi-Fi performance when LTE operates in the same spectrum without any protocolmodifications. Based on indoor office scenario simulations, [11], [12] show that Wi-Fi is mostoften blocked by the LTE interference and that the throughput performance of Wi-Fi decreasessignificantly. In order to achieve fair coexistence with Wi-Fi, several modifications of LTE havebeen proposed. A simple approach which requires minimal changes to the current LTE protocolis to adopt a discontinuous transmission pattern, also known as LTE-U [13], [14]. By using thealmost-blank subframes (ABS) feature to blank a certain fraction of LTE transmissions, Wi-Fithroughput can be effectively increased [12], [15], [16]. This discontinuous transmission ideawas previously adopted to address the coexistence issues of WiMax and Wi-Fi [17]. Coexistencemethodologies using the LBT feature, also known as licensed-assisted access (LAA) in 3GPP [5],have been considered in [16], [18]. In [16], a random backoff mechanism with fixed contentionwindow size is proposed in addition to LBT. The LAA operation of LTE in unlicensed spectrumis investigated in [18], which shows that the load-based LBT protocol of LAA with a backoffdefer period can achieve fair coexistence. When LTE users adopts the LBT feature, [4] showsLTE can deliver significant uplink capacity even if it coexists with Wi-Fi.All the aforementioned works are based on extensive system level simulations, which is usuallyvery time-consuming due to the complicated dynamics of the overlaid LTE and Wi-Fi networks.
Therefore, a mathematical approach would be helpful for more efficient performance evaluationand transparent comparisons of various techniques. A fluid network model is used in [19] toanalyze the coexistence performance when LTE has no protocol modifications. However, the fluidnetwork model is limited to the analysis of deterministic networks, which do not capture themulti-path fading effects and random backoff mechanism of Wi-Fi. A centralized optimizationframework is proposed in [20] to optimize the aggregate throughput of LTE and Wi-Fi. However,the analysis of [20] is based on Bianchi’s model for CSMA/CA [21], which relies on the idealizedassumption that the collision probability of the contending APs is “constant and independent”.In recent years, stochastic geometry has become a popular and powerful mathematical toolto analyze cellular and Wi-Fi systems. Specifically, key performance metrics can be derived bymodeling the locations of base stations (BSs)/access points (APs) as a realization of certain spatialrandom point processes. In [22], the coverage probability and average Shannon rate were derivedfor macro cellular networks with BSs distributed according to the complete spatial randomPoisson point process (PPP). The analysis has been extended to several other cellular networkscenarios, including heterogeneous cellular networks (HetNets) [23]–[25], MIMO [26], [27], andcarrier aggregation [28], [29]. More realistic macro BS location models than PPP are investigatedin [30]–[32]. Stochastic geometry can also model CSMA/CA-based Wi-Fi networks. A modifiedMat´ern hard-core point process, which gives a snapshot view of the simultaneous transmittingCSMA/CA nodes, has been proposed and validated in [33] for dense 802.11 networks. ThisMat´ern CSMA model is also used for analyzing other CSMA/CA based networks, such as ad-hoc networks with channel-aware CSMA/CA protocols [34], and cognitive radio networks [35].Due to its tractability for cellular and Wi-Fi networks, stochastic geometry is a naturalcandidate for analyzing LTE and Wi-Fi coexistence performance. In [36], the coverage andthroughput performance of LTE and Wi-Fi were derived using stochastic geometry. However,the analytical Wi-Fi throughput in [36] does not closely match the simulation results. Also, theeffect of possible LTE coexistence methods, including discontinuous transmission and LBT withrandom backoff, were not investigated in [36]. These shortcomings are addressed in this paper.
B. Contributions
In this work, a stochastic geometry framework is proposed to evaluate the coexistence per-formance of the neighboring Wi-Fi network and LTE network. Specifically, three coexistence scenarios are studied depending on the mechanism adopted by LTE, including: (1) LTE withcontinuous transmission and no protocol changes (i.e., conventional LTE); (2) LTE with fixedduty-cycling discontinuous transmission (i.e., LTE-U); and (3) LTE with LBT and randombackoff mechanism (i.e., LAA). Several key performance metrics, including the MAP, the SINRcoverage probability, the DST, and the rate coverage probability are derived under each scenario.The accuracy of the analytical results is validated against simulation results using SINR coverageprobability. The main design insights of this paper can be summarized as follows:(1) When LTE transmits continuously with no protocol changes, Wi-Fi performance is signif-icantly impacted. Specifically, compared to the baseline scenario where Wi-Fi network coexistswith an additional Wi-Fi network from another operator, the SINR coverage probability, DST,and rate coverage probability of Wi-Fi are severely degraded due to the persistent transmittingLTE eNBs. In contrast, LTE performance is shown to be relatively robust to Wi-Fi’s presence.(2) When LTE transmits discontinuously with a fixed duty cycle, Wi-Fi generally has betterDST and rate coverage under a synchronous muting pattern among LTE eNBs compared to theasynchronous one; and a short duty cycle for LTE transmission is required in both cases toprotect Wi-Fi. Specifically, Wi-Fi achieves better performance under the synchronous case ingeneral since it provides a much cleaner channel to Wi-Fi when LTE is muted. In contrast, sinceall eNBs transmit simultaneously under the synchronous case, LTE experiences stronger LTEinterference and therefore worse DST and rate coverage compared to the asynchronous case.(3) When LTE follows the LBT and random BO mechanism, LTE needs to accept either lowerchannel access priority or more sensitive CCA threshold to protect Wi-Fi. Specifically, Wi-Fiachieves better DST and rate coverage performance compared to the baseline scenario whenLTE has either the same channel access priority (i.e., same contention window size) as Wi-Fiwith more sensitive CCA threshold (e.g., -82 dBm), or lower channel access priority (i.e., largercontention window size) than Wi-Fi with less sensitive sensing threshold (e.g., -77 dBm). Underboth scenarios, LTE is shown to maintain acceptable rate coverage performance.II. S
YSTEM M ODEL
In this section, we present the spatial location model for Wi-Fi APs and LTE eNBs, the radiopropagation assumptions, and the channel access model for Wi-Fi and LTE.
A. Spatial locations
We focus on the scenario where two operators coexist in a single unlicensed frequency bandwith bandwidth B . Operator 1 uses Wi-Fi, while operator 2 uses LTE, which may implementcertain coexistence methods to better coexist with operator 1. Both Wi-Fi and LTE are assumedto have full buffer downlink only traffic. The LTE eNBs are assumed to be low power small celleNBs, such as femto-cell eNBs [37]. The locations for APs and eNBs are modeled as realizationsof two independent homogeneous PPPs. Specifically, the AP process Φ W = { x i } i has intensity λ W , while the eNB process Φ L = { y k } k has intensity λ L . Therefore, the number of APs andeNBs in any region with area A are two independent Poisson random variables with mean λ W A and λ L A respectively (resp.). The PPP assumption for APs is reasonable due to the unplannednature of most Wi-Fi deployments [33], while the PPP assumption for eNBs will exhibit similarSINR trend with a constant SINR gap compared to more accurate eNB location models [32].Both Wi-Fi stations (STAs) and LTE user equipments (UEs) are also assumed to be distributedaccording to homogeneous PPPs. Each STA/UE is associated with its closest AP/eNB, whichprovides the strongest average received power. We assume the STA/UE intensity is much largerthan the AP/eNB intensity, such that each AP/eNB has at least one STA/UE to serve. Sinceboth STAs and UEs are homogeneous PPPs, we can analyze the performance of the typicalSTA/UE, which is assumed to be located at the origin. This is guaranteed by the independenceassumption and Slyvniak’s theorem [10]. Index is used for the serving AP/eNB to the typicalSTA/UE, which will be referred to as the closest or tagged AP/eNB for the rest of the paper.In addition, the link between the typical STA/UE and the tagged AP/eNB is referred to as thetypical Wi-Fi/LTE link. Since Φ W is a PPP with intensity λ W , the probability density function(PDF) of the distance from the typical STA to the tagged AP is f W ( r ) = λ W πr exp( − λ W πr ) .Similarly, the PDF from the typical UE to the tagged eNB is f L ( r ) = λ L πr exp( − λ L πr ) . B. Propagation Assumptions
The transmit power for each AP and eNB is assumed to be P W and respectively P L . Acommon free space path loss model with reference distance of 1 meter is used for both Wi-Fi Note in any given time slot, not all Wi-Fi APs will be necessarily scheduled by CSMA/CA. Wi-Fi STA and Wi-Fi users, as well as LTE UE and LTE users, are used interchangeably in this paper. For any event A and PPP Φ , a heuristic interpretation of the Slyvniak’s theorem is: P (Φ ∈ A | o ∈ Φ) = P (Φ ∪ { o } ∈ A ) . TABLE I: Notation and Simulation Parameters
Symbol Definition Simulation Value Φ W , λ W Wi-Fi AP PPP and intensity Φ L , λ L LTE eNB PPP and intensity P W , P L Wi-Fi AP, LTE eNB transmit power 23 dBm, 23 dBm Γ cs , Γ ed Carrier sensing and energy detection thresholds -82 dBm, -62 dBm e Wi , e Lk Medium access indicator for AP x i , eNB y k x , y The tagged AP and tagged eNB (i.e., the AP and eNB closestto the typical STA and UE resp.) f W ( r ) , f L ( r ) PDF of the distance from tagged AP/eNB to typical STA/UE f c , B Carrier frequency and bandwidth of the unlicensed band 5 GHz, 20 MHz α Path loss exponent 4 µ Parameter for Rayleigh fading channel 1 σ N Noise power 0 B ( x, r ) ( B o ( x, r ) ) Closed (open) ball with center x and radius rB c ( x, r ) Complement of B ( x, r ) F Li, ( F Wi, , F LWi, , F WLi, ) Fading of the channel from eNB y i to typical UE (AP x i totypical STA, eNB y i to typical STA, AP x i to typical UE) exponentially distributedwith parameter µG Li,j ( G Wi,j , G LWi,j , G WLi,j ) Fading of the channel from eNB y i to eNB y j (AP x i to AP x j , eNB y i to AP x j , AP x i to eNB y j ) exponentially distributedwith parameter µ and LTE links, which is given by l [ dB ]( d ) = 20 log ( πλ c ) + 10 α log ( d ) . Here λ c denotes thewavelength, α denotes the path loss exponent, and d denotes the link length. The large-scaleshadowing effects are neglected for simplicity. All the channels are assumed to be subject toi.i.d. Rayleigh fading, with each fading variable exponentially distributed with parameter µ . Thethermal noise power is σ N . Notations and system parameters are listed in Table I. C. Modeling Channel Access for Wi-Fi
In contrast to LTE, Wi-Fi implements the distributed CSMA/CA protocol for channel accesscoordination among multiple APs. The CSMA/CA protocol consists of the physical layer clearchannel assessment (CCA) process and a random backoff mechanism, such that two nearbynodes will never transmit simultaneously. In particular, the Wi-Fi device will hold CCA as busyif any valid Wi-Fi signal that exceeds the carrier sense (CS) threshold Γ cs is detected, or if anysignal that exceeds the energy detection threshold (ED) Γ ed is received [38]. Similar to [19],we assume Wi-Fi devices detect the eNB transmission with the energy detection threshold Γ ed since an LTE signal is not decodable. As soon as a CSMA/CA device observes an idle channel,it needs to follow a random back-off period before transmission. This back-off period is chosenrandomly from a set of possible values called the contention window.To model the locations of Wi-Fi APs which simultaneously access the channel at a given time,we adapt the formulation of [33] to account for the coexisting LTE network. We can define the contender of a Wi-Fi AP x i as the other Wi-Fi APs and the LTE eNBs whose power received by x i exceeds the threshold Γ cs and Γ ed respectively. Each Wi-Fi AP x i has an independent mark t Wi to represent the random back-off period, which is uniformly distributed on [0 , . Each Wi-FiAP obtains channel access for packet transmission if it chooses a smaller timer, i.e., back-offperiod, than all its contenders. A medium access indicator e Wi is assigned to each AP, whichis equal to 1 if the AP is allowed to transmit by the CSMA/CA protocol, and 0 otherwise.Depending on the specific coexistence mechanism of LTE, the medium access indicator for eachAP is determined differently. The Palm probability [10, p.131] that the medium access indicatorof a Wi-Fi AP is equal to 1 is referred to as the medium access probability, or MAP for short.The considered channel access mechanism has some limitations, such as it has a fixed con-tention window size which does not capture the exponential backoff, and it is also more suitablefor synchronized and slotted version of CSMA/CA. Nevertheless, it is able to model the keyfeature of CSMA/CA in IEEE 802.11 standard [38], such that each CSMA/CA device transmitsif it does not carrier sense any other CSMA/CA device with a smaller back-off timer. In addition,through comparisons with simulation results, [33], [39] show this simplified model provides areasonable conservative representation of transmitting APs in the actual CSMA/CA networks. D. Definition of Performance Metrics
The main performance metrics that are analyzed include the MAP of the tagged AP and eNB,as well as the SINR coverage probability for the typical Wi-Fi STA and LTE UE. Specifically,given the tagged AP x transmits (i.e., e W = 1 ), the received SINR of the typical Wi-Fi STA is:SINR W = P W F W , / l ( k x k ) P x j ∈ Φ W \{ x } P W F Wj, e Wj / l ( k x j k ) + P y m ∈ Φ L P L F LWm, e Lm / l ( k y m k ) + σ N , (1)where e Wj and e Lm represent the medium access indicator for AP x j and eNB y m respectively. TheSINR coverage probability of the typical STA with SINR threshold T is defined as P ( SINR W >T | e W = 1) , which gives the instantaneous SINR performance of the typical Wi-Fi link. Similarly,the received SINR of the typical LTE UE given the tagged eNB y transmits is:SINR L = P L F L , / l ( k y k ) P x j ∈ Φ W P W F W Lj, e Wj / l ( k x j k ) + P y m ∈ Φ L \{ y } P L F Lm, e Lm / l ( k y j k ) + σ N , (2)and the SINR coverage probability is P ( SINR L > T | e L = 1) .Based on the MAP and the SINR distribution, we will compare different LTE coexistence mechanisms using the density of successful transmission and the rate coverage probability, whichare defined as follows. Definition
For decoding SINR requirement T , thedensity of successful transmission, or DST for short, is defined as the mean number of successfultransmission links per unit area [7]. Since the typical Wi-Fi/LTE link is activated only when thetagged AP/eNB accesses the channel, the DST for Wi-Fi and LTE are given by: d Wsuc ( λ W , λ L , T ) = λ W E [ e W ] P ( SINR W > T | e W = 1) ,d Lsuc ( λ W , λ L , T ) = λ L E [ e L ] P ( SINR L > T | e L = 1) . (3) Definition
The rate coverage probability with threshold ρ is defined as theprobability for tagged Wi-Fi AP/LTE eNB to support an aggregate data rate of ρ , given by : P Wrate ( λ W , λ L , ρ ) = P ( B log(1 + SINR W ) E [ e W ] > ρ | e W = 1) ,P Lrate ( λ W , λ L , ρ ) = P ( B log(1 + SINR L ) E [ e L ] > ρ | e L = 1) . (4)The E [ e W ] and E [ e L ] in (4) accounts for the fact that the tagged AP and tagged eNB havechannel access for E [ e W ] and E [ e L ] fraction of time respectively. Equivalently, the rate coverageprobability gives the fraction of Wi-Fi APs/LTE eNBs (or Wi-Fi/LTE cells) that can support anaggregate data rate of ρ for the rest of the paper. Remark Since both Φ W and Φ L are stationary and isotropic, the above performance metricsare invariant with respect to (w.r.t.) the angle of the tagged AP x and tagged BS y . Withoutloss of generality, the angle of x and y are assumed to be . In addition, the PDF of k x k and k y k are given by f W ( · ) and f L ( · ) respectively, which are defined in Table I.Finally, we define several functions that will be used throughout this paper in Table II.Specifically, N L ( y, r, Γ) and N W ( y, r, Γ) represent the expected number of eNBs and APsrespectively in R \ B (0 , r ) , whose signal power received at y ∈ R exceeds Γ . In addition, C L ( y , Γ , y , Γ ) and C W ( y , Γ , y , Γ ) represent the expected number of eNBs and APs re-spectively in R \ B (0 , k y k ) , whose signal powers received at y ∈ R and y ∈ R exceed Γ and Γ respectively. Moreover, M , V and U are functions helping to calculate the conditionalMAP in the following sections. The user-perceived data rate distribution can be obtained from (4) by considering the average fraction of resource that eachuser achieves.
TABLE II: Notations and Definitions of Special FunctionsNotation Definition N L ( y, r, Γ) λ L R R \ B (0 ,r ) exp( − µ Γ P L l ( k x − y k ))d xN W ( y, r, Γ) λ W R R \ B (0 ,r ) exp( − µ Γ P W l ( k x − y k ))d xN L ( r, Γ) , N W ( r, Γ) N L ( y, r, Γ) , N W ( y, r, Γ) (polar coordinates of y = ( r, ) N L ( r ) , N W ( r ) N L ( y, r, Γ ed ) , N W ( y, r, Γ cs ) (polar coordinates of y = ( r, ) N L (Γ) , N W (Γ) N L ( o, , Γ) , N W ( o, , Γ) N L , N W N L ( o, , Γ ed ) , N W ( o, , Γ cs ) C L ( y , Γ , y , Γ ) λ L R R \ B (0 , k y k ) exp( − µ Γ P L l ( k x − y k ) − µ Γ P L l ( k x − y k ))d xC W ( y , Γ , y , Γ ) λ W R R \ B (0 , k y k ) exp( − µ Γ P W l ( k x − y k ) − µ Γ P W l ( k x − y k ))d xC L ( y , y ) , C W ( y , y ) C L ( y , Γ ed , y , Γ ed ) , C W ( y , Γ cs , y , Γ cs ) C L ( y ) , C W ( y ) C L ( y , Γ ed , o, Γ ed ) , C W ( y , , Γ cs , o, Γ cs ) M ( N , N , N ) ( − exp( − N ) N − − exp( − N − N + N ) N + N − N ) / ( N − N ) V ( x, s , s , N , N , N ) (1 − exp( − µs l ( k x k ))) M ( N , N , N ) + (1 − exp( − µs l ( k x k ))) M ( N , N , N ) U ( x, s, N ) − exp( − N ) N − exp( − µs l ( k x k ))( − exp( − N ) N − exp( − N ) N ) III. LTE
WITH C ONTINUOUS T RANSMISSION AND N O P ROTOCOL C HANGE
In this section, the MAP and SINR coverage performance for the LTE and Wi-Fi networksare investigated when LTE transmits continuously without any protocol modifications.
A. Medium Access Probability
From the CSMA/CA protocol described in Section II-C, a Wi-Fi AP will not transmit wheneverit has an LTE eNB as its contender, i.e., the power it receives from any LTE eNB exceeds theenergy detection threshold Γ ed . Therefore, the medium access indicator e Wi for AP x i is: e Wi = Y y k ∈ Φ L G LWki / l ( k y k − x i k ) ≤ Γ edPL Y x j ∈ Φ W \{ x i } (cid:18) t Wj ≥ t Wi + t Wj The MAP of Wi-Fi AP x i is the Palm probability that its medium access indicatoris equal to 1. Given its timer t Wi = t , the MAP can be derived as: E x i Φ W (cid:20) Y y k ∈ Φ L G LWki / l ( k y k − x i k ) ≤ Γ edPL Y x j ∈ Φ W \{ x i } (cid:18) t Wj ≥ t + t Wj Corollary When LTE transmits continuously with no protocol modifications, the MAP forthe tagged Wi-Fi AP is given by: ˆ p W , MAP ( λ W , λ L ) = Z ∞ − exp( − N W ( r )) N W ( r ) exp( − N L ) f W ( r )d r , (7)where f W is defined in Table I, while N L and N W are defined in Table II. Proof: According to Remark 1, given the tagged AP is located at x = ( r , , we have: P ( e W = 1 | x = ( r , E x Φ W Y y k ∈ Φ L G LWki / l ( k y k − x k ) ≤ Γ edPL Y x j ∈ Φ W \{ x } (cid:16) t Wj ≥ t W + t Wj Since LTE eNBs transmit con-tinuously with no protocol modifications, the medium access indicator for each LTE eNB is 1almost surely. The medium access indicator e Wj in (5) depends on both Φ L and Φ W . So thereexists a correlation between the interference from LTE eNBs and that from the Wi-Fi APs.Later we will show that if we substitute Φ L by another independent PPP Φ ′ L with intensity λ L in (5), the corresponding SINR coverage is an accurate approximation. This means the correlationbetween the interference from eNBs and APs is mostly captured by the statistical effect of Φ L on determining the MAP for Wi-Fi APs. Given the tagged AP is located at x , we first derivethe conditional MAP for another Wi-Fi AP and x to transmit simultaneously. Corollary Conditionally on the fact that the tagged AP x = ( r , transmits, the proba-bility for another AP x ∈ Φ W ∩ B c (0 , r ) to transmit is: h ( r , x ) = V ( x − x , Γ cs P W , Γ cs P W , N W ( r ) , N W ( x, r , Γ cs ) , C W ( x, x )) U ( x − x , Γ cs P W , N W ( r )) exp( N L − C L ( x − x )) , (8)where B c (0 , r ) is defined in Table I.The proof of Corollary 2 is provided in Appendix A. Then the SINR coverage performanceof the typical STA, denoted by p W ( T, λ W , λ L ) , is obtained as follows: Lemma The SINR coverage probability of the typical Wi-Fi STA with the SINR threshold T can be approximated as: p W ( T, λ W , λ L ) ≈ Z ∞ exp (cid:18) − µT l ( r ) σ N P W (cid:19) exp (cid:18) − Z R T l ( r ) λ LP W P L l ( k x k ) + T l ( r ) d x (cid:19) × exp (cid:18) − Z R \ B (0 ,r ) T l ( r ) λ W h ( r , x ) l ( k x k ) + T l ( r ) d x (cid:19) f W ( r )d r . (9) Proof: The conditional SINR coverage of the typical Wi-Fi STA is derived as follows: P ( SINR W > T | x = ( r , , e W = 1) ( a ) = P x Φ W ( F W , / l ( k x k ) P x j ∈ Φ W \{ x } F Wj, e Wj / l ( k x j k ) + P y m ∈ Φ L P L P W F LWm, / l ( k y m k ) + σ N P W > T | Φ W ( B o (0 , r )) = 0 , e W = 1) ( b ) = P ( F W , / l ( k x k ) P x j ∈ Φ W ∩ B c (0 ,r ) F Wj, ˆ e Wj / l ( k x j k ) + P y m ∈ Φ L P L P W F LWm, / l ( k y m k ) + σ N P W > T | ˆ e W = 1) SINR Threshold (dB) -10 -5 0 5 10 15 20 S I NR C o v e r age Simulation: λ W = 400, λ L = 0Theory: λ W = 400, λ L = 0Simulation: λ W = 200, λ L = 0Theory: λ W = 200, λ L = 0Simulation: λ W = 400, λ L = 200Theory: λ W = 400, λ L = 200Simulation: λ W = 400, λ L = 400Theory: λ W = 400, λ L = 400Simulation: λ W = 200, λ L = 200Theory: λ W = 200, λ L = 200Simulation: λ W = 200, λ L = 400Theory: λ W = 200, λ L = 400 Fig. 3: SINR coverage for the typical Wi-Fi STA. ( c ) ≈ exp( − µT l ( r ) σ N P W ) E (cid:20) − µT l ( r )( X x i ∈ Φ W ∩ B c (0 ,r ) P W P L F W Li, ˆ e Wi l ( k x i k ) ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ e W = 1 (cid:21) E (cid:20) − µT l ( r )( X y m ∈ Φ L F Lm, l ( k y m k ) ) (cid:21) , where (a) follows from Baye’s rule by re-writing x = ( r , as x ∈ Φ W and Φ W ( B o (0 , r )) =0 . Here B o (0 , r ) is defined in Table I. Step (b) is derived from Slyvniak’s theorem and by de-conditioning on Φ W ( B o (0 , r )) = 0 . The modified medium access indicator for AP x i ∈ (Φ W ∩ B c (0 , r )+ δ x ) is given by (26). The conditional probability for the Wi-Fi AP x j ∈ Φ W ∩ B c (0 , r ) to transmit given x transmits, i.e., P (ˆ e Wi = 1 | ˆ e W = 1) , is derived in Corollary 2. Step (c) usesthe assumption that the interference from LTE eNBs is independent of the Wi-Fi network.Since the interfering AP process is a non-independent thinning of Φ W , the Laplace transformof Wi-Fi interference (i.e., the second term in step (c)) is not known in closed-from. Therefore,similar to [7], [33], we approximate the Wi-Fi interferers as a non-homogeneous PPP withintensity λ W h ( r , x ) , which gives (9). Remark For the rest of the paper, given the tagged AP or tagged eNB located at ( r , transmits, we use “non-homogeneous PPP approximation” to refer to the process of approxi-mating Wi-Fi/LTE interferers as a non-homogeneous PPP with intensity λ W h ( r , x ) / λ L h ( r , x ) ,where h denotes the conditional MAP of the AP/eNB located at x . This will be used to derivethe SINR distribution under various coexistence scenarios.Based on the parameters in Table I, Fig. 3 gives the SINR coverage performance of the typicalWi-Fi STA, where the simulation results are obtained from the definition of SINR in (1). It can beobserved from Fig. 3 that Lemma 2 provides an accurate estimate of the actual SINR coverage.When λ L = 0 , Wi-Fi achieves good SINR performance due to the carrier sensing for Wi-Fiinterferers. However, when coexisting with LTE, the additional interference contributed by the consistently transmitting LTE eNBs significantly impacts the SINR coverage of the typical Wi-FiSTA. The smaller the AP intensity λ W , the more significant the LTE interference, which willlead to worse Wi-Fi SINR coverage performance. In Fig. 3, given λ L , the Wi-Fi SINR coveragefor λ W = 200 is worse than the case when λ W = 400 . 2) SINR Coverage Probability of Typical LTE UE: The SINR coverage probability of thetypical UE, which is denoted by p L ( T, λ W , λ L ) , is given in the following lemma: Lemma The SINR coverage probability for a typical LTE UE with SINR threshold T canbe approximated by: p L ( T, λ W , λ L ) ≈ Z ∞ exp (cid:18) − µT l ( r ) σ N P L (cid:19) exp (cid:18) − Z R \ B (0 ,r ) T λ L l ( r )d yT l ( r ) + l ( k y k ) (cid:19) exp (cid:18) − Z R T l ( r ) λ W h W ( r , x ) T l ( r ) + P L P W l ( k x k ) (cid:19) × f L ( r )d r (10)where h W ( r , x ) = − exp( − N W ) N W exp( − N L ( x, r , Γ ed ))(1 − exp( − µ Γ ed P L l ( k y − x k ))) denotes theconditional MAP for AP x given the tagged eNB y = ( r , transmits. Proof: According to Remark 1, given the tagged eNB is located at y = ( r , , denotingthe conditional SINR coverage probability by p L ( r , T, λ W , λ L ) , we have: p L ( r , T, λ W , λ L ) ( a ) = E (cid:20) exp( − µT l ( r ))( σ N P L + X y m ∈ Φ L \{ y } F Lm, l ( k y m k ) + X x j ∈ Φ W P W P L F W Lj, e Wj l ( k x j k ) ) (cid:12)(cid:12)(cid:12)(cid:12) y ∈ Φ L , Φ L ( B (0 , r )) = 0 (cid:21) ( b ) = E (cid:20) exp( − µT l ( r ))( σ N P L + X y Lm ∈ Φ L ∩ B c (0 ,r ) F Lm, l ( k y m k ) + X x j ∈ Φ W P W P L F W Lj, ˆ e Wj l ( k x j k ) ) (cid:21) ( c ) ≈ exp( − µT l ( r ) σ N P L ) E (cid:20) exp (cid:18) − µT l ( r ) X y m ∈ Φ L ∩ B c (0 ,r ) F Lm, l ( k y m k ) (cid:19)(cid:21) E (cid:20) exp (cid:18) − µT l ( r ) X x j ∈ Φ W P W P L F W Lj, ˆ e Wj l ( k x j k ) (cid:19)(cid:21) , where (a) is because the channels have Rayleigh fading and y L is the closest eNB to the typicaluser. Step (b) is obtained by using Slyvniak’s theorem and de-conditioning on Φ L ( B (0 , r )) = 0 .The modified medium access indicator for each AP in step (b) is given by: ˆ e Wj = Y y k ∈ Φ L ∩ B c (0 ,r ) (cid:18) G LWkj / l ( k y k − x j k ) ≤ Γ edPL G LW j / l ( k y − x j k ) ≤ Γ edPL (cid:19) Y x i ∈ Φ W \{ x j } (cid:18) t Wi ≥ t Wj + t Wi A straightforward scheme to guarantee the fair-coexistence between Wi-Fi and LTE is to letLTE adopt a discontinuous, duty-cycle transmission pattern, which is also know as LTE-U [13],[14]. Specifically, LTE transmits for a fraction η of time ( ≤ η ≤ ), and is muted for thecomplementary 1- η fraction.The LTE transmission duy cycle η can be fixed or adaptively adjusted based on Wi-Fi mediumutilization [13]. Generally, η needs to be chosen in such a way that LTE shall not impact Wi-Fimore than an additional Wi-Fi network w.r.t. SINR coverage probability, rate coverage, etc. Weconsider a static muting pattern for LTE, where all the eNBs follow the same muting patterneither synchronously or asynchronously. If the eNBs are muted synchronously, they transmit andmute at the same time. If the eNBs are muted asynchronously, the neighboring eNBs could adopta shifted version of the muting pattern [16]. For simplicity, we assume each eNB is transmittingwith probability η at a given time under the asynchronous scheme. In the rest of this section,the time-averaged DST and rate coverage performance when LTE transmits discontinuously arederived. A. LTE with Synchronous Discontinuous Transmission Pattern In this case, since all eNBs transmit and mute at the same time, the MAP for the tagged Wi-Fi AP during LTE “On” and “Off” period are ˆ p W ,MAP ( λ W , λ L ) and ˆ p W ,MAP ( λ W , respectively,where ˆ p W , MAP is given in (7). Similarly, the SINR coverage probability of the typical Wi-Fi STA(resp. LTE UE) with threshold T is p W ( T, λ W , λ L ) (resp. p L ( T, λ W , λ L ) ) and p W ( T, λ W , (resp.0) during LTE “On” and “Off” period respectively, where p W and p L are provided in Lemma 2and Lemma 3. Define the time-averaged DST with SINR threshold T as the time-averagedfraction of links that can support SINR level T . Lemma When LTE adopts a synchronous discontinuous transmission pattern with dutycycle η , the time-averaged DST with threshold T for the Wi-Fi and LTE network are given by: d W ,suc ( λ W , λ L , T, η ) = ηλ W ˆ p W ,MAP ( λ W , λ L ) p W ( T, λ W , λ L ) + (1 − η ) λ W ˆ p W ,MAP ( λ W , p W ( T, λ W , ,d L ,suc ( λ W , λ L , T, η ) = ηλ L p L ( T, λ W , λ L ) . (12) Proof: Since LTE transmits for η fraction of time and silences for − η fraction time, thetime-averaged DST performance for Wi-Fi and LTE can be obtained directly from Definition 1.In addition, the time-averaged rate coverage probability with threshold ρ is defined as the time-averaged fraction of eNBs/APs that can support an aggregate data rate of ρ . Since each LTEeNB transmits for η fraction of time, we treat the MAP of the tagged eNB as η in (4). Lemma When LTE adopts a synchronous discontinuous transmission pattern with dutycycle η , the time-averaged rate coverage probability with rate threshold ρ for Wi-Fi and LTEare given by: P W ,rate ( λ W , λ L , ρ, η ) = ηp W (2 ρB ˆ pW ,MAP ( λW ,λL ) − , λ W , λ L ) + (1 − η ) p W (2 ρB ˆ pW ,MAP ( λW , − , λ W , ,P L ,rate ( λ W , λ L , ρ, η ) = p L (2 ρBη − , λ W , λ L ) . (13) Proof: The time-averaged Wi-Fi rate coverage can be derived since the fraction of Wi-Fi APsthat can support data rate ρ is p W (2 ρB ˆ pW ,MAP ( λW ,λL ) − , λ W , λ L ) and p W (2 ρB ˆ pW ,MAP ( λW , − , λ W , during LTE “on” and “off” period respectively. In addition, the time-averaged LTE rate coverageis derived by noting that the typical LTE link is active for η fraction of time.It is straightforward from (12) and (13) that better DST and rate coverage can be achievedby Wi-Fi when η decreases. By contrast, since p L ( T, λ W , λ L ) is a decreasing function w.r.t. the SINR Threshold (dB) -10 -5 0 5 10 15 20 A v e r age nu m be r o f s u cc e ss f u l li n ks pe r k m η = 33.3%, Synchronous η = 33.3%, Asynchronous η = 50%, Synchronous η = 50%, Asynchronous η = 66.7%, Synchronous η = 66.7%, Asynchronous( λ W , λ L ) = (400,400) (a) DST for Wi-Fi SINR Threshold (dB) -10 -5 0 5 10 15 20 A v e r age nu m be r o f s u cc e ss f u l li n ks pe r k m η = 66.7%, Asynchronous η = 66.7%, Synchronous η = 50%, Asynchronous η = 50%, Synchronous η = 33.3%, Asynchronous η = 33.3%, Synchronous( λ W , λ L ) = (400,400) (b) DST for LTE Fig. 5: DST comparison of the synchronous and asynchronous muting pattern.SINR threshold T , LTE achieves better DST and rate coverage when η increases. B. LTE with Asynchronous Discontinuous Transmission Pattern Since each eNB transmits independently with probability η at a given time, the eNBs con-tributing to the interference of Wi-Fi form a PPP with intensity ηλ L . Therefore, the MAP forthe tagged AP is ˆ p W , MAP ( λ W , ηλ L ) , and the SINR coverage probability with threshold T forthe typical Wi-Fi STA is p W ( T, λ W , ηλ L ) . Correspondingly, the time-averaged DST of Wi-Fi isgiven by: d W ,suc ( λ W , λ L , T, η ) = λ W ˆ p W ,MAP ( λ W , ηλ L ) p W ( T, λ W , ηλ L ) , (14)and the time-averaged rate coverage probability of Wi-Fi is given by: P W ,rate ( λ W , λ L , ρ, η ) = p W (2 ρB ˆ pW ,MAP ( λW ,ηλL ) − , λ W , ηλ L ) . (15)According to (14) and (15), Wi-Fi achieves better DST and rate coverage when η decreases.For LTE, during the η fraction of time that the tagged eNB transmits, the interfering eNBsform a PPP with intensity ηλ L . Thus, the time-averaged DST of LTE is given by: d L ,suc ( λ W , λ L , T, η ) = λ L η Z ∞ p L ( r , T, λ W , ηλ L )2 πλ L r exp( − λ L πr )d r , (16)and the time-averaged rate coverage probability is given by: P L ,rate ( λ W , λ L , ρ, η ) = Z ∞ p L ( r , ρBη − , λ W , ηλ L )2 πλ L r exp( − λ L πr )d r , (17)where p L ( r , T, λ W , λ L ) is derived in Lemma 3. Rate Threshold (Mbps) R a t e C o v e r age P r obab ili t y η = 33.3%, Synchronous η = 33.3%, Asynchronous η = 50%, Synchronous η = 50%, Asynchronous η = 66.7%, Synchronous η = 66.7%, Asynchronous( λ W , λ L ) = (400, 400) (a) Rate coverage for Wi-Fi Rate Threshold (Mbps) R a t e C o v e r age P r obab ili t y η = 66.7%, Asynchronous η = 66.7%, Synchronous η = 50%, Asynchronous η = 50%, Synchronous η = 33.3%, Asynchronous η = 33.3%, Synchronous( λ W , λ L ) = (400, 400) (b) Rate coverage for LTE Fig. 6: Rate coverage comparison of the synchronous and asynchronous muting pattern. C. Comparison of Synchronous and Asynchronous Muting Patterns Fig. 5 and Fig. 6 show the analytical time-averaged DST and rate coverage performancewhen λ W = 400 APs/km and λ L = 400 eNBs/km . In terms of Wi-Fi DST and rate coverageperformance, the synchronous LTE muting pattern generally outperforms the asynchronous one.This is due to fact that when all LTE eNBs are muted, Wi-Fi APs observe a much cleanerchannel and therefore benefit more from LTE muting compared to the asynchronous scheme.Since LTE interferers form an independent thinning of the eNB process under the asynchronousmuting pattern, the latter outperforms the synchronous pattern in terms of DST and rate coverage.In addition, Fig. 5 and Fig. 6 also indicate that LTE needs to adopt a short transmission dutycycle η (e.g., less than 50%) to protect Wi-Fi. However, LTE is also more sensitive to thetransmission duty cycle compared to Wi-Fi, which means that a very small η leads to muchdegraded performance of LTE. Therefore, a synchronous muting pattern with a reasonably shortLTE transmission duty cycle (e.g., within 33.3% to 50%) is suggested to protect Wi-Fi.V. LTE WITH L ISTEN - BEFORE - TALK AND R ANDOM B ACKOFF Besides LTE with discontinuous transmission, another fair coexistence method is to let LTEimplement the listen-before-talk (LBT) and random backoff (BO) mechanism similar to Wi-Fi. Specifically, we consider each eNB implements carrier sense mechanism to detect stronginterfering LTE and Wi-Fi neighbors with a common threshold Γ L . In addition, each eNBimplements a random back off timer, which is uniformly distributed between a and b . Thevalue of ( a, b ) determines how aggressively LTE eNBs access the channel. The medium access indicators for AP x i and eNB y k are given as follows: e Wi = Y x j ∈ Φ W \{ x i } (cid:18) t Wj ≥ t Wi + t Wj Corollary Conditionally on the fact that the tagged eNB y = ( r , transmits, the prob-ability for another AP x i ∈ Φ W ∩ B c (0 , r ) to transmit is: h W ( r , x i ) = V ( x i − y , Γ ed P L , Γ L P W , N , N , N ) U ( x i − y , Γ L P W , N ) , where N = N W (Γ L )+ N L ( r , Γ L ) , N = N W + N L ( x i , r , Γ ed ) , and N = C W ( y − x i , Γ L , o, Γ cs )+ C L ( x i , Γ ed , y , Γ L ) . Corollary Conditionally on the fact that the tagged eNB y = ( r , transmits, the prob-ability for another AP x i ∈ Φ W ∩ B c (0 , r ) to transmit is: h L ( r , y k ) = V ( y k − y , Γ L P L , Γ L P L , N , N , N ) U ( y k − y , Γ L P L , N ) , where N = N W (Γ L ) + N L ( r , Γ L ) , N = N W (Γ L ) + N L ( y k , r , Γ L ) , and N = C W ( y − y k , Γ L , o, Γ L ) + C L ( y k , Γ L , y , Γ L ) .Based on Corollary 6 and Corollay 7, the SINR coverage of the typical UE can also be derivedusing the non-homogeneous PPP approximation of the interfering eNBs and APs: Lemma When LTE implements listen-before-talk and random backoff mechanism with ( a, b ) = (0 , , the approximate SINR coverage probability of the typical LTE UE is: p L ( T, λ W , λ L ) ≈ Z ∞ exp (cid:18) − µT l ( r ) σ N P W (cid:19) exp (cid:18) − Z R T l ( r ) λ W h W ( r , x ) P L P W l ( k x k ) + T l ( r ) d x (cid:19) × exp (cid:18) − Z R \ B (0 ,r ) T l ( r ) λ L h L ( r , y ) l ( k y k ) + T l ( r ) d y (cid:19) f L ( r )d r . B. LTE with Lower Channel Access Priority as Wi-Fi when (a,b) = (1,2) In this case, since the random backoff timer for each LTE eNB is always larger than that ofWi-Fi APs, LTE has a lower channel access priority. Specifically, the medium access indicatorfor each Wi-Fi AP and LTE eNB in (18) can be simplified to: e Wi = Y x j ∈ Φ W \{ x i } (cid:18) t Wj ≥ t Wi + t Wj Lemma When LTE implements the listen-before-talk and random backoff mechanismwith ( a, b ) = (1 , , the approximate SINR coverage probability of a typical Wi-Fi STA is: p W ( T, λ W , λ L ) ≈ Z ∞ exp (cid:18) − µT l ( r ) σ N P W (cid:19) exp (cid:18) − Z R T l ( r ) λ L h L ( r , y ) P W P L l ( k y k ) + T l ( r ) d y (cid:19) × exp (cid:18) − Z R \ B (0 ,r ) T l ( r ) λ W h W ( r , x ) l ( k x k ) + T l ( r ) d x (cid:19) f W ( r )d r . Next, given the tagged eNB of the typical UE is located at y = ( r , , the two conditionalprobabilities P x i Φ W ( e Wi = 1 | e L = 1 , y = ( r , and P y k Φ L ( e Lk = 1 | e L = 1 , y = ( r , , denotedby h W ( r , x i ) and h L ( r , y k ) respectively, are given in (23) and (24): h W ( r , x i ) = 1 − exp( − N W + C W ( y − x i , Γ L , o, Γ cs )) N W − C W ( y − x i , Γ L , o, Γ cs ) , (23) h L ( r , y k ) = (1 − exp( − µ Γ L P L l ( k y k − y k )))( M ( N , N , N ) + M ( N , N , N ))exp( N W (Γ L ) − C W ( y − y k , Γ L , o, Γ L )) U ( y k − y , Γ L P L , N ) , (24)where N = N L ( r , Γ L ) , N = N L ( y k , r , Γ L ) and N = C L ( y k , Γ L , y , Γ L ) in (24).Based on h W and h L , the SINR coverage probability of the typical UE can be derived byapplying the non-homogeneous PPP approximation: Lemma When LTE implements the listen-before-talk and random backoff mechanism with ( a, b ) = (1 , , the approximate SINR coverage probability of the typical LTE UE is: p L ( T, λ W , λ L ) ≈ Z ∞ exp (cid:18) − µT l ( r ) σ N P W (cid:19) exp (cid:18) − Z R \ B (0 ,r ) T l ( r ) λ L h L ( r , y ) l ( k y k ) + T l ( r ) d y (cid:19) × exp (cid:18) − Z R T l ( r ) λ W h W ( r , x ) P L P W l ( k x k ) + T l ( r ) d x (cid:19) f L ( r )d r . The SINR coverage performance of the typical STA and UE under two LTE channel accesspriority schemes is plotted in Fig. 7 and Fig. 8, where the simulation results are obtained fromthe definition of SINR in (1) and (2). The accuracy of the approximations can be validated forvarious LTE sensing thresholds and AP/eNB densities. Since both Wi-Fi/LTE adopt the LBTand random backoff mechanism, a good overall SINR coverage probability can be achieved forWi-Fi and LTE. In addition, given LTE contention window size ( a, b ) , both Wi-Fi STA and LTEUE can achieve better SINR performance with a more sensitive threshold Γ L , which is due toless LTE interference. It can also be observed that when LTE has lower channel access priority,a less sensitive threshold Γ L is needed to obtain a similar Wi-Fi SINR performance as in thecase when LTE has the same channel access priority as Wi-Fi. SINR Threshold (dB) -10 -5 0 5 10 15 20 S I NR C o v e r age P r obab ili t y Theory: (a,b) = (0,1), Γ L = -82 dBmSimulation: (a,b) = (0,1), Γ L = -82 dBmTheory: (a,b) = (1,2), Γ L = -77 dBmSimulation: (a,b) = (1,2), Γ L = -77 dBmTheory: (a,b) = (0,1), Γ L = -77 dBmSimulation: (a,b) = (0,1), Γ L = -77 dBmTheory: (a,b) = (1,2), Γ L = -72 dBmSimulation: (a,b) = (1,2), Γ L = -72 dBm ( λ W , λ L ) = (400, 400) SINR Threshold (dB) -10 -5 0 5 10 15 20 S I NR C o v e r age P r obab ili t y Theory: (a,b) = (1,2), Γ L = -77 dBmSimulation: (a,b) = (1,2), Γ L = -77 dBmTheory: (a,b) = (0,1), Γ L = -82 dBmSimulation: (a,b) = (0,1), Γ L = -82 dBmTheory: (a,b) = (1,2), Γ L = -72 dBmSimulation: (a,b) = (1,2), Γ L = -72 dBmTheory: (a,b) = (0,1), Γ L = -77 dBmSimulation: (a,b) = (0,1), Γ L = -77 dBm ( λ W , λ L ) = (200, 200) Fig. 7: Wi-Fi SINR performance under different LTE channel access priorities and Γ L .VI. P ERFORMANCE C OMPARISONS OF D IFFERENT C OEXISTENCE S CENARIOS In this section, the DST and rate coverage performance for each coexistence scenario arecompared through numerical evaluations. In particular, we use Wi-Fi + LTE (Wi-Fi + LTE-U, andWi-Fi + LAA respectively) to denote the scenario when Wi-Fi operator 1 coexists with anotheroperator 2, which uses LTE with no protocol change (LTE with discontinuous transmission, andLTE with LBT and random BO respectively). The baseline performance of Wi-Fi operator 1 is SINR Threshold -10 -5 0 5 10 15 20 S I NR C o v e r age P r obab ili t y Theory: (a,b) = (0,1), Γ L = -82 dBmSimulation: (a,b) = (0,1), Γ L = -82 dBmTheory: (a,b) = (0,1), Γ L = -77 dBmSimulation: (a,b) = (0,1), Γ L = -77 dBmTheory: (a,b) = (1,2), Γ L = -77 dBmSimulation: (a,b) = (1,2), Γ L = -77 dBmTheory: (a,b) = (1,2), Γ L = -72 dBmSimulation: (a,b) = (1,2), Γ L = -72 dBm ( λ W , λ L ) = (400, 400) SINR Threshold -10 -5 0 5 10 15 20 S I NR C o v e r age P r obab ili t y Theory: (a,b) = (0,1), Γ L = -82 dBmSimulation: (a,b) = (0,1), Γ L = -82 dBmTheory: (a,b) = (0,1), Γ L = -77 dBmSimulation: (a,b) = (0,1), Γ L = -77 dBmTheory: (a,b) = (1,2), Γ L = -77 dBmSimulation: (a,b) = (1,2), Γ L = -77 dBmTheory: (a,b) = (1,2), Γ L = -72 dBmSimulation: (a,b) = (1,2), Γ L = -72 dBm ( λ W , λ L ) = (200, 200) Fig. 8: LTE SINR performance under different channel access priorities and Γ L . SINR Threshold (dB) -10 -5 0 5 10 15 20 A v e r age nu m be r o f s u cc e ss f u l li n ks pe r k m Wi-Fi + LTE-U: η = 50%, SynchronousWi-Fi + LAA: (a,b) = (1,2), Γ L = -77 dBmWi-Fi + LAA: (a,b) = (0,1), Γ L = -82 dBmWi-Fi + Wi-FiWi-Fi + LAA: (a,b) = (0,1), Γ L = -62 dBmWi-Fi + LTE ( λ W , λ L ) = (400,400) (a) DST for Wi-Fi SINR Threshold (dB) -10 -5 0 5 10 15 20 A v e r age nu m be r o f s u cc e ss f u l li n ks pe r k m Wi-Fi + LTEWi-Fi + LAA: (a,b) = (0,1), Γ L = -62 dBmWi-Fi + LAA: (a,b) = (0,1), Γ L = -82 dBmWi-Fi + LAA: (a,b) = (1,2), Γ L = -77 dBmWi-Fi + LTE-U : η = 50%, Synchronous( λ W , λ L ) = (400,400) (b) DST for LTE Fig. 9: DST comparisons under different coexistence scenarios.when operator 2 also uses Wi-Fi (i.e., Wi-Fi + Wi-Fi). The Wi-Fi MAP and SINR coverage ofthe baseline scenario can be obtained directly from Lemma 6 and Lemma 7 by setting all thesensing thresholds to Γ cs . In addition, we focus on a dense network deployment where λ W =400 APs/km and λ L = 400 eNBs/km . Based on the MAP and approximate SINR coverageprobability, we have investigated the DST and rate coverage probability of Wi-Fi/LTE under allthe coexistence scenarios in Fig. 9 and Fig. 10.Fig. 9a shows that when coexisting with LTE, Wi-Fi has the worst DST performance sinceit experiences strong interference from the persistent transmitting LTE eNBs. In addition, Wi-Fiachieves similar DST performance when operator 2 implements one of the following mechanisms:(1) LTE-U with a short duty cycle (e.g., 50%); (2) LAA with same channel access priority asWi-Fi and a more sensitive sensing threshold (e.g., ( a, b ) = (0 , , Γ L = − dBm); and (3)LAA with lower channel access priority than Wi-Fi and a less sensitive sensing threshold (e.g., Rate Threshold (Mbps) R a t e C o v e r age P r obab ili t y Wi-Fi + LTE-U: η = 50%, SynchronousWi-Fi + LAA: (a,b) = (1,2), Γ L = -77 dBmWi-Fi + LAA: (a,b) = (0,1), Γ L = -82 dBmWi-Fi + Wi-FiWi-Fi + LAA: (a,b) = (0,1), Γ L = -62 dBmWi-Fi + LTE( λ W , λ L ) = (400, 400) (a) Rate coverage for Wi-Fi Rate Threshold (Mbps) R a t e C o v e r age P r obab ili t y Wi-Fi + LTEWi-Fi + LAA: (a,b) = (0,1), Γ L = -62 dBmWi-Fi + LAA: (a,b) = (0,1), Γ L = -82 dBmWi-Fi + LAA: (a,b) = (1,2), Γ L = -77 dBmWi-Fi + LTE-U : η = 50%, Synchronous( λ W , λ L ) = (400, 400) (b) Rate coverage for LTE Fig. 10: Rate coverage comparisons under different coexistence scenarios. ( a, b ) = (1 , , Γ L = − dBm). Compared to the baseline scenario, Wi-Fi has better DST underthe above scenarios, especially in the low SINR threshold regime. Furthermore, when operator 2implements LAA with the -62 dBm energy detection threshold, the DST performance of Wi-Fiis not much improved over Wi-Fi + LTE. Therefore, the -62 dBm energy detection threshold istoo conservative to protect Wi-Fi, and a more sensitive threshold Γ L is recommended for LAA.In contrast, Fig. 9b shows that operator 2 has significantly lower (around 50%) DST when usingLTE-U or LAA with a sensitive sensing threshold (e.g., -82 dBm or -77 dBm), which is mainlydue to the decreased MAP for eNBs.In terms of rate coverage, it can be observed from Fig. 10a that when operator 2 adoptsLTE-U with a 50% duty cycle or LAA with a sensitive sensing threshold (e.g., -82 dBm or -77dBm), Wi-Fi has similar performance as the baseline scenario in low rate threshold regime (e.g.less than 5 Mbps), and better performance with medium to high rate threshold (e.g., more than10 Mbps). If LAA uses the -62 dBm energy detection threshold, the rate coverage of Wi-Fihas negligible improvement over the Wi-Fi + LTE scenario, which means the energy detectionthreshold does not suffice to protect Wi-Fi. In addition, due to the degraded SINR performance,Wi-Fi has worse rate performance under Wi-Fi + LTE than the baseline scenario. Meanwhile,when the sensing threshold of LAA is -82 dBm or -77 dBm, Fig. 10b shows that the rate lossof operator 2 under Wi-Fi + LAA is around 30% to 40% compared to Wi-Fi + LTE. In contrast,the rate loss under Wi-Fi + LTE-U is slightly more than 50% for most rate thresholds.Overall, under Wi-Fi + LTE, the DST and rate coverage probability of Wi-Fi decreasessignificantly compared to the baseline performance, which makes it an impractical scenario to operate LTE in unlicensed spectrum. Under Wi-Fi + LTE-U, LTE-U operator 2 has the flexibilityto guarantee good DST and rate coverage performance for Wi-Fi operator 1 by choosing a lowLTE transmission duty cycle. In addition, LTE-U has low implementation cost due to its simplescheme. However, LTE-U also has the following disadvantages: (1) the LTE-U operator has muchdegraded DST and rate coverage performance under low transmission duty cycle; (2) LTE-U isonly feasible in certain regions and/or unlicensed bands where LBT feature is not required, suchas the 5.725-5.825 GHz band in U.S. [5]; (3) LTE-U transmissions are more likely to collide withWi-Fi acknowledgment packets due to the lack of a CCA procedure, which means Wi-Fi SINRcoverage under Wi-Fi + LTE-U may not as easily translate into the rate performance as Wi-Fi +LAA; and (4) LTE-U has practical cross-layer issues, such that the frequent on and off switchingof LTE will trigger the Wi-Fi rate control algorithm to lower the Wi-Fi transmission rate [41].In contrast, under Wi-Fi + LAA, by choosing an appropriate LAA channel access priority (i.e.,contention window size) and sensing threshold, Wi-Fi operator 1 also achieves better DST andrate coverage performance compared to the baseline scenario, while LAA operator 2 can maintainacceptable rate coverage performance. Additionally, LAA also meets the global requirement foroperation in the unlicensed spectrum. The main disadvantage of LAA versus LTE-U is that LAArequires more complicated implementation for the LBT and random BO feature. Therefore, interms of performance comparisons and practical constraints, LTE with LBT and random BO (i.e.,LAA) is more promising than LTE with discontinuous transmission (i.e., LTE-U) to provide aglobal efficient solution to the coexistence issues of LTE and Wi-Fi in unlicensed spectrum.VII. C ONCLUSION This paper proposed and validated a stochastic geometry framework for analyzing the coex-istence of overlaid Wi-Fi and LTE networks. Performance metrics including the medium accessprobability, SINR coverage probability, density of successful transmission and rate coverageprobability have been analytically derived and numerically evaluated under three coexistencescenarios. Although Wi-Fi performance is significantly degraded when LTE transmits continu-ously without any protocol changes, we showed that LTE can be a good neighbor to Wi-Fi bymanipulating the LTE transmission duty cycle, sensing threshold, or channel access priority. Theproposed analytical framework validates and complements the ongoing system level simulationstudies of LTE-U and LAA, and can be utilized by both academia and industry to rigorously study LTE and Wi-Fi coexistence issues.Future works could include: (1) extending the single unlicensed band analysis into multiplebands, and incorporating channel selection schemes to further improve Wi-Fi/LTE performance;(2) charactering Wi-Fi/LTE delay performance by extending the full-buffer traffic assumption tonon full-buffer traffic; (3) analyzing the SINR or rate coverage when both downlink and uplinktraffic exist; and (4) investigating Wi-Fi/LTE performance when LTE adopts both listen-before-talk and discontinuous transmission features, which will be standardized by LTE-U forum [14]in the future. A PPENDIX AP ROOF OF C OROLLARY x i ∈ Φ W , the conditional MAP of x i given the tagged AP x = ( r , transmits is: P x i Φ W ( e Wi = 1 | e W = 1 , x ∈ Φ W , Φ W ( B o (0 , r )) = 0) ( a ) = P x i ,x Φ W ( e Wi = 1 , e W = 1 | Φ W ( B o (0 , r )) = 0) P x i ,x Φ W ( e W = 1 | Φ W ( B o (0 , k x k )) = 0) ( b ) = E x i Φ W (ˆ e Wi ˆ e W ) E x i Φ W (ˆ e W ) , (25)where (a) follows from the Baye’s rule, and (b) is derived by applying the Slyvniak’s theoremand de-conditioning. The modified medium access indicators for x i and x are: ˆ e Wi = Y x j ∈ (Φ W ∩ B c (0 ,r )+ δ x ) \{ x i } (cid:18) t Wj ≥ t Wi + t Wj IEEE GLOBECOM Workshop on Heterogeneous and Small Cell Networks , Dec. 2015.[2] FCC, “Revision of part 15 of the commission’s rules to permit unlicensed national information infrastructure (U-NII)devices in the 5 GHz band,” Feb. 2013.[3] Qualcomm, “Extending LTE Advanced to unlicensed spectrum,” white paper , Dec 2013.[4] R. Ratasuk, M. Uusitalo, N. Mangalvedhe, A. Sorri, S. Iraji, C. Wijting, and A. Ghosh, “License-exempt LTE deploymentin heterogeneous network,” in International Symposium on Wireless Communication Systems , pp. 246–250, Aug. 2012. [5] 3GPP TR 36.889, “Study on licensed-assisted access to unlicensed spectrum,” 2015.[6] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks: Volume 1: THEORY . Now Publishers Inc,2010.[7] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks: Volume II-Applications . Now PublishersInc, 2010.[8] M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for theanalysis and design of wireless networks,” IEEE Journal on Selected Areas in Communications , vol. 27, pp. 1029–1046,Sep. 2009.[9] M. Haenggi, Stochastic geometry for wireless networks . Cambridge University Press, 2013.[10] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic geometry and its applications . John Wiley & Sons, 2013.[11] A. M. Cavalcante, E. Almeida, R. D. Vieira, F. Chaves, R. C. Paiva, F. Abinader, S. Choudhury, E. Tuomaala, andK. Doppler, “Performance evaluation of LTE and Wi-Fi coexistence in unlicensed bands,” in IEEE 77th VehicularTechnology Conference (VTC Spring) , pp. 1–6, Jun. 2013.[12] T. Nihtila, V. Tykhomyrov, O. Alanen, M. Uusitalo, A. Sorri, M. Moisio, S. Iraji, R. Ratasuk, N. Mangalvedhe, et al. ,“System performance of LTE and IEEE 802.11 coexisting on a shared frequency band,” in IEEE Wireless Communicationsand Networking Conference (WCNC) , pp. 1038–1043, Jun. 2013.[13] Qualcomm, “LTE in unlicensed spectrum: Harmonious coexistence with Wi-Fi,” white paper , Jun. 2014.[14] LTE-U Forum, “LTE-U technical report: coexistence study for LTE-U SDL v1.0,” Technical Report , Feb. 2015.[15] E. Almeida, A. M. Cavalcante, R. C. Paiva, F. S. Chaves, F. M. Abinader, R. D. Vieira, S. Choudhury, E. Tuomaala, andK. Doppler, “Enabling LTE/WiFi coexistence by LTE blank subframe allocation,” in IEEE International Conference onCommunications (ICC) , pp. 5083–5088, Jun. 2013.[16] J. Jeon, H. Niu, Q. C. Li, A. Papathanassiou, and G. Wu, “LTE in the unlicensed spectrum: Evaluating coexistencemechanisms,” in IEEE Globecom Workshops (GC Wkshps) , pp. 740–745, Dec. 2014.[17] T. Kondo, H. Fujita, M. Yoshida, and T. Saito, “Technology for WiFi/bluetooth and WiMax coexistence,” Fujitsu Sci.Technol. J , vol. 46, pp. 72–78, 2010.[18] A. Mukherjee, J.-F. Cheng, S. Falahati, L. Falconetti, A. Furusk¨ar, and B. Godana, “System architecture and coexistenceevaluation of licensed-assisted access LTE with IEEE 802.11,” in IEEE ICC Workshop on LTE in Unlicensed Bands:Potentials and Challenges , Jun. 2015.[19] J. Jeon, Q. C. Li, H. Niu, A. Papathanassiou, and G. Wu, “LTE in the unlicensed spectrum: A novel coexistence analysiswith WLAN systems,” in IEEE Global Communications Conference (GLOBECOM) , pp. 3459–3464, Dec. 2014.[20] S. Sagari, S. Baysting, D. Saha, I. Seskar, W. Trappe, and D. Raychaudhuri, “Coordinated dynamic spectrum managementof LTE-U and Wi-Fi networks,” in ,pp. 209–220, Sept 2015.[21] G. Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination function,” IEEE Journal on Selected Areasin Communications , vol. 18, pp. 535–547, March 2000.[22] J. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Transactionson Communications , vol. 59, pp. 3122–3134, Nov. 2011.[23] H. Dhillon, R. Ganti, F. Baccelli, and J. Andrews, “Modeling and analysis of K-tier downlink heterogeneous cellularnetworks,” IEEE Journal on Selected Areas in Communications , vol. 30, pp. 550–560, Apr. 2012. [24] H. Dhillon, R. Ganti, and J. Andrews, “Load-aware modeling and analysis of heterogeneous cellular networks,” IEEETransactions on Wireless Communications , vol. 12, pp. 1666–1677, Apr. 2013.[25] S. Mukherjee, “Distribution of downlink SINR in heterogeneous cellular networks,” IEEE Journal on Selected Areas inCommunications , vol. 30, pp. 575–585, Apr. 2012.[26] R. Heath, M. Kountouris, and T. Bai, “Modeling heterogeneous network interference using Poisson point processes,” IEEETransactions on Signal Processing , vol. 61, pp. 4114–4126, Aug. 2013.[27] H. Dhillon, M. Kountouris, and J. Andrews, “Downlink MIMO HetNets: modeling, ordering results and performanceanalysis,” IEEE Transactions on Wireless Communications , vol. 12, pp. 5208–5222, Oct. 2013.[28] X. Lin, J. G. Andrews, and A. Ghosh, “Modeling, analysis and design for carrier aggregation in heterogeneous cellularnetworks,” IEEE Transactions on Communications , vol. 61, pp. 4002–4015, Sept. 2013.[29] R. Zhang, M. Wang, Z. Zheng, X. S. Shen, and L.-L. Xie, “Stochastic geometric performance analysis for carrier aggregationin LTE-A systems,” in IEEE International Conference on Communications (ICC), , pp. 5777–5782, Jun. 2014.[30] N. Miyoshi and T. Shirai, “A cellular network model with Ginibre configurated base stations,” Research Rep. on Math.and Comp. Sciences (Tokyo Inst. of Tech.) , Oct. 2012.[31] Y. Li, F. Baccelli, H. Dhillon, and J. Andrews, “Statistical modeling and probabilistic analysis of cellular networks withdeterminantal point processes,” IEEE Transactions on Communications , vol. 63, pp. 3405–3422, Sept. 2015.[32] A. Guo and M. Haenggi, “Asymptotic deployment gain: A simple approach to characterize the sinr distribution in generalcellular networks,” IEEE Transactions on Communications , vol. 63, pp. 962–976, Mar. 2015.[33] H. Nguyen, F. Baccelli, and D. Kofman, “A stochastic geometry analysis of dense IEEE 802.11 networks,” in IEEEINFOCOM 2007 , pp. 1199–1207, May 2007.[34] Y. Kim, F. Baccelli, and G. de Veciana, “Spatial reuse and fairness of ad hoc networks with channel-aware CSMAprotocols,” IEEE Transactions on Information Theory , vol. 60, pp. 4139–4157, Jul. 2014.[35] T. V. Nguyen and F. Baccelli, “A stochastic geometry model for cognitive radio networks,” The Computer Journal , vol. 55,pp. 534–552, Apr. 2012.[36] A. Bhorkar, C. Ibars, and P. Zong, “On the throughput analysis of LTE and WiFi in unlicensed band,” in AsilomarConference on Signals, Systems and Computers , pp. 1309–1313, Nov. 2014.[37] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, “Femtocell networks: a survey,” IEEE Communications Magazine ,vol. 46, pp. 59–67, Sept. 2008.[38] IEEE Std 802.11-2012, “IEEE standard for information technology-telecommunications and information exchange betweensystems local and metropolitan area networksspecific requirements part 11: Wireless lan medium access control (MAC)and physical layer (PHY) specifications,” Mar. 2012.[39] A. Busson, G. Chelius, and J.-M. Gorce, “Interference Modeling in CSMA Multi-Hop Wireless Networks,” ResearchReport RR-6624, INRIA, Feb. 2009.[40] M. Haenggi and R. K. Ganti, Interference in large wireless networks . Now Publishers Inc, 2009.[41] N. Jindal and D. Breslin, “LTE and Wi-Fi in unlicensed spectrum: A coexistence study,”