Impact of Spatial Multiplexing on the Throughput of Ultra-Dense mmWave AP Networks
11 Impact of Spatial Multiplexing on theThroughput of Ultra-Dense mmWave APNetworks
Shuqiao Jia and Behnaam Aazhang
Abstract
The operating range of a single millimeter wave (mmWave) access point (AP) is small due tothe high path loss and blockage issues of the frequency band. To achieve the coverage similar toconventional sub-6GHz networks, the ultra-dense deployments of APs are required by the mmWavenetwork. In general, the mmWave APs can be categorized into backhaul-connected APs and relay APs.Though the spatial distribution of backhaul-connected APs can be captured by the Poison point process(PPP), the desired locations of relay APs depend on the transmission protocol operated in the mmWavenetwork. In this paper, we consider modeling the topology of mmWave AP network by incorporatingthe multihop protocol. We first derive the topology of AP network with the spatial multiplexing disabledfor each transmission hop. Then we analyze the topology when the spatial multiplexing is enabled atthe mmWave APs. To derive the network throughput, we first quantify the improvement in latency andthe degradation of coverage probability with the increase of spatial multiplexing gain at mmWave APs.Then we show the impact of spatial multiplexing on the throughput for the ultra-dense mmWave APnetwork.
I. I
NTRODUCTION
The use of millimeter wave (mmWave) frequencies in access points (APs) becomes a trend inthe emerging fifth generation network [1–6]. Despite the large available bandwidth in mmWavefrequencies, the small wavelength experiences a high path loss and a severe penetration loss,which limits the coverage of a single mmWave AP [7]. To achieve the same size of networkcoverage as the sub-6GHz network, the ultra-dense deployment of APs appears to be the solutionfor mmWave network [3, 5, 8]. Note that the topology of sub-6GHz network is relatively simple,where all the APs are connected to the Internet backhaul and each cell is covered by one APin the network [9]. However, owing to its ultra-density, only a small portion of APs in the a r X i v : . [ c s . I T ] F e b mmWave network have direct access to the Internet backhaul. Other mmWave APs are used asrelays to extend the coverage for the network, as shown in Fig.1 [3]. Accordingly, the topologyof mmWave AP network varies with different multihop transmission protocols, which lead todifferent network performances.Several aspects of the ultra-dense mmWave AP network have been studied. In [4], the mmWavemodeling was comprehensively studied. The hybrid precoding for mmWave network was pro-posed in [10]. In [3], the optimal intensity of the ultra-dense mmWave AP network was derivedunder the impact of blockage. The SINR coverage probability and rate analysis for mmWavenetworks were presented in [2, 5]. We remark that all the previous work assumed mmWave APs tobe uniformly distributed in the mmWave network. Such an assumption has been validated for thesub-6GHz network [9]. However, the ultra-density of mmWave APs results in a complicated andflexible network topology, which cannot be captured by simply applying a uniformly distributedspatial model.We approach the topology of ultra-dense mmWave AP network by introducing the tieredmodel, rather than modeling the network as a whole. In the tiered mmWave AP network, thebackhaul-connected APs are considered as the th tier. Other AP tiers in the mmWave networkare used as relay, which extend the coverage of backhaul-connected APs to the whole network.Note that the spatial distribution of backhaul-connected APs is determined by the infrastructure.However, the AP locations in other tiers are decided by the topological structures of previoustiers and the transmission protocol. Consequently, the topology of mmWave AP network haslarge flexibility and is dependent on the transmission protocol.Our key contribution is to analyze the performance of ultra-dense mmWave AP networkwith considering the impact of transmission protocol. Specifically, we analyze how the hybridprecoding scheme affects the performance of ultra-dense mmWave AP network. Note that theimplementation of hybrid precoding combines the analog beamforming and baseband spatialmultiplexing [10]. In this paper, we focus on the impact of spatial multiplexing on the perfor-mance of mmWave AP networks. In section II, we first introduce the tiered structure for themmWave APs. We then derive the topology of mmWave AP network concerning the differentspatial multiplexing gains in Section III. The performance analysis is provided in section IV,where we characterize the latency, the coverage probability and the throughput for the mmWaveAP network. (a) (b) (c) (d) Fig. 1: The mmWave AP network with multihop transmission protocol: (a) the whole AP network including backhaul-connect APs (tier 0) atblue squares, relay APs with hop count 1 (tier 1) at yellow dots, relay APs with hop count 2 (tier 2) at green dots; (b) transmission hop 1 fromtier 0 to tier 1; (c) a subset of tier 1 (highlight in red circle) scheduled to transmit at hop 2; (d) transmission hop 2 from tier 1 to tier 2.
II. T
IERED M ODEL OF MM W AVE AP S The mmWave APs have an inherently tiered structure due to the multihop protocol operatedin the network, as shown in Fig.1. Therefore, instead of modeling the whole network, we modelthe spatial distribution of mmWave APs for each transmission hop. The APs directly connectedto the backhaul are defined as the th tier. The other mmWave APs are used as relay, which canbe further divided into multiple tiers with respect to ’the distance’ to the th tier.In the mmWave AP network, we measure the distance of an AP to tier 0 by the number ofrelays between the AP and tier , which is termed as hop count. Based on the hop count, the i th AP tier or tier i is defined as the subset of mmWave APs with hop count equal to i .Consider a mmWave AP network of density Λ A , where the transmission protocol contains M hops. It follows that mmWave APs can be divided into M + 1 tiers, namely one backhaul-connected tier and M relay tiers. The locations of mmWave APs in tier i are modeled by thepoint process Φ i = { x , x , · · · } , x j ∈ R . We then define the sequence of point processes { Φ i } Mi =0 as the topology of the mmWave AP network.Note that the locations of backhaul-connected mmWave APs are restricted by the networkinfrastructure. Thus, we model tier 0 by a homogeneous Poison point process (PPP) Φ withintensity Λ . Unlike tier 0, the spatial distribution of Φ i +1 is determined by the point process Φ i and the transmission protocol applied in the mmWave AP network. Denote φ i as the subset of Φ i , where φ i consists of the APs which are scheduled to transmit at hop i + 1 . For the i + 1 th transmission hop, the AP located at x ∈ φ i transmits to a cluster of points B xi = { y , y , · · · } ,where B xi is centered at x . It follows that tier i + 1 can be expressed as Φ i +1 = (cid:83) x ∈ φ i B xi . Here,the points of B xi are assumed to be independently and identically distributed (i.i.d.) around the cluster center x . III. T OPOLOGY OF MM W AVE AP NETWORK
Assume that the hybrid precoding is implemented at the mmWave APs, which consists ofanalog beamforming and spatial multiplexing. Note that analog beamforming is mandatory for ammWave AP to combat the high path loss. However, spatial multiplexing is required only whenthe mmWave AP needs to support multiple data streams.
Definition . Spatial Multiplexing Gain.
The spatial multiplexing gain for a mmWave AP isdefined as the number of data streams supported by the AP.Assume the mmWave AP to be equipped with K RF chains. Then the spatial multiplexing gainof the mmWave AP is upper bounded by K . Let all the transmitters of the same hop employ theidentical hybrid precoding process. It follows that all APs in φ i have the same spatial multiplexinggain, which is denoted by k i . Next, we characterize the topology of mmWave AP network withrespect to the spatial multiplexing gain k i . A. Topology with Spatial Multiplexing Disabled
Given that the spatial multiplexing is disabled, we then have k i = 1 , ∀ i . It implies that foreach x ∈ φ i , the cluster B xi contains only one point y ∈ Φ i +1 , where the probability densityfunction (PDF) of y conditioning on x is denoted by f i ( y | x ) . Assume that the mmWave APs in φ i transmit at the same power. Let each AP of tier i + 1 be deployed to receive the maximumaverage power from φ i . Given that φ i is a PPP, we then derive the conditional PDF f i ( y | x ) . Lemma . In a multihop mmWave AP network, assume x ∈ φ i and y ∈ Φ i +1 to be a pair oftransmitter and receiver at the i + 1 th hop. If φ i is a PPP with intensity λ i , then y is isotropicallydistributed around x with the conditional PDF: f i ( y | x ) = f L ( y | x ) e − πλ i (cid:82) | y − x | α L /α N P N ( r ) r d r + f N ( y | x ) e − πλ i (cid:82) | y − x | α N /α L P L ( r ) r d r , (1)where f L ( y | x ) = 2 π | y − x | λ i P L ( | y − x | ) e − πλ i (cid:82) | y − x | P L ( r ) rdr ,f N ( y | x ) = 2 π | y − x | λ i P N ( | y − x | ) e − πλ i (cid:82) | y − x | P N ( r ) rdr . Here, the constant α L and α N represent the path loss exponent for line-of-sight (LOS) and non-line-of-sight (NLOS) mmWave link, respectively. P L ( r ) refers to the probability with that ammWave link of length r is LOS. It follows the NLOS probability P N ( r ) = 1 − P L ( r ) . Proof.
We start to derive the PDF of the distance from y ∈ Φ i +1 to its nearest LOS AP in φ i . Without loss of generality, let y be the origin of the coordinate system. Denote the disc ofradius z centered at the origin as D zo . Following the network model, the APs in φ i which areLOS to the origin form an inhomogeneous PPP φ iL with density λ iL ( z ) = λ i P L ( z ) . Thus the nullprobability of φ iL in D zo is given by P ( φ iL ∩ D zo = ∅ ) = e − π (cid:82) z xλ iL ( r ) d r . The PDF of z for φ iL can then be derived as f L ( z ) = d (1 − P ( φ iL ∩ D zo = ∅ )) d z = 2 πzλ i P L ( z ) e − πλ i (cid:82) z P L ( r ) rdr . (2)The PDF of z for NLOS APs in φ i i.e. φ iN can be obtained by the same steps, where f N ( z ) = d (1 − P ( φ iN ∩ D zo = ∅ )) d z = 2 πzλ i P N ( z ) e − πλ i (cid:82) z P N ( r ) rdr . (3)Following the Bayes’ rule, we have P ( x ∈ φ iL | | x | = z ) ∝ f ( | x | = z | x ∈ φ iL ) P ( x ∈ φ iL ) and P ( x ∈ φ iN | | x | = z ) = f ( | x | ∝ z | x ∈ φ iN ) P ( x ∈ φ iN ) . The deductions of P ( x ∈ φ iL ) and P ( x ∈ φ iN ) follow the similar steps in [5, Theorem 2]. In [5, Theorem 3], f ( | x | = z | x ∈ φ iL ) and f ( | x | = z | x ∈ φ iN ) are derived as a function of f L ( z ) in (2) and f N ( z ) in (3).Lemma 1 implies that each point y ∈ Φ i +1 can be considered as the isotropic displacementof some point x ∈ φ i , where the distance between y and x follows the PDF f i ( y | x ) in (1). Itfollows from [11] that if φ i is a homogeneous PPP, then Φ i +1 is a PPP with the same intensityof φ i .Consider a mmWave AP network of density Λ A , where all APs are scheduled to transmitto the following tier i.e. φ i = Φ i , ∀ i . By repeatedly using the displacement property of thePPP [11], the topology of mmWave AP network can then be written as { Φ i } Mi =0 , where Φ i isa homogeneous PPP of intensity Λ , ∀ i . It follows that the total number of transmission hops M = Λ A Λ − . Note that (cid:83) Mi =0 Φ i is the superposition of M + 1 homogeneous PPPs, thus is alsoa homogeneous PPP. It implies that the mmWave APs are uniformly distributed in the networkif the spatial multiplexing at APs is disabled by the transmission protocol. -5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1012345 backhaul-connected mmWave APsrelay mmWave APs (a) The total number of transmission hops equal to M = 12 . Thespatial multiplexing gain for each hop k i = 1 . -5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1012345 backhaul-connected mmWave APsrelay mmWave APs (b) The total number of transmission hops equal to M = 2 . Thespatial multiplexing gain for each hop k i = 6 .Fig. 2: The spatial distribution of mmWave APs in the network with spatial multiplexing disabled (left) and enabled (right). B. Topology with Spatial Multiplexing Enabled
By enabling the spatial multiplexing of APs in φ i , the cluster B xi , x ∈ φ i becomes a sequenceof i.i.d. points { y , · · · , y k i } with the PDF f i ( y j | x ) . Following Lemma 1, if φ i is a PPP ofintensity λ i , then y j is isotropically located around x with f i ( y j | x ) in (1). We remark that thelocation distribution of y j only depends on x , which is irrelevant to the size of B xi .For tier i + 1 , the mmWave APs are spatially distributed following the point process Φ i +1 = (cid:83) x ∈ φ i B xi . Given that φ i is a PPP of intensity λ i , we then have that Φ i +1 is a Poisson clusterprocess (PCP), more specifically, a Neyman-Scott process with intensity Λ i +1 = k i λ i [11].Note that Φ i is a collection of clusters B xi − with size k i − . Let φ i be generated by taking onepoint from each cluster B xi − , x ∈ φ i − . It follows from Lemma 1 that if φ i − is a PPP, then φ i is a PPP, where φ i and φ i − are of the same intensity. Note that Φ represents the transmitters athop 1, which is a PPP of intensity Λ . Therefore, each φ i is a PPP with intensity Λ . It followsthat Φ i +1 is a Neyman-Scott point process with intensity k i Λ for all i .In Fig.2b, we illustrate the spatial distribution of mmWave APs with spatial multiplexingenabled. Note that the mmWave AP networks in Fig.2a and Fig.2b are of the same density Λ A . Moreover, tier 0 is assumed to be identical for the two AP networks. However, the spatialmultiplexing is disabled in Fig.2a. It can be observed from Fig.2 that given k i = 1 , ∀ i , themmWave relays are uniformly distributed in the area. By enabling the spatial multiplexing, the locations of mmWave relays become clustering. Such a clustering pattern is consistent with thedistribution of Neyman-Scott point process.IV. P ERFORMANCE OF MM W AVE
AP N
ETWORK
To characterize the impact of spatial multiplexing on the throughput, we first derive the latencyand coverage probability for the mmWave AP network. Note that the latency of mmWave APnetwork indicates the delay of packet transmission. While the reliability of packet transmissioncan be captured by the coverage probability.
A. Network Latency
In a mmWave AP network, tier is always the source tier for a packet, whereas the destinationof the packet can be a mmWave AP in any tier. Note that the delay of a packet depends onthe hop count from the source AP to the destination AP. Accordingly, the worst-case delay ofa packet is equivalent to the maximum hop count of APs in the network. It follows that thelatency of a mmWave AP network can be defined as the total number of transmission hops M in the network. Theorem . For a mmWave AP network of density Λ A , the latency M is bounded by Λ A K Λ − K ≤ M ≤ Λ A Λ − , (4)where Λ is the intensity of tier 0. Each mmWave AP is equipped with K RF chains, thus K represents the maximum spatial multiplexing gain for each transmission hop. Proof.
The network latency M satisfies (cid:80) Mi =1 Λ i = Λ A − Λ . Note that Λ ≤ Λ i ≤ K Λ , theresult immediately follows.Theorem 1 demonstrates that the latency of mmWave AP network decreases linearly withthe increase of spatial multiplexing gain. Note that the network latency reaches its upper boundwhen the spatial multiplexing is disabled. By setting the spatial multiplexing gain to K for eachtransmission hop, the minimum latency of mmWave AP network can be achieved. B. Coverage probability
To calculate the coverage probability for tier i + 1 , we need to first derive the signal-to-interference-noise ratio (SINR) for APs in Φ i +1 . Note that φ i and Φ i +1 represent the transmitters and receivers of the i + 1 th transmission hop, respectively. At hop i + 1 , the hybrid precodingis employed by two stages [4]. In the first stage, the mmWave AP located at x ∈ φ i assigns aunique analog beam to each AP in B xi . Denote θ A as the main lobe width of the analog beam,where the beamforming gains within and outside the main lobe are denoted by G A and g A ,respectively. We use G ( k i ) to denote the transceiver beamforming gain between two APs. Itfollows that G ( k i ) equals to G A , G A g A and g A with probabilities (cid:0) θ A k i π (cid:1) , θ A k i π (cid:0) − θ A k i π (cid:1) and (cid:0) − θ A k i π (cid:1) , respectively.In the second stage of hybrid precoding, the spatial multiplexing is performed in the baseband,where the AP at x ∈ φ i transmits a different data stream for each y ∈ B xi as well as cancels theinter-stream interference. Let a randomly selected AP at y ∈ Φ i +1 be the origin of the coordinatesystem, where y belongs to the cluster B xi . It follows that the coordinate of x is translated to x = x − y . The SINR of the AP at the origin can then be expressed asSINR ( k i ) (cid:44) h G A (cid:96) ( | x | ) σ + I i ( k i ) = h G A (cid:96) ( | x | ) σ + (cid:80) x b ∈ φ i \{ x } h b G b ( k i ) (cid:96) ( | x b | ) , (5)where h b represents the channel fading from x b to the origin; G b ( k i ) is the transceiver beam-forming gain between x b and the origin; σ denotes the noise power; (cid:96) ( · ) denotes the path lossof the mmWave link [4] (cid:96) ( r ) = βr − α L , with probability P L ( r ) βr − α N , with probability P N ( r ) , (6)where β is a constant representing the intercept of path loss model [4]; the LOS probability P L ( r ) and NLOS probability P N ( r ) are introduced in Lemma 1.As discussed in Section III, φ i is formed by taking one point from each cluster of Φ i . Thus, φ i is always a PPP with intensity Λ regardless of the intensity of Φ i . It means that the SINRdistribution in (5) depends only on k i and τ . Based on (5), the coverage probability of tier i + 1 is defined as C ( τ, k i ) (cid:44) P ( SINR ( k i ) > τ ) (7)with that the AP in tier i + 1 has a SINR larger than the threshold τ . To calculate the coverageprobability, we first provide the characteristic function of the interference at the origin. Given that the AP at the origin is connected to a LOS AP located at x , the characteristic function canbe written as L I L ( s, k i ) = exp (cid:18) − π Λ (cid:90) ∞| x | [1 − G L ( s, r, k i )] P L ( r ) r d r (cid:19) × exp (cid:18) − π Λ (cid:90) ∞| x | α L /α N [1 − G N ( s, r, k i )] P N ( r ) r d r (cid:19) , (8)where s is the value on that the characteristic function is evaluated and G L ( s, r, k i ) = E h, G (cid:104) e − sβh G ( k i ) r − α L (cid:105) .If the AP at x is in the NLOS state, the characteristic function of the interference is given as L I N ( s, k i ) = exp (cid:18) − π Λ (cid:90) ∞| x | [1 − G N ( s, r, k i )] P N ( r ) r d r (cid:19) × exp (cid:18) − π Λ (cid:90) ∞| x | α N /α L [1 − G L ( s, r, k i )] P L ( r ) r d r (cid:19) (9)with G N ( s, r, k i ) = E h, G (cid:104) e − sβh G ( k i ) r − α N (cid:105) .Next, we show the main result on the coverage probability of tier i + 1 for a mmWave APnetwork. Theorem . In a mmWave network, the coverage probability of a randomly selected AP intier i + 1 is given by C ( τ, k i ) = (cid:90) r> f L ( r ) f N ( r α L /α N )2 πr α L /α N Λ P N ( r α L /α N ) C L ( τ, r, k i ) d r + (cid:90) r> f L ( r α N /α L ) f N ( r )2 πr α N /α L Λ P L ( r α N /α L ) C N ( τ, r, k i ) d r, (10)where k i is the spatial multiplexing gain at hop i ; f L ( · ) and f N ( · ) are the PDF of distancedistributions given in (1). Assume that channel follows the Rayleigh fading model, then C L ( τ, r, k i ) = exp (cid:18) − r α L τ σ G A β (cid:19) L I L (cid:18) r α L τG A β , k i (cid:19) , C N ( τ, r, k i ) = exp (cid:18) − r α N τ σ G A β (cid:19) L I N (cid:18) r α N τG A β , k i (cid:19) . Proof.
For the AP of Φ i +1 at the origin, the SINR coverage probability can be written as C ( τ, k i ) (cid:44) P ( SINR ( k i ) > τ ) = E r [ C ( τ, k i , r )] , where C ( τ, k i , r ) (cid:44) P ( SINR ( k i ) > τ | | x | = r ) is the conditional SINR coverage probability.According to the LOS AP or NLOS AP at x , C ( τ, k i , r ) can be further expanded as C ( τ, k i , r ) = P ( x ∈ φ iL | | x | = r ) C L ( τ, k i , r ) + P ( x ∈ φ iN | | x | = r ) C N ( τ, k i , r ) , where C L ( τ, k i , r ) (cid:44) P ( SINR > τ | | x | = r, x ∈ φ iL ) = P (cid:18) h > τ ( σ + I L ) G A G U βr − α L (cid:19) and C N ( τ, k i , r ) (cid:44) P ( SINR > τ | | x | = r, x ∈ φ iN ) = P (cid:18) h > τ ( σ + I N ) G A G U βr − α N (cid:19) . The conditional probabilities C L ( τ, k i , r ) and C N ( τ, k i , r ) can be expressed as a function of thecharacteristic functions in (8) and (9) [11]. The details in deriving the characteristic function isshown in [9]. In Lemma 1, we prove the expressions of conditional probabilities P ( x ∈ φ iL | | x | = r ) = f L ( r ) e − πλ i (cid:82) rα L /α N P N ( z ) z d z (11)and P ( x ∈ φ iN | | x | = r ) = f N ( r ) e − πλ i (cid:82) rα N /α L P L ( z ) z d z , (12)where f L ( · ) and f N ( · ) are also given in Lemma 1. The coverage probability C ( τ, k i ) then follows.In Fig.3, we numerically evaluate (10) by showing C ( τ, k i ) with respect to k i . The main lobewidth of the analog beam is set as θ A = 30 ◦ with the main lobe gain G A = 20 dB and side lobegain g A = 0 dB. The noise power is assumed to be negligible. Each mmWave AP is assumed tobe equipped with K = 12 RF chains, which indicates the spatial multiplexing gain k i ≤ . InSection III, we show that φ i , ∀ i , is of the same intensity as tier 0. Here, Λ is represented by theinter-AP distance r = (cid:112) /π Λ [5]. For the path loss model, we use β = 1 , α L = 2 and α N = 4 [6]. It can be observed from Fig.3 that given the SINR threshold τ , the decrease of coverageprobability C ( τ, k i ) is close to linear with the increase of k i . The other observation is that φ i with r = 100 m and r = 200 m results in the similar coverage probability of tier i + 1 . Sincea higher density of transmitters results in higher power of desired signal, but leads to higherstrength of interference. C. Network Throughput
For a mmWave AP network, we define the throughput as the aggregate data rate of all relaytiers i.e. (cid:83) Mi =1 Φ i . Consider the network, where each transmission hop follows the same protocol.It follows that k i = k, ∀ i . Consequently, the coverage probability of each tier is equal to C ( τ, k ) . Spatial Multiplexing Gain k i C o ve r a g e P r ob a b ili t y C ( = , k i ) r =100m, = =0dBr =100m, = =10dBr =100m, = =20dBr =200m, = =0dBr =200m, = =10dBr =200m, = =20dB Fig. 3: The coverage probability C ( τ, k i ) with different spatialmultiplexing gain k i . The intensity Λ is measured by inter-APsdistance r , where r = (cid:113) π Λ . Here, θ A = 30 ◦ , G A = 20 dBand g A = 0 dB. Spatial Multiplexing Gain k N e t w o r k Th r oughpu t ( G bp s ) r =100m, = =0dBr =100m, = =10dBr =100m, = =20dBr =200m, = =0dBr =200m, = =10dBr =200m, = =20dB Fig. 4: Network throughput with different spatial multiplexing gain k . The intensity Λ is denoted by r = (cid:113) π Λ . Here, θ A = 30 ◦ , G A = 20 dB and g A = 0 dB. Assume that each AP in the network is assigned with the same bandwidth. We then derive thenetwork throughput with respect to spatial multiplexing gain.
Theorem . Consider a mmWave AP network of density Λ A , where the distribution of backhaul-connected APs follows a homogeneous PPP with intensity Λ . Assume that each hop followsthe identical transmission protocol with the spatial multiplexing gain k , the network throughputis then given by T ( k ) (cid:44) W (Λ A − Λ ) C ( τ, k ) log (1 + τ ) M = W k Λ C ( τ, k ) log (1 + τ ) , (13)where τ is the SINR threshold; M = (Λ A − Λ ) /k Λ is the network latency; W denotes thebandwidth of each AP; the coverage probability C ( τ, k ) is given in (10). Proof.
By combining Theorem 1 and 2, the result immediately follows.We remark that the throughput of a mmWave AP network T ( k ) is dependent on the intensityof backhaul-connected APs i.e. Λ . However, T ( k ) is independent of Λ A .In Fig.4, we show the network throughput derived in Theorem 3. The parameters used inFig.3 and Fig.4 are the same. For a fixed intensity Λ , Fig.4 demonstrates the trade-off betweenthe latency and coverage probability when choosing the spatial multiplexing gain k . It canbe observed from Fig.4 that the network throughput is not monotonically varying with thespatial multiplexing gain. As the network latency can be improved by increasing k , whereas the degradation of coverage probability is considerable as k increases. Moreover, the optimal spatialmultiplexing gain is different depending on the SINR threshold. When the SINR threshold islow, the latency dominates the network throughput. Consequently, the higher spatial multiplexinggain corresponds to the higher throughput. In the region of the high SINR threshold, the coverageprobability becomes the major challenge of network performance. Therefore, the smaller spatialmultiplexing gain results in the higher throughput for the mmWave AP network. By comparingthe throughputs of networks with different Λ , Fig.4 then illustrates that the ultra-dense mmWaveAP network can benefit from the densification of tier 0.V. C ONCLUSION
We proposed to incorporate the multihop transmission protocol in modeling the ultra-densemmWave AP network. Moreover, we exploited the spatial distributions of mmWave APs withdifferent transmission protocols. Our analysis indicated that the mmWave AP network can bemodeled by a PPP if spatial multiplexing is disabled. However, the topology of mmWave APnetwork is a collection of Neyman-Scott processes when the spatial multiplexing is enabled.We then analyzed the performance for mmWave AP networks with different topologies. Weshowed that the uniform distribution of mmWave APs experiences the largest latency but hasthe highest coverage probability. Moreover, the latency of mmWave AP network decreases asthe spatial multiplexing gain increases, while the coverage probability drops with the increaseof spatial multiplexing gain. The numerical results showed the optimal spatial multiplexing gainto maximize the throughput of the ultra-dense mmWave AP network.R
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