Stochastic Geometry-Based Modeling and Analysis of Beam Management in 5G
Sanket S. Kalamkar, Fuad M. Abinader Jr., François Baccelli, Andrea S. Marcano Fani, and Luis G. Uzeda Garcia
aa r X i v : . [ c s . I T ] J un Stochastic Geometry-Based Modeling and Analysisof Beam Management in 5G
Sanket S. Kalamkar, Fuad M. Abinader Jr., Franc¸ois Baccelli, Andrea S. Marcano Fani, and Luis G. Uzeda Garcia Affiliation: INRIA-ENS, Paris, France, Nokia Bell Labs, Paris, France
Abstract —Beam management is central in the operation ofdense 5G cellular networks. Focusing the energy radiated tomobile terminals (MTs) by increasing the number of beams percell increases signal power and decreases interference, and hashence the potential to bring major improvements on area spectralefficiency (ASE). This benefit, however, comes with unavoidableoverheads that increase with the number of beams and the MTspeed. This paper proposes a first system-level stochastic geome-try model encompassing major aspects of the beam managementproblem: frequencies, antennas, and propagation; physical layer,wireless links, and coding; network geometry, interference, andresource sharing; sensing, signaling, and mobility management.This model leads to a simple analytical expression for the effectiveASE that the typical user gets in this context. This in turn allowsone to find, for a wide variety of 5G network scenarios includingmillimeter wave (mmWave) and sub-6 GHz, the number of beamsper cell that offers the best global trade-off between these benefitsand costs. We finally provide numerical results that discussthe effects of different systemic trade-offs and performances ofmmWave and sub-6 GHz 5G deployments.
I. I
NTRODUCTION
A. Motivation
The ever-increasing demand in capacity for mobile commu-nications makes it necessary to consider new implementationapproaches/techniques that can significantly boost data ratesand the area spectral efficiency (ASE) of mobile networks.One key enabler considered in 5G [1] to face this demand isthe use of the spectrum beyond the sub-6 GHz frequencies,known as millimeter wave (mmWave). More specifically, 5Grelies on the spectrum above 20 GHz, where bandwidth ofup to 400 MHz can be used to offer very high data rates(above 10 Gbps peak rates) and increase the network capacity;nevertheless, the sub-6 GHz bands, with up to 100 MHz ofbandwidth, are still needed to ensure wide area coverage anddata rates up to a few Gbps.One of the key difficulties that mmWave frequencies face istheir challenging propagation characteristics: they are subjectto high path loss, penetration loss, and diffraction loss dueto their millimetric wavelength, thus reaching short distancestypically within a few hundred meters. But what was onceconsidered a limitation makes nowadays mmWave a suitablecandidate for small cells deployments, which can be used fornetwork densification and capacity boosting.To overcome the propagation challenges at mmWave fre-quencies, steerable arrays with a large number of antenna
E-mail: [email protected], [email protected] elements are used to create highly directional beams thatconcentrate the transmitted energy to achieve high gains andmake the signal more robust to increase coverage. Moreover,mmWave communications must be designed to operate undermobility conditions, covering users in LOS (Line-of-Sight)and NLOS (Non-LOS) at pedestrian and vehicular speeds.This can become quite challenging since mmWave frequenciesare highly sensitive to changes in the environment. Thus,any mmWave-based system relies on beam management tech-niques to select the best beam during a base station (inter-cell)handover and quickly switch/reselect a new beam during intra-cell mobility to avoid beam misalignment and performancelosses. Although more important for mmWave frequencies,beamforming techniques with highly directional narrow beamscan also be used in sub-6 GHz frequencies to enhance thenetwork performance.Nevertheless, beam-based communications come with im-plementation challenges, beam management procedures suchas beam refinement and beam failure detection and recoverythat introduce different signaling and delay overheads. Also,an efficient beam management relies on capturing systemictrade-offs that depend on the mobile terminal (MT) mobility,cell sizes, and the number/width of the beams. For instance,a large number of narrow beams improves the signal-to-interference-plus-noise ratio (SINR), but it also leads to morefrequent service interruption due to beam switching and beammisalignment, which degrade the network performance. Givensuch a complex nature of beam management, the system-levelsimulations, although crucial, are often time-consuming andexpensive. Hence, to complement system-level simulations, inthis paper, we aim to provide a tractable mathematical frame-work that permits a system-level analysis of beam managementin 5G.
B. Contributions
1) We provide a first mathematical optimization frameworkfor the beam management problem in 5G. Using thewell-established tools from stochastic geometry [2], [3],our proposed optimization yields the number of beamsthat maximizes the area spectral efficiency.2) The optimization permits a system-level analysis in 5G.In particular, it takes into account the average speedof the MTs, mobility-induced beam-misalignment error,beam selections during base station handovers and beamreselections within a cell, and time overheads associatedith these beam (re)selections. For instance, given anaverage speed of the mix of MTs (pedestrians, bikes, andcars), the mix of geometries (some cells are bigger, othersmaller, with the MTs either close or far from the servingbase station), the mix for fading and blockages presentin the network, our optimization gives the number ofbeams that provides the best “sum rate” for MTs of alltypes in a large ball.3) While the proposed framework is generic and coversa variety of related problems, e.g., other beam-centricsystems, we give numerical results to evaluate a 5G NewRadio (NR)-compliant radio access network operatingin a dense urban macro/picocell scenario, for both sub-6 GHz and mmWave frequencies. The results reveal thekey inter-dependencies between network parameters andprovide insights into the associated trade-offs.
C. Related Work
The study of user handovers in cellular networks and theireffect on various performance metrics such as throughput is arich area (see e.g., [4]–[6] and the references therein). Mostof the works have focused on only base station handovers(or, equivalently, cell handovers) in cellular networks [7]–[12]. The work in [13] is probably the closest one to ourwork. This work studies both initial beam selections duringbase station handovers and beam reselections within a cell andtheir associated overheads in a beam-centric mmWave cellularnetwork. In particular, authors in [13] have obtained analyticalexpressions of inter-cell and inter-beam handover rates, basedon which, the average spectral efficiency is calculated subjectto overheads due to handovers. But, the work in [13] considersonly noise-limited scenario and ignores interference fromother base stations, while our work considers both noise andinterference in the analysis. Second, our work also studiesthe effect of handovers on optimizing the number of beams,while [13] assumes a fixed number of beams, i.e., no beammanagement. Third, unlike [13], we consider blockages andbeam misalignment due to mobility. Overall, there is a lackof understanding on how the MT mobility affects the beammanagement in the presence of interference, blockages, andbeam misalignment error.II. S
YSTEM M ODEL
A. Network Model
As shown in Fig. 1, we consider a downlink cellularnetwork, where the base station (BS) locations are modeledas a homogeneous Poisson point process (PPP) Φ ⊂ R withintensity λ . We assume that the omnidirectional MT moveson a randomly oriented straight line with speed v . Withoutloss of generality, thanks to the isotropy and the stationarityof the PPP [2], this line of MT motion can be considered to bealong the X -axis and passing through the origin. We assumethat each BS always has an MT to serve. Also, a BS servesone MT at a time per resource block. The MT associates itselfwith the nearest BS. Such an association results in BS cellsforming a Poisson Voronoi tessellation as shown in Fig. 1. Let Fig. 1.
A snapshot of a Poisson cellular network with directionalbeamforming to an MT. In this case, each BS has = 8 beams,i.e., n = 3 . △ : Base station (BS), red square : MT location, brownfilled circle : Beam reselection location, yellow filled circle : BShandover location, solid black lines : Cell boundaries, dashed lines :Beam boundaries, and dotted line : Path of the MT. X ∈ Φ denote the location of the serving BS of the MT ata given time. Without loss of generality, we can focus on theMT that is located at the origin at that time. After averagingover the PPP, this MT becomes the typical MT.
B. Beamforming Model
A BS at location X ∈ Φ uses directional beamforming tocommunicate with the typical MT. As shown in Fig. 1, weapproximate the actual antenna pattern by a sectorized one,where each sector corresponds to one beam of the BS. Forsimplicity, we assume that each BS has n beams with n ∈ N , which corresponds to n − beam boundaries. Hence, thebeamwidth is ϕ n = 2 π n = π n − . (1)We focus on a simple antenna pattern model where the mainlobe is restricted to the beamwidth. Both the main and sidelobe gains depend on the number n of beams. In particular,the antenna gain G n is expressed as G n ( ψ ) = ( G m ,n if | ψ | ≤ ϕ n / G s ,n otherwise , (2)where G m ,n and G s ,n denote the antenna gains within themain lobe and the side lobe, respectively, as a function of thenumber n of beams, and ψ is the angle off the boresightdirection. Here, a beam boundary is a line segment that connects two points on thecell boundary and passes through the location of the BS, as shown in Fig. 1. . Mobility-induced Beam Misalignment
In 5G NR, the beam (re)selection occurs during a synchro-nization signal burst (SSB) [14], which is done periodicallywith period τ . During an SSB, the beam that has the MTwithin its beamwidth is selected for communication. We callthis beam the reference beam . Due to the mobility of the MT,it is possible that the MT has moved out of the beamwidthof the reference beam without selecting the new beam, andis still connected to the reference beam until the next SSBwhen the new reference beam is selected. Such a mobility-induced beam misalignment reduces the signal strength atthe MT as it then lies within the side lobe of the referencebeam chosen during a previous SSB. At a given time, thebeam misalignment probability p bm depends on the speed v of the MT, the duration τ between two SSBs, and the averagedistance between two beam reselections. Here, we propose asimple yet effective formula that captures the effects of theseparameters on the probability of beam misalignment due toMT mobility in a snapshot of the network. We evaluate theprobability p bm that the MT is outside the beamwidth of thereference beam by the: p bm = 1 − exp (cid:18) − vτ /µ s , b (cid:19) , (3)where /µ s , b is the average distance between two consecutivebeam reselections, with µ s , b being the linear intensity of beamreselection given in Theorem 2. We can interpret (3) as − p bm = P ( T > τ ) , where T is an exponential random variable with mean vµ s , b ,with T interpreted as the time between two consecutive SSBs.Note that vµ s , b is the average time between two consecutivebeam reselections. Hence, the average time between two SSBsis equal to the average time between two consecutive beamreselections.Based on the probability of the beam misalignment, theantenna gain G of the serving BS is given as G = ( G m ,n w . p . − p bm G s ,n w . p . p bm . (4) D. Blockage and Channel Model
The presence of obstacles leads to LOS and NLOS propaga-tion between the typical MT and a BS. We adopt a LOS ballmodel [15] to capture the effect of blockages. In particular,the propagation between a BS and the typical MT separatedby distance d is LOS if d < R c where R c is the maximumdistance for LOS propagation. The LOS and NLOS channelconditions induced by the blockage effect are characterizedby different path loss exponents, denoted by α L and α N ,respectively. As shown in [16], typical values of these pathloss exponents are α L ∈ [1 . , . and α N ∈ [2 . , . .The channel follows Rayleigh fading with unit mean powergain. Let h X denote the channel power gain from the BS at location X ∈ Φ to the MT. Note that, the random variables h X are i.i.d. exponential with unit mean, i.e., h X ∼ exp(1) .Let | X | denote the distance between a BS at X ∈ Φ and thetypical MT located at the origin. We consider the standardpower-law path loss model with path loss function as l ( X ) = ( K | X | − α L if | X | < R c K | X | − α N if | X | ≥ R c , (5)where K = (cid:16) c πf c (cid:17) is a frequency-dependent constant with c being the speed of light and f c the carrier frequency. Weassume that the typical MT can receive from its serving BS(also the nearest BS) in both LOS and NLOS conditions. E. Signal-to-Interference-Noise Ratio (SINR)
When the typical MT is associated with the BS located at X ∈ Φ with antenna gain G as in (4), the SINR at thetypical MT is given by SINR n = P G h X l ( | X | ) σ + I n , (6)where P is the transmit power of a BS. Also, σ = W N isthe noise power, where W and N are the bandwidth and thenoise spectral density, respectively. In the denominator of (6), I n is the interference power at the typical MT given by I n = X X ∈ Φ \{ X } P G n h X l ( | X | ) . (7)Since the beams of all BSs are oriented towards their respec-tive MTs, the direction of arrivals between interfering BSs andthe typical MT is distributed uniformly in [ − π, π ] . Thus, thegain G n of an interfering BS is equal to G m ,n with probability ϕ n / π and G s ,n with probability − ϕ n / π , where ϕ n is thebeamwidth given by (1).III. E RGODIC S HANNON R ATE
We are interested in calculating the downlink ergodic Shan-non rate at the typical MT, which is given as R n = W E [log (1 + SINR n )] , (8)where E ( · ) denotes expectation. Definition 1 (Success Probability):
The success probability p s of the typical MT is the probability that the SINR at thetypical MT exceeds a predefined threshold. Mathematically, p s ( n, β ) , P ( SINR n > β ) , (9)where β > is the predefined SINR threshold, which alsoparametrizes the transmission rate. Lemma 1:
Let F ( α S , α I , w ) , πλ − p I,n βr α S G m ,n G w α I − − p I,n βr α S G s ,n G w α I ! w, Although the LOS channels are better modeled by Nakagami- m fading,Rayleigh fading allows us much better analytical tractability. The assumption of the nearest-BS association irrespective of LOS/NLOSconditions of the nearest BS is due to analytical tractability. here p I,n = ϕ n / π . The success probability p s ( n, β ) is p s ( n, β ) = (1 − p bm ) q s ( n, β, G m ,n ) + p bm q s ( n, β, G s ,n ) , (10)where p bm is given by (3) and q s ( n, β, G ) = Z R c f R ( r ) exp (cid:18) − βr α L σ P KG (cid:19) × exp (cid:18) − Z R c r F ( α L , α L , w )d w − Z ∞ R c F ( α L , α N , w )d w (cid:19) d r + Z ∞ R c exp (cid:18) − βr α N σ P KG − Z ∞ r F ( α N , α N , w )d w (cid:19) f R ( r )d r (11) with f R ( r ) = 2 πλre − λπr where R is the distance to thenearest BS. Proof:
The proof is given in Appendix A.
Theorem 1:
Imposing the limit Q max on the maximumachievable SINR stemming from RF imperfections and mod-ulation schemes, the ergodic Shannon rate per unit time is R n = W Z Q max p s ( n, z ) z + 1 d z, (12)where p s ( n, z ) is given by (11). Proof:
The proof follows directly from R n = W E [log(1 + min( SINR n , Q max ))] . IV. B
EAM (R E )S ELECTION
A. Beam Selection during BS handovers
When the MT performs a BS handover, it selects a beamwith the new BS. We know from [17] that, for the Poisson-Voronoi tessellation, the linear intensity of cell boundarycrossings, i.e., BS handovers, is µ s , c = √ λπ . Hence, the timeintensity of BS handover (or, equivalently, beam selection) is µ c = 4 √ λπ v. (13) B. Beam Reselection
The beam reselection occurs at a beam boundary within theVoronoi cell of a BS. In Fig. 1, the locations of beam rese-lections are denoted by brown filled circles. We are interestedin calculating the average number of beam reselections thetypical MT performs per unit length. We also calculate thetime intensity of beam reselection, i.e., the average rate ofbeam reselections.
Theorem 2 (Intensity of beam reselection):
For n beamsand PPP of intensity λ , the linear intensity µ s , b of beamreselection for the typical MT moving on a straight line withspeed v is n √ λπ . The time intensity µ t , b of beam reselectionis µ t , b ( n ) = n √ λπ v . Proof:
The proof is given in Appendix B.
Remark 1:
When taken into account the average number ofbeam reselections that are skipped between two consecutiveSSBs, the effective time intensity of beam reselection becomes µ b ( n ) = 1max (cid:16) τ, µ t , b ( n ) (cid:17) , (14) TABLE IV
ALUES OF NETWORK PARAMETERS [20]Parameter FR1 FR2Carrier frequency ( f c ) . Bandwidth ( W )
100 MHz 400 MHz
Noise density ( N ) −
174 dBm / Hz −
174 dBm / Hz Transmit power ( P )
43 dBm 36 dBm
Beam reselection overhead ( T b )
23 ms 23 ms
Cell handover overhead ( T c )
43 ms 43 ms
SSB periodicity ( τ )
20 ms 20 ms
MT speed ( v ) [3 , , / h [3 ,
30] km / h Inter-site distance (ISD) [250 , , , , Maximum SINR ( Q max )
30 dB 30 dB
Path loss exponent ( α ) 3.5 α L = 1 . , α N = 3 . Blockage model Implicit (NLOS) LOS ballLOS ball radius ( R c ) −
75 m where τ is the SSB periodicity.V. T IME O VERHEAD DUE TO B EAM (R E )S ELECTION
BS handovers and beam reselections may result in signifi-cant overheads in terms of time as a consequence of the timespent in beam sweeping and alignment, respectively. Such anoverhead reduces the time available for data transmissions, inturn reducing the ergodic Shannon rate.The typical MT moves on a straight line across differentbeams within the Voronoi cell of a BS as well as acrossVoronoi cells of different BSs. Hence, the two componentsthat contribute to the time overhead are:1) The time T c for beam sweeping after each BS handover,i.e., cell boundary crossing, which includes the periodicSSB measurement, receiver processing time for theSSBs, and the handover interruption time due to cellswitch and radio resource control (RRC) reconfiguration.2) The time T b for beam alignment after each beam reselec-tion within the Voronoi cell of a BS, which includes theperiodic SSB measurement and the receiver processingtime for the SSBs.Thus, the total average overhead per unit time is T o ( n ) = µ b ( n ) T b + µ c T c . (15)The effective area spectral efficiency (ASE) per unit time is R eff ( n ) = λ R n (1 − T o ( n )) + , (16)where R n is given by (12) and ( A ) + = max(0 , A ) .Our objective is to find the integer n that maximizes theeffective ASE per unit time, i.e., n ∗ = arg max n ∈ N R eff ( n ) . (17)The value of n ∗ can easily be found by a linear search.VI. N UMERICAL R ESULTS AND D ISCUSSIONS
To validate our proposed model, we consider a G NR-compliant radio access network operating in a dense urbanmacro/pico cell scenario. A summary of the model parametersfor FR1 (sub- ) and FR2 (above ) networkdeployments is provided in Table I.
Fig. 2.
FR1 : Effect of average inter-site distance (ISD) on the effectiveASE for different n . Dashed line: v = 3 km / h , Dashed-dotted line: v = 30 km / h , and Solid line: v = 120 km / h . To explore the best potential of G NR networks in op-erational bands FR1 and FR2, we use the maximum band-width allowed as per G NR Release 15 [18], [19], namely
100 MHz for FR1 and
400 MHz for FR2. As the inter-site (or,equivalently inter-base station) distances (ISDs) for FR1 areexpected to be larger due to lower frequency-dependent pathloss, we choose a transmission power P =
43 dBm . For FR2,we can expect smaller ISDs due to (a) higher attenuation lossat higher carrier frequencies, and (b) because massive MIMOand high beamforming gains imposes challenges in terms ofRF exposure and EMF limitations. Thus we decrease P to
36 dBm .One important network planning decision is related to thenumber of beams per BS. In our proposed model, this isrelated to the choice of the values for n . The main and sidelobe gains depend on the number n of beams. In particular,with the increase in the number of beams, the main lobe gainincreases while the side lobe gain decreases. Thus, withoutloss of generality, we assume the following antenna gains.The main lobe gain is G m ,n = 2 n , while the side lobe gain is G s ,n = n . Another important parameter is the intensity λ of BSs,which is directly related to the average cell size and the ISD.For an intensity λ of BSs, the average cell size is /λ [2].The average cell radius in a network is defined as the radius r cell of a ball having the same average area as the cell: /λ .Then, the average ISD is r cell = 2 / ( √ πλ ) . Hence we simplyadjust the value of the intensity λ of BSs to represent differentISD scenarios. For FR1 bands, we analyze the following ISDs: ,
500 m , and
250 m . As for FR2 bands, the effect ofpropagation attenuation requires us to decrease the ISD. Hencewe consider the following ISDs:
250 m ,
125 m , and
75 m . As seen from previous sections, our model is amenable to differentmodeling of antenna gain patterns without loss of insights obtained in thispaper.
Fig. 3.
FR2 : Effect of average inter-site distance (ISD) on the effectiveASE for different n . Dashed line: v = 3 km / h and Solid line: v =30 km / h . In Figs. 2, 3, and 4, we evaluate the performances of FR1and FR2 deployments, for different MT speeds v and ISDs.The following discussions are valid for Figs. 2, 3, and 4.In both FR1 and FR2 deployments, for a given ISD, as n (and hence the number n of beams) increases, the beam-forming gain increases and the interference from interferingBSs decreases due to narrower beams. But, at the same time,for a given v , the typical MT performs more frequent beamreselections due to a smaller beamwidth, increasing the timeoverhead. The negative impact of the increased time overheadgradually dominates the beamforming gain. Also, with anincrease in n , the probability of beam misalignment (givenby (3)) increases as the MT is more likely to move into thebeamwidth of another beam between two consecutive SSBmeasurements. As a result of this tussle, we observe that thereis a value n ∗ of n that maximizes the effective ASE.Similarly, with an increase in v , the MT crosses beam andcell boundaries more frequently resulting in a higher numberof beam reselections and BS handovers, respectively. Also, theprobability of beam misalignment increases. Hence, with anincrease in v , the optimal value n ∗ decreases, and for a fixed n , the effective ASE decreases.Note that a smaller ISD means a denser cellular network.As the ISD increases, the average cell size increases. Thus,BSs need to increase the number n of beams for a higherbeamforming gain (equivalently, a smaller beamwidth) andsmaller interference. Hence, the value of n ∗ increases withISD. For a smaller n , the time overhead associated with beamreselections is small. Hence, a smaller ISD results in a highereffective ASE due to a smaller distance between the typicalMT and its serving BS. This dominates the negative effectsof increased interference power due to a denser deploymentof BSs and increased time overhead due to more frequent BShandovers. But, for a large n , both beam reselections and BShandovers happen more frequently for a smaller ISD, and the Fig. 4.
Comparison between FR1 and FR2 deployments for differentISDs with v = 30 km / h . network with a larger ISD achieves a higher effective ASE.When comparing beamwidths for FR1 with those for FR2given the same n value, we can expect that, even though theangular beamwidth for a given n is the same, the linear widthof the beams will be larger in FR1 than that in FR2 due to a usually larger cell radius in the former. Thus, it is expectedthat MTs operating in FR1 are less penalized due to overheadsassociated with beam reselections and BS handovers. Also, dueto a larger ISD, the distance-based path loss when operatingin FR1 is higher than when operating in FR2. Hence for thesame value of n , we can expect that the interference in FR1is smaller than that in FR2 despite a bit higher transmit powerin FR1. This indicates that the additional received interferencefrom a higher number of beams will not be as intense in FR2as in FR1. As shown in Fig. 4, this behavior is captured byour model, where the optimal value n ∗ for the ISD = 250 m in FR1 ( n ∗ = 8 ) is higher than that in FR2 ( n ∗ = 6 ) with ISD = 75 m or in FR2 with ISD = 125 m ( n ∗ = 7 ).Also, as Fig. 4 shows, when we compare the performancesof FR1 and FR2 deployments for the same ISD, we observean interesting trade-off. Smaller ISDs (e.g.,
75 m ) benefitFR2 irrespective of the value of n as, for a given criticalLOS distance R c , it is more likely that the serving BS hasLOS propagation to the MT (so smaller path loss) boostingsignal power. Recall that, in FR1, even the serving BS alwayshas NLOS propagation to the MT. As a result, the impactof increased time overheads with n on the effective ASEis less in FR2 than that in FR1. As the ISD increases, theprobability of serving BS lying outside the LOS ball increases,in turn, reducing signal power significantly in FR2. Thus,when combined with higher path loss at higher frequenciesin FR2, for higher values of n , the impact of time overheadsis relatively higher in FR2 than FR1. Note that, as a result of Although nothing prohibits small ISDs for FR1 (e.g., Wi-Fi operation in . and ), pico/femto cell deployment models are the best fit formmWave deployments. the tussle between competing effects, the values of n ∗ in FR1and FR2 are close to each other for the same ISD.Overall, the numerical results discussed in this sectionvalidate the claim that the proposed model fits as a toolfor system-level evaluation of network planning decisions inbeam-based access networks such as G NR.VII. C
ONCLUSIONS AND F UTURE D IRECTIONS
We presented a system-level stochastic geometry model fora G NR radio access network that allows one to captureessential scenario characteristics (e.g., operational frequency,blockage characteristics), technological features (e.g., beam-forming configuration, delay overheads), and network deploy-ment choices (e.g., ISD, carrier bandwidth). We analyzed theapplication of the model for a dense urban macro/pico cellscenario for FR1 and FR2. We demonstrated that the modelaccurately captures the existing trade-offs between ISDs, pathloss, interference, and signaling overhead due to beam man-agement. This also shows that the stochastic geometry modelcan be effective for conducting system-level analysis of beam-based radio access networks such as G NR.In the future, we will extend our model to capture system-level impact of multiple MT panels and/or multiple MTbeams. We also aim to improve the model to incorporatesectorized BSs with antenna panels and their effects on beamshaping/gains, MT blockage models, and others. Finally, weintend to extend our model to capture other trade-offs involvedin designing a beam-based radio access network. For instance,with an increase in the number of beams, the average beamtime-of-stay (ToS), i.e., the time a user remains with a givenbeam before switching to another beam, decreases, whichimpacts the available time for performing channel estimation,link adaptation, and power control. Also, increasing the peri-odicity of monitored resources may improve the effectivenessand response time of beam refinement, but it results in reducedspectral efficiency due to the control overhead.R
EFERENCES[1] H. Holma, A. Toskala, and T. Nakamura, 5G Technology: 3GPP NewRadio. John Wiley & Sons Ltd., 2020.[2] F. Baccelli and B. Błaszczyszyn, Stochastic Geometry and WirelessNetworks, NoW Publishers, 2009.[3] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge,U.K.: Cambridge University Press, 2012.[4] M. Polese, M. Giordani, M. Mezzavilla, S. Rangan and M. Zorzi,“Improved handover through dual connectivity in 5G mmWavemobile networks,” in IEEE Journal on Selected Areas in Commu-nications, vol. 35, no. 9, pp. 2069-2084, Sept. 2017.[5] M. Tayyab, X. Gelabert and R. J¨antti, “A survey on handovermanagement: From LTE to NR,” in IEEE Access, vol. 7, 2019.[6] H. Tabassum, M. Salehi and E. Hossain, “Fundamentals of mobility-aware performance characterization of cellular networks: A tutorial,”in IEEE Communications Surveys & Tutorials, vol. 21, no. 3, pp.2288-2308, third quarter 2019.[7] K. Tokuyama and N. Miyoshi, “Data rate and handoff rate analysis foruser mobility in cellular networks,” IEEE Wireless Communicationsand Networking Conference (WCNC), Barcelona, 2018, pp. 1-6.[8] A. Chattopadhyay, B. Błaszczyszyn and E. Altman, “Two-tier cellularnetworks for throughput maximization of static and mobile users,” inIEEE Transactions on Wireless Communications, vol. 18, no. 2, pp.997-1010, Feb. 2019.9] E. Demarchou, C. Psomas and I. Krikidis, “Mobility managementin ultra-dense networks: Handover skipping techniques,” in IEEEAccess, vol. 6, pp. 11921-11930, 2018.[10] R. Arshad, H. ElSawy, S. Sorour, T. Y. Al-Naffouri and M. Alouini,“Handover management in dense cellular networks: A stochasticgeometry approach,” IEEE International Conference on Communi-cations (ICC), Kuala Lumpur, 2016, pp. 1-7.[11] R. Arshad, H. ElSawy, S. Sorour, T. Y. Al-Naffouri and M. Alouini,”Velocity-aware handover management in two-tier cellular networks,”in IEEE Transactions on Wireless Communications, vol. 16, no. 3,pp. 1851-1867, Mar. 2017.[12] Y. Teng, A. Liu and V. K. N. Lau, “Stochastic geometry based han-dover probability analysis in dense cellular networks,” InternationalConference on Wireless Communications and Signal Processing(WCSP), Hangzhou, 2018, pp. 1-7.[13] Z. Li and W. Wang, “Handover performance in dense mmWave cellu-lar networks,” International Conference on Wireless Communicationsand Signal Processing (WCSP), Hangzhou, 2018, pp. 1-7.[14] M. Giordani, M. Polese, A. Roy, D. Castor and M. Zorzi, “A tutorialon beam management for 3GPP NR at mmWave frequencies,” inIEEE Communications Surveys and Tutorials, vol. 21, no. 1, pp. 173-196, First quarter 2019.[15] T. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” in IEEE Transactions on Wireless Commu-nications, vol. 14, no. 2, pp. 1100-1114, Feb. 2015.[16] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun,“Wideband millimeter-wave propagation measurements and channelmodels for future wireless communication system design,” IEEETransactions on Communications, vol. 63, no. 9, pp. 3029-3056, Sep.2015.[17] F. Baccelli and S. Zuyev, “Stochastic geometry models of mobilecommunication networks,” Frontiers in Queueing, CRC Press, pp.227-243, 1997.[18] 3GPP, “NR; User Equipment (UE) radio transmission and reception;Part 1: Range 1 Standalone,” 3rd Generation Partnership Project(3GPP), TS 38.101-1, Version 15.9.0, April 2020.[19] 3GPP, “NR; User Equipment (UE) radio transmission and reception;Part 2: Range 2 Standalone,” 3rd Generation Partnership Project(3GPP), TS 38.101-2, Version 15.9.1, April 2020.[20] https://portal.3gpp.org/desktopmodules/Specifications/SpecificationDetails.aspx?specificationId=319. A PPENDIX AP ROOF OF L EMMA p s ( n, β ) , P ( SINR n > β ) . (18)Depending on whether the serving BS lies within the LOS ballof radius R c or not, we can write (18) as p s ( n, β ) = Z R c P ( SINR n > β | r ) f R ( r )d r + Z ∞ R c P ( SINR n > β | r ) f R ( r )d r, (19)where f R ( r ) = 2 πλr exp( − λπr ) is the probability densityfunction of the distance R of the typical MT to the nearestBS. When < r < R c , we have P ( SINR n > β | r ) = P (cid:18) P KG h X r − α L N + I n > β | r (cid:19) = P (cid:18) h X > βr α L ( N + I n ) P KG | r (cid:19) = e − βrα L N PKG E (cid:20) exp (cid:18) − βr α L I n P KG (cid:19)(cid:21) . (20) We have E (cid:20) exp (cid:18) − βr α L I n P KG (cid:19)(cid:21) = E " exp − βr α L P X ∈ Φ \{ X } P G n h X l ( | X | ) P KG ! (a) = E Y X ∈ Φ \{ X } E (cid:18) exp (cid:18) − βr α L G n h X l ( | X | ) KG (cid:19)(cid:19) (b) = E Y X ∈ Φ \{ X } E
11 + βr α L l ( | X | ) G n KG ! (c) = E Y X ∈ Φ \{ X } p I,n βr α L l ( | X | ) G m , n KG + 1 − p I,n βr α L l ( | X | ) G s , n KG ! , where (a) follows from the i.i.d. nature of fading randomvariables, (b) follows from averaging over the channel powergains on interfering channels, and (c) follows from the factthat the interference from an interfering BS at X ∈ Φ comesfrom its main lobe with probability p I,n and from its side lobewith probability − p I,n with p I,n = ϕ n / π the probabilitythat the typical MT lies within the beamwidth of the mainlobe of an interfering BS.From the probability generating functional (PGFL) of PPP,it follows that E (cid:20) exp (cid:18) − βr α L I n P KG (cid:19)(cid:21) = exp − πλ Z ∞ r − p I,n βr α L l ( w ) G m ,n KG − − p I,n βr α L l ( w ) G s ,n KG ! w d w ! = exp (cid:18) − Z R c r F ( α L , α L , w )d w − Z ∞ R c F ( α L , α N , w )d w (cid:19) , (21) where F ( α S , α I , w ) , πλ − p I,n βr α S G m ,n G w α I − − p I,n βr α S G s ,n G w α I ! w. Similarly, we can obtain an expression for P ( SINR n > β | r ) when R c ≤ r < ∞ . A PPENDIX BP ROOF OF T HEOREM θ denote the angle ofa beam boundary with respect to the direction of the motionof the MT. This angle θ is distributed uniformly at randomin [0 , π ] . In Fig. 5, a ‘brown filled circle’ denotes a pointalong the X -axis where the typical MT reselects the beam. Let Ψ ⊂ R denote the point process of beam reselection events,where the points of Ψ are indicated by ‘brown filled circles.’We are interested in calculating the average number of beamreselections the typical MT performs per unit length, which isequivalent to calculating the intensity of Ψ . -axisY-axis Fig. 5. At most one beam reselection for a BS. △ : Base station (BS), brownfilled circle : Beam reselection location, dashed lines : Beam boundaries, anddotted line : Path of MT along the X -axis. Without loss of generality, let us consider the motion of thetypical MT in the interval [0 , ℓ ] , which corresponds to the mo-tion of the typical MT from (0 , to ( ℓ, along the X -axis.First, we calculate the average number of beam reselectionswhen there is at most one beam reselection corresponding toa BS. This happens when there are two beams of the samesize for a BS. Using the result for two beams, we can obtainthe intensity of beam reselections corresponding n beams ofa BS.During the interval [0 , ℓ ] , the typical MT may move throughthe Voronoi cells of multiple BSs. As shown in Fig. 6, let ω ( X ) denote the point of the beam reselection correspondingto the BS located at X ∈ Φ . The event of beam reselectioncorresponding to a BS at X ∈ Φ occurs when the followingtwo events occur simultaneously:1) the point of the beam reselection lies in the Voronoi cellof the BS at X ∈ Φ , i.e., ω ( X ) ∈ V Φ ( X ) where V Φ ( X ) is the Voronoi cell of the BS located at X ∈ Φ .2) the point of the beam reselection lies on the line con-necting (0 , and ( ℓ, , i.e, ω ( X ) ∈ [0 , ℓ ] .Consequently, conditioning on θ , the average of number ofbeam reselections in [0 , ℓ ] is E (Ψ[0 , ℓ ] | θ ) = E X X ∈ Φ ω ( X ) ∈ V Φ ( X ) ω ( X ) ∈ [0 ,ℓ ] ! , where is the indicator function. Conditioning on the fact thatthere is a BS at location z ∈ R and using the Campbell’stheorem [2], it follows that E (Ψ[0 , ℓ ] | θ ) = λ Z R P z ( ω ( z ) ∈ V Φ ( z )) ω ( z ) ∈ [0 ,ℓ ] d z, (22)where P z ( · ) denotes the Palm probability. Event ω ( z ) ∈ V Φ ( z ) : This event occurs when there is noBS closer to the typical MT than the one at location z . This isequivalent to the event that there is no BS in the ball of radius Y-axis X-axis
Fig. 6. Strip S (shaded area) between two blue solid lines with angle θ with the X -axis. △ : BS, brown filled circle : Beam reselection location, anddashed line : Beam boundary. k z − ω ( z ) k centered at ω ( z ) . Since the BS point process is aPPP with intensity λ , we have P z ( ω ( z ) ∈ V Φ ( z )) = exp (cid:0) − λπ k z − ω ( z ) k (cid:1) = exp (cid:18) − λπy sin θ (cid:19) , (23)where k z − ω ( z ) k = y sin θ since z = ( x, y ) . Event ω ( z ) ∈ [0 , ℓ ] : This event occurs if the BS at z lieswithin the strip S between two lines passing through the origin (0 , and ( ℓ, at angle θ as shown in Fig. 6. Equivalently,by representing z in Euclidean coordinates as z = ( x, y ) , itfollows that ω ( z ) ∈ [0 ,ℓ ] d z = z ∈S d z = ( x,y ) ∈S d x d y. (24)The left line passing through the origin at an angle θ can begiven as y = − x tan θ , while the right line passing through ( ℓ, at an angle θ can be given as y = ( ℓ − x ) tan θ . Thus,the BS at z lies in the strip S if −∞ < y < ∞ and − y tan θ ≤ x ≤ ℓ − y tan θ . (25)From (23), (24), and (25), we can express (22) as E (Ψ[0 , ℓ ] | θ ) = λ Z ∞ y = −∞ d y Z ℓ − y tan θ x = − y tan θ exp (cid:18) − λπy sin θ (cid:19) d x = ℓ (cid:16) √ λ | sin θ | (cid:17) . (26)Since θ is uniformly distributed in [0 , π ] , averaging over θ yields E (Ψ[0 , ℓ ]) = √ λπ ! ℓ. (27)Hence, for the case of two beams, the linear intensity of beamreselections is √ λπ .For n beams with n ∈ N , there are n − possibilities ofbeam reselections corresponding to the serving BS. In thiscase, we can obtain the average number of beam reselectionsby superimposing the beam reselection events correspondingto each beam boundary crossing considered earlier in thisroof. Hence, the linear intensity of beam reselections for n beams is µ s , b ( n ) = 2 n − √ λπ = 2 n √ λπ . (28)By considering the speed v of the typical MT, we get the timeintensity µ t , b of beam reselections as µ t , b ( n ) = 2 n √ λπ v.λπ v.