Performance Analysis for Multi-layer Unmanned Aerial Vehicle Networks

Performance Analysis for Multi-la yer Unmanned Aerial V ehicle Netw orks Dongsun Kim † , Jemin Lee † , and T ony Q. S. Que k ∗ † Daegu Gyeongbuk Institute of Science and T echnology (DGIST), K orea ∗ Singapor e Univ ersity of T echnolo g y an d Design (SUTD), Sing apore Email: yida ever@dgist.ac.kr, jmnlee@dgist.ac.kr , tonyq uek@sutd.ed u .sg Abstract —In this pap er , we provide th e model of the multi - layer aerial network (MAN), composed unmanned aerial vehicles (U A Vs) that distributed in Poisson point process (PPP) with different densities, heights, and transmission power . In our model, we consid er the li n e of sight (LoS) and non- l ine of sight (NLoS) channels which is probabilistically fo rmed. W e fi rstly derive th e probability di stribution fun ction (PDF) of th e main link distance and th e Laplace transform of interference of M AN considering strongest a vera ge receive d p ower -based association. W e then analyze the successful transmission probability (STP) of the MAN and prov ide the u pper bound of th e optimal d ensity that maximizes the STP of the MAN. Through the numerical results, we show the existence of the opti mal height of U A V due to a p erf ormance t radeoff caused by the height of the aerial network (AN ) , and also show th e upper bounds of the optimal densities in terms of the S TP, wh ich decrease wi th the height of the ANs. Index T erms —Stochastic geometry , aerial networks, multi- layer , Po isson p oint p rocess, association rule, optimal density . I . I N T RO D U C T I O N Recent dev elopments in the unman ned aerial vehicle ( U A V) technolog y incr e ase payloads capacity , av erage flight time, and battery ca pacity that enab le s the UA V to play an im- portant role in wireless networks. In the area which n eeds quick d eployment of the b a se station ( BS) due to d isaster or ev ents, UA Vs are expected to act as a tempo ral BS as well [1]. Furtherm ore, the data collection fro m the d evices under certain energy constraints c an be don e by using U A Vs [2]. In additio n, demands on the data acq uisition using UA V in crowd surveillance h av e ar isen [3]. T o utilize U A Vs for the aforemen tioned applications an d services, the resear ch on the establishment o f reliable aerial network (AN) is required . The UA V based wireless co mmunicatio ns has been stud ied in [4]– [6] after mode ling the wireless channel an d th e mob il- ity , which are different from those o f the te r restrial n e tworks. In [4], the probab ility that a link for ms line of sight (LoS), i.e., the Lo S probab ility , is modeled, which is d etermined by th e angle fro m the gro und, and also prop osed the optimal UA V deployment that maximizes the coverage area. In addition, U A V relay networks in cellular networks an d device-to-device commun ications are co nsidered in [5] and [6], respectively . Howe ver, the studies m e ntioned above have considered on ly the small number of UA Vs, wh ich can show the perfo r mance only f or the limited scenar io s. Recently , the research on the ANs, which consist of UA Vs, is p resented in [7], [8] using stochastic geome try , which is a widely-used tool for randomly distributed no des [9]. The coexistence o f AN and the terrestrial cellular networks is studied by considering the distribution of U A Vs as Poisson point pro cess (PPP) in [8]. The coverage p r obability of UA V by using binomial point process (BPP)-based distribution is presented in [ 7]. In additio n, [1 0] studies the multi-tier U A V networks and shows the d ownlink spectr al efficiency of the network via simulations. Howe ver , except f or [10], most of these work s did not co nsider the multiple lay er structu re of ANs. Nev e r theless, no analytical app roach is provided in [10]. In ANs, the UA V hav e limitation on heig ht due to the hardware a nd the law [11]. Further more, as a number of UA V have mobility for serving different service , to avoid the collision between U A Vs and to efficiently manage the resource, it is req uired to have the multiple layer structure in AN which d ifferentiate th e height accordin g to the ro les and typ es of UA V. Therefo re, in this paper, we in vestigate th e perform ance of the mu lti-layer aerial network (MAN) w ith various ty pes of U A V. Specifically , the MAN is composed of K layer ANs that ha ve UA Vs with different transmission power , spa tial densities, and heights. Note that the multip le lay er structure has been con sidered fo r terr e strial networks, wh ich is c a lled as the heterog eneous networks [1 2]–[14]. Howe ver, d ifferent to those works, the heights of no des need to be considered together with the chann e l model which has the LoS probab ility determined by the heigh t of node. This leads to new analysis on the interferenc e a n d the successfu l transmission pr obability (STP) of the MAN considering association rule. T o our best knowledge, there is no analy sis on the STP for mu lti- layer U A V networks consider ing assoc ia tio n rule, LoS, and no n- line of sight (NLoS ) channel. Furthe r more, our analysis on the up per bo und of the optimal d ensity g iv e s u seful insig h ts for futur e MAN im plementation . Our co ntribution can be summarized as follows: • u sing stochastic ge ometry , we newly an alyze the La p lace transform of interferen ce of MAN by considerin g NLoS and LoS chann els with the height- depend ent Lo S pro b- ability; • w e derive the STP when a groun d node selects a U A V ݄ ଵ Ȱ ଵ Ȱ ଶ Ȱ ଴ ݄ ଶ 2-layer Aerial Network 1-layer Aerial Network T errestrial Network (0-layer) Fig. 1. An exa mple of two layer AN with ground nodes (i.e., 0 -layer). T he black lines represent the main link from a transmitte r to a recei ver and red dashed lines repre sent interfe rence links which comes from other UA Vs. with association r ules con sidering stron gest av erage re- ceiv ed power for co mmunica tio ns; • to give in sight o n the effect of UA V d ensity on STP, we provide th e up per bound s of the op timal U A V d e nsities of each layer AN that maximize the STP; and • w e show the effects of channel and network par ameters on the op timal heights o f ANs a n d the com patibility of the upp er boun d of op timal den sities. I I . S Y S T E M M O D E L In this section, we pre sen t th e system mode l of MAN with UA V inclu ding the network description an d the channel model. Fur th ermore , we describe the a ssociation ru le which is used to ob tain the p robability distribution f unction (PDF) of the main link distance . A. Multi-layer Aerial Networks W e consider a MAN which con sists of K lay ers of ANs at dif f erent altitudes with a terrestrial network as shown in Fig. 1. W e den ote K as the set of AN lay e r indexes, i.e., K = { 1 , · · · , K } , and layer 0 as the terrestrial network. W e assume U A V in ANs and the gro und nodes in the terrestrial network are distributed according to PPPs [9]. Altho ugh each U A V has giv en p ath scheduled by controller, lo cation of U A Vs at given time is r andom f rom the p erspective of other layers, hence, we use the PPP for the locatio n of the UA V as [8]. Specifically , in the k -layer, the node locations follows a homog e neous PPP Φ k with density λ k and they are at th e fixed altitude h k and transmit with the power P k . Note that the altitude of nodes in the 0 -layer (i.e. , the ter restrial layer) is h 0 = 0 and altitu des of oth er layer s are h k > 0 for k ∈ K . In the MAN, we consider the commun ication fro m a U A V to a ground node. 1 In the communicatio n with U A Vs, we should consider both LoS and NL oS channels since the existence o f obstacles (e.g., buildings) between the transmitter 1 Note that we omitted analysis in cluding the air-to-ai r chan nel in the paper , ev en though our framew ork is readily expa ndable for communication in between U A V which will be presented in the future research. and the receiver can be changed with the altitu d e of U A V. In [ 4], the pro bability of f orming LoS chanel is modeled by a signomial appro ximation of the p r obability of ha ving obstruction s b etween tra nsmitter an d receiver . When a n ode in the k - layer transmits to a g round node, the LoS p robability is defin e d as [ 4] ρ (L) k ( x ) = 1 1 + a exp( − b [sin − 1  h k x  − a ]) (1) where a an d b ar e the param eters related with en vironm e nts, and x is the link distanc e between the tran smitter and the receiver . In real environment, U A Vs can act as obstacles, e.g., UA V in 1-layer can block the channel betwe e n ground and 2-la y er AN. Here, we assume existence of UA V d oes not affect the air - to-gro und channel since the density of AN is low , therefo re, the obstruction caused by U A V is negligible. From (1), we can see tha t the LoS pro bability increases with x which means high er altitude gives h igher Lo S prob ability since there will be fewer obstru ctions. The NLoS prob ability is then given as ρ (N) k ( x ) = 1 − ρ (L) k ( x ) . Since each link between a transmitter and a receiver can be in either L oS or NLoS with the probab ilities, ρ (L) k ( x ) and ρ (N) k ( x ) , respec tively , we can divide a set of the k -layer transmitters into th e ones in LoS an d th e ones in NLoS as Φ (L) k and Φ (N) k , respe c ti vely , which are non- homog eneous PPPs. The den sities of no des in Φ (L) k and Φ (N) k with the distance x from a rec eiv e r are, respe c ti vely , defin ed acc o rding to the link d istance x as λ (L) k ( x ) = 2 π xλ k ρ (L) k ( x ) and λ (N) k ( x ) = 2 π xλ k − λ (L) k ( x ) . W e also con sider different chan nel models for links in LoS and NLoS. The pathloss exponen ts for Lo S and the NLoS links are den oted b y α (L) and α (N) , respe c ti vely , a n d gene r ally , 2 ≤ α ( L ) ≤ α ( N ) ≤ 6 . W e co n sider th e Nakagami- m fading for the ch annels of LoS an d the NLo S links, of wh ic h channel gains are respectively presented by G (L) ∼ Γ( m (L) , 1 m (L) ) and G (N) ∼ Γ( m (N) , 1 m (N) ) . Here, we u se m (N) = 1 , which gives Rayleigh fadin g , i.e., G (N) ∼ exp(1 ) , while m (L) ≥ 1 . B. Association Ru le In th is paper, we assume a receiver connects to the transmit- ter , which has the strongest av erage received power d escribed in [1 5]. This can be applied to the scen ario that in th e presen ce of UA V b ased BSs [1], a u ser selects a BS to receive its data. Based on the a ssocia tio n rule, we can pr esent the selected transmitter’ s coo rdinates x main as x main = a rg max x ∈ Φ k ,k ∈K P k k x − x rec k − α x (2) where x rec is the coordin ates the receiver is located, an d α x is the pathloss exponent o f the link between the transmitter at x and the receiver . Based on the association rule above, we can deter mine the PDF of the distance for th e main link fr o m a selected tran s- mitter to a rec e i ver . In co n ven tio nal terrestrial networks, the PDF of main link d istance is determined by the transmission power , the p athloss exponent, an d the link distance. Howe ver , in ANs, we need to addition ally consider the LoS/NLoS probab ilities for all links to the tran sm itter s. W e de note th e channel environmen t by c ∈ { N , L } , where c = N and c = L, respectively , means the L oS and NLoS en vir onmen ts of the link. The PDF of main link distance in c channel cond ition is presented in following lemma. Lemma 1: When a transmitter in the j - la y er u n der the channel en vir onment c is selected , the PDF of m a in link distance Y j ( c ) is giv e n by f Y ( c ) j ( y ) = f V ( c ) j ( y ) A ( c ) j Y k ∈K ,c o ∈ { L, N } , ( k,c o ) 6 =( j,c ) ¯ F V ( c o ) k   P k y α ( c ) P j ! 1 α ( c o )   (3) where A ( c ) j is association proba b ility giv en by A ( c ) j = Z y > 0 f V ( c ) j ( y ) Y k ∈K ,c o ∈ { L,N } , ( k,c o ) 6 =( j,c ) ¯ F V ( c o ) k   P k y α ( c ) P j ! 1 α ( c o )   .dy (4) Here, V ( c ) j is the distance to the nearest nod e among the nodes in the k - layer u nder the ch annel en viro nment c . ¯ F V ( c ) j ( v ) and f V ( c ) j ( v ) are th e c o mplemen tary cumu lati ve distribution function ( CCDF) and the PDF of V ( c ) j , given by ¯ F V ( c ) j ( v ) = exp " − Z max( v ,h j ) h j 2 π λ j xρ ( c ) j ( x ) dx # , (5) f V ( c ) j ( v ) = 2 π λ j v ρ ( c ) j ( v ) exp " − Z v h j 2 π λ j xρ ( c ) j ( x ) dx # . The PDF f V ( c ) j is 0 when v < h j . Pr oof: The cu mulative distrib u tion func tio n (CDF) of V ( c ) j is giv en by F V ( c ) j ( v ) ( a ) = 1 − exp " − Z max( v ,h j ) h j λ ( c ) j ( x ) dx # (6) where ( a) is fr om th e v o id probability of PPP, and from (6), we have (5).Since the main link h av e smallest pa thloss, probab ility that main link distance is smaller than y when x main ∈ Φ ( c ) j is given by P  Y ( c ) j ≤ y ∩ x main ∈ Φ ( c ) j  = Z y 0 f V ( c ) j ( v ) P  x main ∈ Φ ( c ) j ∩ V ( c ) j = v  dv (7) ( a ) = Z y 0 f V ( c ) j ( v ) Y k ∈ K ,c o ∈ { L, N } , ( k,c o ) 6 =( j,c ) P h P j v − α ( c ) ≥ P k  V ( c o ) k  − α ( c o ) i dv where (a) f rom (2). Th erefore, we der ived the association probab ility as ( 4 ) by y → ∞ . Fu rthermo re, we ca n der i ved the PDF of the m a in link distance as (6). I I I . I N T E R F E R E N C E A N A L Y S I S A N D S U C C E S S F U L LY T R A N S M I S S I O N P RO B A B I L I T Y In this section, we an a lyze the Laplace tran sform of the interferen ce considerin g association rules. Then, we der i ve the STP of the MAN and the upper bo und of the density o f AN that m aximize the STP o f the M AN. A. Laplace T ransform of the Interfer en ce In th e MAN, we first consider interference fro m specific layer and ch annel en viro n ment. Since there is no interferer which have stronger power than main link transmitter, we an- alyze inter ference from specific layer and chann e l to analyze total inter ference. Here, the inter ference from transmitters in the k -layer un der the channe l environment c o is giv e n by I ( c o ) k = X x ∈ Φ ( c o ) k P ( c o ) k ( k x − x rec k ) (8) where P k ( c o ) is the received power from a transmitter which is given by P ( c o ) k ( x ) = P k G ( c o ) x − α ( c o ) . (9) Here, we re present distance b e tween th e transmitter and the receiver as x = k x − x rec k . The Laplace transfo r m of the interferen ce is given in the f o llowing lemma. In the lemm a, we use x ( c ) j ( y ) to r epresent the main link u nder channel en vironmen t c with distanc e y is selected. Lemma 2: When a transmitter in the j - la y er u n der the channel en v ironme n t c with distance y is selected, the L a place transform of the interferen ce fr om the tr ansmitting nodes in the k -lay e r un der the channel environment c o is given by (10), which p r esented on the top of next pa ge, where R ( c,c o ) j,k ( y ) is R ( c,c o ) j,k ( y ) =  P k y α ( c ) /P j  1 /α ( c o ) . (11) Pr oof: The Lap la c e transform o f the interfer ence is L I ( c o ) k | x ( c ) j ( y ) ( s ) ( a ) = E Φ ( c o ) k    Y x ∈ Φ ( c o ) k   1 1 + sP k x − α ( c o ) m ( c o )   m ( c o )     x ( c ) j ( y )    (12) where (a) is fro m the expec tatio n over chan n el G ( c o ) which giv es the moment- generatin g function (MGF) of Gamma dis- tribution a s [15]. Since λ ( c o ) k ( x ) = 2 π xλ k ρ ( c o ) k ( x ) , the pro ba- bility gener ating function al (PGFL) o f non- homog eneous PPP needs to be o btained as [9] E    Y x ∈ Φ ( c o ) k f ( x )     x ( c ) j ( y )    = exp  − 2 π λ k Z ∞ R ′ x (1 − f ( x )) ρ ( c o ) k ( x ) dx  . (13) In (1 3), selecting a transmitter in th e j -lay er u n der the chann el en vironmen t c b y (2) m e ans there is no inter fering node in th e k -layer under chan nel en vironme nt c o , clo ser th an R ′ = max  R ( c,c o ) j,k ( y ) , h k  . Combined with ( 12) a nd (13), L I ( c o ) k | x ( c ) j ( y ) ( s ) = exp    − 2 π λ k      Z ∞ max  R ( c,c o ) j,k ( y ) ,h k  xρ ( c o ) k ( x )    1 −   1 1 + sP k x − α ( c o ) m ( c o )   m ( c o )    dx         (10) we obtain the Lap lace transfor m interfer e nce under co ndition x ( c ) j ( y ) as ( 10) From Lemma 2 and pr operty o f the Laplace tran sform, we can obtain the Laplace transfor m of the sum of the interferenc e and n o ise as L I | x ( c ) j ( y ) ( s ) = exp( − sσ 2 ) Y k ∈K ,c o ∈{ L,N } L I ( c o ) k | x ( c ) j ( y ) ( s ) (14) where I = P c o ∈{ L,N } k ∈K I ( c o ) k + σ 2 and σ 2 is for the noise power . B. Successful T ransmission Pr obability In this subsection, we define th e STP when the d istance and the channel environment is giv en. Then, we derive the STP of the MAN by using association pro bability and the PDF of the main link distanc e . When the main link is in th e chann el en vironmen t c with the link distance y , th e STP is defined using sign al to interferen ce plus noise ratio ( SINR) as p ( c ) j ( y ) = P h SINR ( c ) j ( d ) > β i (15) where SINR ( c ) j ( d ) = P ( c ) j ( d ) / I , and β is the target SINR, which is related with the tran smission rate. When the associ- ation r ule in (2) is used, the STP o f MAN is p resented in th e following lemma. Lemma 3: The STP of the MAN is given by P = X j ∈K , c ∈{ L, N } Z ∞ h j p ( c ) j ( y ) f Y ( c ) j ( y ) A ( c ) j dy (16) where f Y ( c ) j ( y ) and A ( c ) j is in (3) and (4), p ( c ) j ( y ) is p ( c ) j ( y ) = m ( c ) − 1 X n =0 ( − s ) n n ! d n ds n L I | x ( c ) j ( y ) ( s )       s = S ( c ) j ( y ) , (17) S ( c ) j ( y ) = m ( c ) β P j y − α ( c ) , (18) and L I | x ( c ) j ( y ) ( s ) is in ( 1 4). Pr oof: From the definition of STP, we ha ve the con - ditional STP when a tr ansmitter in j - layer under channel en vironmen t c is selected with the main link distance j represented as p ( c ) j ( y ) ( a ) = E  1 − 1 Γ( m ( c ) ) γ  m ( c ) , m ( c ) β P j y − α ( c ) I      x ( c ) j ( y )  ( b ) = E   m ( c ) − 1 X n =0 ( s I ) n n ! exp ( − s I )     x ( c ) j ( y )         s = S ( c ) j ( y ) (19) where (a) follows from the Gam ma distrib ution of ch annel gain and (b) follows from th e pro perty of lower in c omplete Gamma function. Notice that we derived (18) fro m (b). Using the p roper ty of the f ollowing Laplace transform ( −I ) n L I ( s ) = d n ds n L I ( s ) (20) we o btain ( 17). Furthermo re, fr om th e PDF an d association probab ility in the Lemma 1, we o btain (1 6). When UA Vs are deployed as BSs which serve for gro und receiver , it is impo rtant to maximize STP. In ou r works, we analyze op timal den sity of the tran sm itter . As shown in (16), it is hard to present STP in a closed form, hence, hard to obtain the optimal d e nsities of each AN layer that m aximize ST P. Howe ver, in th e f ollowing co rollary , we p r esent the closed- form upper bou n d of the optimal densities for a special case. Cor ollary 1 : For the case of m (N) = m (L) = 1 , when the optimal d ensity of the j -la y er AN is λ ∗ j , its upper b ound is λ ∗ j ≤ λ b j = 1 2 π ǫ j ( S (L) j ( h j )) (21) where S (L) j ( h j ) is in ( 18), and ǫ j ( s ) is given by ǫ j ( s ) = Z ∞ h j x 1 − ρ (L) j ( x ) 1 + sP j x − α (L) − ρ (N) j ( x ) 1 + sP j x − α (N) ! dx. (22) Pr oof: See App endix A. In Coro llary 1 , the upper bound λ b j is on ly affected by th e network parameter of the j -laye r AN such as h j , but inde- penden t with the d e nsity , height, an d transmission p ower of other ANs. Hen ce, the up per bound of each layer’ s density in MAN can be deter mined ind epende n tly each othe r . Althou gh function ǫ j ( s ) is n ot in the closed fo rm, it is easy to evaluate and analyze. 2 Furthermo re, due to the above inde p endency , we can also obtain the upper bo und of the total d ensity of MAN as λ b T = P k ∈K λ b k . I V . N U M E R I C A L R E S U LT S In this section, we pr esent the numerical results to ev aluate our analysis on the STP o f MAN with single o r two layers of ANs u n der the interferenc e-limited en viro nments. i.e., σ 2 = 0 . For the num e rical results, we use β = 0 . 7 , P k = 1 for all k , α (L) = 2 . 5 , an d α (N) = 3 . 5 . Ex cept for Fig. 2, m (N) = 1 a nd m (L) = 1 are u sed. Mo reover , we use a = 12 . 4231 and b = 0 . 1202 are used for the L oS prob ability , which are determin ed for th e urban ar e a environment in [4]. Fig. 2 shows the STP as a f unction o f the altitude h 1 in single AN f or different values of chan n el coe fficient 2 Note that the upper bound of optimal density λ b j is decrease with height h j under condition β h α (L) j > 1 , h j > 1 which is omitte d in this paper . The conditi ons are readily achie veabl e for aerial netw orks. 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P S f r a g r e p l a c e m e n t s m (L) = 1 m (L) = 3 m (L) = 2 0 m (L) = 100 ρ (L) j ( x ) = 1 h 1 (m) Successful Transmission Probability Fig. 2. STP of single AN according to the al titude h 1 and the LoS coef ficient m (L) when λ 1 = 10 − 5 . ρ (L) j ( x ) = 1 represent the STP when e very link is under L oS channe l. m (L) = { 1 , 3 , 20 , 10 0 } , where λ 1 = 10 − 5 [nodes/ m 2 ]. W e shows the ST P when LoS probab ility is 1 which rep resented with ρ (L) j ( x ) = 1 . Simulation results, obtained f r om Monte Carlo simulatio ns, are presented by the dashed lines with filled markers, while analysis results for m (L) = { 1 , 3 } are presented by the solid lines with unfilled markers. From Fig. 2, we c a n first see that the simulation r esults match well with the an a lysis. W e can also see that for all m (L) , the STP first increases an d th e n de c reases with h 1 . F or small h 1 , as the height increases, the LoS ch annel probability increases, which makes th e main link power stronger and results in the hig her STP. Howe ver, as h 1 keeps in creasing, the main link distance also increases, which makes the main lin k power smaller . As a result, th e optimal value of h eight can be o btained from the tradeoff between the link distance the the LoS probab ility . When the Lo S proba b ility is 1 , th ere is n o tr adeoff, the STP of the AN decr e ase with height, which is same for ρ (L) j ( x ) = 0 . Furthermo re, b y compa ring th e results fo r different m (L) , we can see that for small h 1 , larger L oS coefficient m (L) giv e high e r STP. In sm a ller h 1 , p ower from th e main link is dominan t and mostly LoS ch annel. Since larger LoS chann el coefficient gi ves smaller variance to th e main link channel in smaller h 1 , larger m (L) can give higher STP. Con trary , in larger h 1 , power from the in terferenc e is domin ant. Th erefor e , larger LoS ch annel coefficient g iv es smaller variance to the interferen ce cha n nel which gives lower STP. Howe ver, trends of STP accordin g to th e height are the same for different m (L) . Therefo re, we use m (L) = 1 in the following numer ical r esults because it can g iv e sufficient insigh ts on the per forman ce of MAN. Fig. 3 sho ws the contour of STP as a function of the altitude h 1 and th e d ensity λ 1 of U A V in single AN where m (L) = 1 . W e represent the optimal density as a white line with circles an d the up per bou nd of o ptimal d ensity , obtained from Coro llary 1, as a red lin e with diam o nds. By comparin g the optimal density an d the u p per bound of the 0 50 100 150 200 250 300 10 -6 10 -5 10 -4 0.1 0.2 0.3 0.4 0.5 0.6 P S f r a g r e p l a c e m e n t s h 1 (m) λ 1 (number / m 2 ) Fig. 3. ST P of single AN as functions of the density λ 1 and the height h 1 when m (L) = 1 . T he white line marked by circle s represent s the optimal density when the height is giv en and the red line marked by diamonds represent s the upper bound of the optimal densit y . 10 -6 10 -5 10 -4 10 -6 10 -5 0.1 0.2 0.3 0.4 0.5 0.6 P S f r a g r e p l a c e m e n t s λ 1 (number / m 2 ) λ 2 (number / m 2 ) Fig. 4. STP of 2-layer MAN as functions of the density of 1-laye r λ 1 and the den sity of 2-lay er λ 2 of U A V when m (L) = 1 , h 1 = 100 , and h 2 = 200 . The white line represen ts the optimal λ 2 in giv en λ 1 . The magenta, yello w , and c yan lin es wit h symbol s r epresent s the area whic h ha ve same total d ensity λ T = λ 1 + λ 2 = { 10 − 6 , 10 − 5 . 3 , 10 − 4 . 6 } optimal density , we can notice that the trends according to h 1 are the same. Fr o m Fig. 3, w e can a lso see that as the h eight increases, th e optimal density and its up per bound dec r ease. This is b ecause the n umber of inter fering links in LoS incre a se with h 1 , so the in terferenc e beco mes larger . Fig. 4 sho ws the contour of STP of MAN with two AN as a fun ction of the density of 1 - layer λ 1 and the d e nsity o f 2 -layer λ 2 of U A V, wher e m (L) = 1 , h 1 = 100 , h 2 = 200 . I n this figure, the optim al d e nsity o f 2 -layer λ 2 is presented by a wh ite lin e with circles. The lines with color an d symb ol represent th e STPs of the MAN that h ave th e total density as λ T = λ 1 + λ 2 = { 10 − 6 , 10 − 5 . 3 , 10 − 4 . 6 } [nod es/ m 2 ], respectively . W e can see when density of corre sponding A N is low , the STP incre a se with the den sity , an d when density is high, the STP d ecrease with den sity w h ich is same with th e result of single-layer M AN. Furthermore, fr om the colored lines, we can get th e relationship of o p timal ratio o f densities when the total d ensity of MAN is gi ven. When the total density of the MAN is small, e.g., the ma g enta lin e , the optimal density is λ 2 = λ T . Howe ver, for the large total density , e . g., the cyan line, th e optimal d e n sity is λ 1 = λ T . In other case, as shown in yellow line , we can see the o ptimal is neither λ 1 = λ T nor λ 2 = λ T . Since optimal heig ht of larger AN is low as sho wn in sing le AN, optimal ratio of lower lay er is incre ased with the total density . V . C O N C L U S I O N This paper models the MAN which is wireless networks consist of multi-lay ers o f UA Vs that distributed in PPP with different densities, heights, and powers. W e consider LoS and NLoS chan nel and the stro ngest transmitter association fo r the A N . Our approac h is to derive the PDF of the m ain lin k distance and the Lap lace transfor m of the inter ference for the STP analysis. By ana ly zing the STP, we show that ea ch AN in the MAN have the uppe r boun d of optim al de nsity wh ich is g iven by th e fu n ction of the height of correspo nding AN. In ad dition, our numer ic a l r esults show the tradeoff cau sed by height of the AN, affection of Lo S co efficient, significance of the u p per boun d of the optimal d ensity , and optim al densities and optimal ratio o f densities in th e 2-laye r MAN. Specially , our results show h ig her altitude AN ha ve sparser optimal density and show th at the optimal ratio o f d ensities in the 2-layer MAN is ch anged with the to tal d e nsity of the M AN. A P P E N D I X A. Pr oof of Cor ollary 1 From Lem ma 3, the ST P can be rep resented by P = X j ∈K Z ∞ h j  ϕ (L) j ( y ) + ϕ (N) j ( y )  dy , (23) ϕ ( c ) j ( y ) = 2 π ρ ( c ) j ( y ) λ j y exp " − 2 π X k ∈K λ k φ ( c ) j,k  y , S ( c ) j ( y )  # , φ ( c ) j,k ( y , s ) = X c o ∈{ L,N } " Z R ′ h k xρ ( c o ) k ( x ) dx + Z ∞ R ′ xρ ( c o ) k ( x )  1 − 1 1 + sP k x − α ( c o )  dx  , and R ′ = max  R ( c,c o ) j,k ( y ) , h k  , Notice that φ ( c ) j,k ( y , s ) is in- crease with y and s . In add ition, we can deriv e the differential of the total STP with density λ j as ∂ ∂ λ j P = X k ∈K Z ∞ h k ∂ ∂ λ j  ϕ (L) k ( y ) + ϕ (N) k ( y )  dy (24) where co mpone n ts inside the in tegral are given b y ∂ ∂ λ j ϕ ( c ) j ( y ) = ϕ ( c ) j ( y ) λ j  1 − 2 πλ j φ ( c ) j,j ( y , S ( c ) j ( y ))  (25) ∂ ∂ λ j ϕ ( c ) j ′ ( y ) = ϕ ( c ) j ′ ( y )  − 2 π φ ( c ) j ′ ,j ( y , S ( c ) j ( y ))  (26) where j ′ used to represen t j ′ 6 = j . No tice that (25) is differential of th e ST P when main link is j -layer . On the other hand, (26) is d ifferential with λ j for the STP when main link is no t j -lay er , hence the STP always de c r ease with λ k . From (2 5) an d (26), we derive th e ran g e of λ j that m akes the total STP d ecreased with th e λ j . Her e, ϕ ( c ) j ( y ) and λ j is positive for all d omain. Hence, th e total STP decreases with λ j , if following inequ ality holds. max y ,c " 1 2 π φ ( c ) j,j ( y , S ( c ) j ( y )) # ≤ λ j . (27) Furthermo re, a s φ ( c ) j,j ( y , s ) is incre ase with y and s , we can put the minim u m value of y and s which gives φ (L) j,j  h j , S (L) j ( h j )  = ǫ j  S (L) j ( h j )  . (28) Hence, th e upper b o und of optimal d ensity is given b y (22). R E F E R E N C E S [1] Y . Zeng, R. Zhang, and T . J. Lim, “Wir eless communications with un- manned aerial vehic les: opportunit ies and challe nges, ” IEEE Commun. Mag . , vol . 54, no. 5, pp. 36–42, May 2016. [2] H. 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