Interference and Outage in Clustered Wireless Ad Hoc Networks

1 In terferen ce and Outage in Clus tered Wireles s Ad Ho c Net w orks Radha Krishna Ganti and Martin Haenggi Depart ment of Electrica l Engineering Univ ersity of Notre Dame Indiana-4655 6, USA E-mail {rganti,mhaenggi}@nd.edu Abstract In the analysis of large random wireless n etw orks, the u nderlying no de distribution is almost ubiquitously assumed to b e t he homogeneous Pois son p oin t pro cess. I n this pap er, the no de locations are assumed to form a Poisson cluster e d pr o c ess on the plane. W e derive the distribut ional prop erties of the in terference and pro vide upper and lo wer bounds for its C CDF. W e consider the probability of successful transmission in an interference limited c h annel when fading is mo deled as Rayleig h. W e provide a numericall y integra ble expression for the outage probability and closed-form u pp er and lo wer b ounds. W e sh o w that when the transmitter-receiver distance is large, t he su ccess probability is greater than that of a Poi sson arrangemen t. These results characterize the p erformance of th e system under geographical or MAC -induced clustering. W e obtain the maximum in tensity of transmitting no des for a given out age constraint, i.e. , the transmission capacit y (of this sp atial arrangemen t) and show that it is equal to that of a Poi sson arrangement of no des. F or the analysis, techniques from sto c hastic geometry ar e used, in p articular the probability generating functional of P oisson cluster pro cesses, the P alm chara cterization of P oisson cluster pro cesses and the Campb ell-Mec ke theorem. I. Introduction A common and ana lytically conv enient a ssumption for the no de distribution in lar g e wir eless netw or ks is the homogeneo us (or stationar y) Poisson p oint pr o cess (P P P) of in tensity λ , where the num b er o f no des in a certa in area of size A is Poisson with parameter λA , and the num b ers o f no des in tw o disjoint areas are indep endent ra ndom v ar iables. F or sensor netw orks, this assumption is usua lly justified by claiming that sensor no des may b e dr o pp e d from aircra ft in larg e num b er s; for mobile ad ho c netw orks, it may b e argued that ter minals mov e indep endent ly from each other. While this ma y b e the ca se for certain netw or ks, it is m uch more likely that the no de distribution is no t ” completely spa tially random” (CSR), i.e. , that no des Pa rt of the material in this paper has b een presente d at the 2006 As ilomar conference. May 29, 2018 DRAFT 2 are either cluster ed o r more re g ularly distributed. Moreover, even if the complete set of no des cons titutes a PPP , the subset of active no des ( e.g. , trans mitter s in a given time-slot or sentries in a sensor net work), may not b e homog eneously Poisson. Certa inly , it is preferable that s im ultaneous transmitters in an ad ho c net work o r sentries in a sensor netw or k form mor e regular pr o cesses to maximize spatial r euse or c overage resp ectively . On the other hand, many proto co ls hav e b een sugges ted that are based on clustered pro cesses . This motiv a tes the need to extend the rich set of results av a ila ble for PPPs to other no de distributions. The clustering of no des may be due to geog raphical fa ctors, for example communicating no des inside a building or gro ups o f no des moving in a co ordinated fashion. The clustering may also b e “artificia lly” induced by MA C proto cols. W e denote the former as geog raphical clustering and the latter a s logical clus ter ing. A. R elate d W ork There exists a significant b o dy of literature for netw or k s with Poisson distributed no des. In [1] the characteristic function of the interference was obtained when there is no fading and the no des a re Poisson distributed. They also provide the proba bilit y distr ibution function of the int erference as a n infinite series. Mathar et al., in [2 ], ana lyze the interference when the in terference co ntribution by a transmitter lo cated at x , to a rec eiver lo cated a t the origin is exp onentially distributed with pa r ameter k x k 2 . Using this mo del they derive the density function of the interference whe n the no des a re arra nged as a one dimensional lattice. Also the Laplace transform of the interference is obtained when the no des are Poisson distributed. It is known that the interference in a plana r netw or k of no des can b e mo deled as a sho t no ise pro ces s . Let { x j } b e a point pro cess in R . Let { β j ( . ) } b e a sequence of indepe nden t and ident ically distributed random functions on R d , indep endent of { x j } . Then a generaliz e d s hot noise pro cess can b e defined as [3] Y ( x ) = X j β j ( x − x j ) If β j () is the path loss mo del with fading , Y ( x ) is the interference at lo cation x if all nodes x j are transmitting. The shot no ise pr o cess is a very well s tudied pr o cess for noise mo deling. It was first in tro duced b y Schottky in the study of fluctuations in the ano de cur rent of a thermionic dio de and it was s tudied in detail by Rice [4], [5]. Daley in 1971 defined multi-dimensional sho t noise a nd exa mined its existence when the po ints { x j } are Poisson distributed in R d . The existence of gener alized shot-noise pro c e ss, for any p oint pro ces s was studied by W estcott in [3]. W estcott also provides the Laplace transfor m of the shot- no ise when the p oints { x j } are distributed as a Poisson clus ter pr o cess. Normal co nv erg ence of the multidimensional s ho t-noise pro cess is shown by Heinrich a nd Schmidt [6]. They a lso show that when the p oints { x j } for m a Poisson po int pro cess of intensit y λ , the rate of conv e r gence to a normal distribution is √ λ . In [7 ], Ilow a nd Hatzinakos mo del the interference as a shot no is e pro ce s s and show that the interference is a symmetric α - stable pro cess [8] when the no des are Poisso n distributed on the plane. They also show that channel randomness affects the disp ersion of the distribution, while the pa th-loss exp onent affects the exp onent o f the pro cess. The throughput and outa g e in the presence o f interference ar e analyzed in [9 ]–[11]. DRAFT May 29, 2018 3 In [9], the shot-noise pro cess is analyzed us ing sto chastic geometry when the no des are distributed as Poisson and the fading is Ra yleigh. In [12 ] upp er a nd low e r b ounds are obtaine d under gener al fading and Poisson arrang ement of no des. Even in the case of the PP P , the interference distribution is not known fo r all fading distributions a nd all channel atten uation mo dels . Only the characteristic function or the Laplace tra nsform o f the interference can b e o btained in most o f the cas e s. The Laplac e transfor m can be used to ev a luate the o utage pro babilities under Rayleigh fading characteristics [9], [1 3]. In the ana ly sis o f o utage probability , the c onditional La place transform is require d, i.e. , the La place transform given that ther e is a p oint of the pro cess lo cated at the origin. F or the P PP , the conditional Laplac e tra ns form is eq ua l to the unconditional L a place transfo r m. T o the bes t o f our knowledge, we are not aware o f a ny literature p ertaining to the int erference character ization in a clustered netw o rk. [14] intro duces the no tion of t r ansmission c ap acity , whic h is a measure of the ar ea sp ectral efficiency of the success ful tr ansmissions res ulting from the optimal co n tention density as a function of the link distanc e . T rans mission capacity is defined as the pr o duct of the maximum density of suc c e ssful transmiss io ns and their da ta rate, g iven an outage constra int. W eber et al., pr ovide b ounds for the tra nsmission capacity under different mo dels of fading, when the no de lo cation are Poisson distributed. B. Main c ont ributions and or ganization of the p ap er In this work, we mo del the transmitters a s a Poisson cluster pro ces s. T o cir c um ven t tech nical difficulties we as sume that the receivers are not a pa rt of this clustered pr o cess. W e then fo cus on a s p ecific tra nsmit- receive pair at a distance R a pa rt, see Fig 1. W e ev alua te the Laplace tr ansform of the in terference on the plane conditioned o n the even t that there is a transmitter lo cated at the or igin. Upper and low er bo unds are o btained for the CCDF of the interference. F r om these b ounds, it is obse rved that the interference is a heavy-tailed distribution with exp onent 2 /α when the path loss function is k x k − α . When the path-lo ss function ha s no s ingularity at the orig in ( i.e. , remains b ounded), the distr ibution o f interference dep ends heavily on the fading distribution. Using the Laplace transform, the probability of successful transmission betw een a transmitter and receiver in an int erference-limited Rayleigh channel is obtained. W e provide a nu merically integrable express ion for the outage pr o bability and clo sed-form upper and lower b ounds. The clustering gain G ( R ) is defined as the ratio of success proba bilities of the cluster ed pro ces s and the PPP with the same intensit y . It is obse rved that when the transmitter- receiver distance R is large, the clus ter ing gain G ( R ) is gr eater than unit y and bec o mes infinity as R → ∞ . The g ain G ( R ) at small R dep ends on the path lo ss mo del and the total intensit y of tra ns missions. W e pr ovide conditions o n the total intensit y of tr ansmitters under whic h the g ain is gre ater than unity for small R . This is useful to deter mine when logical c lus tering pe r forms b etter than uniform deploymen t of no des . W e also obtain the maximum intensit y of tra nsmitting no des for a given outage co nstraint, i.e. , the trans mis s ion capac it y [12], [14], [1 5] of this spatial arra ngement and show that it is equa l to that of a Poisson ar rangement of no des. W e observe that in May 29, 2018 DRAFT 4 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 λ p =1,c=3, σ =0.2 Transmitter at origin Intended receiver for transmitter at origin All empty circles indicate interfering transmitters R Fig. 1. Illustration of transmitters and receiv ers. Cluster density is 1. T ransmitter densit y in each cluster is 3. Spread of each cluster is Gaussian with standard deviation σ = 0 . 25. Observe that the i n tended receive r for the transmitter at the origin is not a part of the cl uster pro cess. The transmitter at the origin is a part of the cluster l ocated around the origin. a spread- s pe c tr um system, clustering is b eneficial for long range transmissio ns, and we compa re DS-CDMA and FH-CDMA. The pap er is o rganized as follows: in Section I I we pres ent the s ystem mo del and as sumptions, introduce the Neyman-Scott cluster pro cess and derive its conditional g enerating functional. In Sec tio n I I I we derive the prop erties o f interference, outa ge pro bability and the gain function G ( R ). In Section IV, w e derive the transmission ca pacity o f the cluster ed netw o rk. I I. System Mo del and Assu mptions In this section we intro duce the s ystem mo del a nd derive some r e quired r e sults for the Poisson cluster pro cess. A. System mo del and notation The lo cation of tra nsmitting no des is mo deled a s a stationary and isotr opic Poisson cluster pr o cess φ on R 2 . The receiver is no t considere d a part of the pr o cess. See Figure 1. Each trans mitter is assumed to transmit a t unit power. The p ow er r eceived by a r eceiver lo cated at z due to a transmitter a t x is mo deled as h x g ( x − z ), where h x is the p ower fading co efficient (squar e of the amplitude fading co efficient) a sso ciated with the channel b etw een the no des x and z . W e also assume that a ll the fading co efficients ar e indep endent and are dr awn from the sa me distribution. W e will so metimes use h to denote a r andom v a riable tha t is i.i.d with the power fading co efficients. Let { o } denote the o rigin (0 , 0). W e assume that the path loss mo del g ( x ) : R 2 \ { o } → R + satisfies the following co nditions. DRAFT May 29, 2018 5 1) g ( x ) is a contin uous, p o s itive, non- increasing function of k x k and  R 2 \ B ( o,ǫ ) g ( x )d x < ∞ , ∀ ǫ > 0 where B ( o, ǫ ) denotes a ball of radius ǫ around the o rigin. 2) lim k x k→∞ g ( x ) g ( x − y ) = 1 , ∀ y ∈ R 2 (1) g ( x ) is usually taken to b e a p ow er law in the form k x k − α , (1 + k x k α ) − 1 or min { 1 , k x k − α } . T o sa tisfy condition 1 , we r equire α > 2. The interference a t no de z o n the plane is g iven by I φ ( z ) = X x ∈ φ h x g ( x − z ) (2) The conditions requir ed for the existence o f I φ ( z ) are discussed in [3]. Let W denote the a dditive Gaus s ian noise the r eceiver. W e s ay that the communication from a transmitter at the or igin to a receiver situated at z is successful if and o nly if hg ( z ) W + I φ \{ x } ( z ) ≥ T (3) or e q uiv alently , hg ( z ) W + I φ ( z ) ≥ T 1 + T F or the calculation of outage pr obability a nd transmission capacity , the amplitude fading √ h x is assumed to be Ra yleigh with mean µ , but some re s ults are presented for the more ge neral case o f Nak agami- m fading. Hence the p ow ers h x are exp onentially and gamma distributed resp ectively . W e will b e ev aluating the p erfor mance of spread-sp ectr um in some sections o f the pap er. Even though we ev aluate spread-sp ectr um systems (sp ecifically DS-CDMA and FH-CDMA) we will not b e using any p ow e r control, the reason being that ther e is no central base station. Notation: If lim x →∞ f ( x ) /g ( x ) = C , we shall use f ( x ) ∼ g ( x ) if C = 1, f ( x ) . g ( x ) if 0 < C < 1 and f ( x ) & g ( x ) if 1 < C < ∞ . B. Neyman-Sc ott clu s t er pr o c esses Neyman-Scott cluster pro cesses [1 6] are Poisson clus ter pro cesses that res ult fro m homog eneous indep en- dent clus ter ing applied to a s tationary Poisson pro c ess, where the parent p oints form a stationa ry Poisson pro cess φ p = { x 1 , x 2 , . . . } of intensit y λ p . The clus ters a r e of the form N x i = N i + x i for ea ch x i ∈ φ p . The N i are a family of identical and indep endently distributed finite p oint sets with distribution indep endent o f the pa rent pro cess. The co mplete pro ces s φ is given by φ = [ x ∈ φ p N x . (4) Note that the par ent p o ints themselves are not included. The daughter p oints of the representative cluster N 0 are sc a ttered indep endently and with identical distribution F ( A ) =  A f ( x )d x, A ⊂ R 2 , ar ound the May 29, 2018 DRAFT 6 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 Fig. 2. (Left) Thomas cluster pro cess with parameters λ p = 1 , ¯ c = 5 and σ = 0 . 2. The crosses indicate the paren t p oints. (Righ t) PPP with the same i n tensit y λ = 5 for comparison. origin. W e also as sume that the scattering density of the daughter pro cess f ( x ) is isotropic. This makes the pro cess φ is otropic. The intensit y of the cluster pro cess is λ = λ p ¯ c , where ¯ c is the av erage num b er of p o ints in r epresentativ e cluster. W e further fo cus o n more s p ecific mo dels for the repr e sentativ e cluster, namely Matern cluster pr o cesses and T ho mas cluster pro cess es. In these pro c esses the num b er of p o in ts in the r epresentativ e cluster is Poisson distributed with mean ¯ c . F o r the Ma tern c lus ter pro ce s s each p oint is uniformly distributed in a ba ll o f ra dius a around the o rigin. So the density function f ( x ) is given by f ( x ) =      1 π a 2 , k x k ≤ a 0 otherwise . (5) In the Thomas clus ter pro cess ea ch p oint is scattered using a symmetric nor mal distribution with v ariance σ 2 around the o rigin. So the density function f ( x ) is given by f ( x ) = 1 2 π σ 2 exp  − k x k 2 2 σ 2  . A Thomas cluster pro cess is illustra ted in Fig.1. Newma n- Scott cluster pro cesses are also a Cox proces ses [1 6] when the num b er of p oints in the daughter cluster are Poisson distributed. The density of the driving r andom measure in this ca se is π ( y ) = ¯ c X x ∈ φ p f ( y − x ) Let E ! 0 ( . ) denote the exp ectation with r esp ect to the r educed Palm measure [1 6], [17]. It is basically the conditional exp ectation for p oint pro cesses, given the ther e is a p oint o f the pro cess at the or igin but without DRAFT May 29, 2018 7 including the p oint. Let v ( x ) : R 2 → [0 , 1] and  R 2 | 1 − v ( x ) | d x < ∞ . When φ is Poisson o f intensit y λ , the conditional generating functiona l is E ! 0   Y x ∈ φ v ( x )   = E   Y x ∈ φ v ( x )   = exp  − λ  R 2 [1 − v ( x )]d x  (6) The gener ating functional ˜ G ( v ) = E  Q x ∈ φ v ( x )  of the Neyman-Scott cluster pro ces s is g iven by [1 6], [18] ˜ G ( v ) = e xp  − λ p  R 2  1 − M   R 2 v ( x + y ) f ( y )d y  d x  where M ( z ) = P ∞ i =0 p n z n is the moment g e nerating function of the num b er of p oints in the representativ e cluster. When the num b er of p oints in the r epresentativ e cluster is Poisson with mean ¯ c , as in the c a se of Matern a nd Tho mas cluster pro ce s ses, M ( z ) = exp( − ¯ c (1 − z )) . The generating functiona l for the repre s ent ative cluster G c ( v ) is given by [18], [1 9] G c ( v ) = M   R 2 v ( x ) f ( x )d x  The reduced Palm distribution P ! 0 of a Neyman-Scott c lus ter pro c ess φ is given by [16]– [18], [20] P ! 0 = P ∗ ˜ Ω ! 0 (7) where P is the distribution of φ , a nd ˜ Ω ! 0 is the reduce d Palm distribution of the finite repres e ntative cluster pro cess N 0 . ” ∗ ” denotes the conv olution of distributions, which corr esp onds to the sup erp os itio n of φ and N 0 . The r educed Palm distribution ˜ Ω ! 0 is given b y ˜ Ω ! 0 ( Y ) = 1 ¯ c E  X x ∈ N 0 1 Y ( φ − x \ { 0 } )  (8) where φ x = φ + x , is a translated p oint pro ces s . W e require the following lemma to ev aluate the conditional Laplace transfor m of the interference. Let G ( v ) denote the conditiona l gener ating functional of the Neyman- Scott cluster pr o cess, i.e. , G ( v ) = E ! 0   Y x ∈ φ v ( x )   (9) W e will use a dot to indicate the v ariable which the functional is a cting o n. F or ex ample G ( v ( · − y )) = E ! 0 [ Q x ∈ φ v ( x − y )]. L emma 1: Le t 0 ≤ v ( x ) ≤ 1. The conditional gener ating functional o f Thomas and Mater n clustered pro cesses is G ( v ) = ˜ G ( v )  R 2 G c ( v ( · − y )) f ( y )d y . May 29, 2018 DRAFT 8 Pr o of: Let Y x = Y + x . F rom (8 ), we have ˜ Ω ! 0 ( Y ) = 1 ¯ c E  X x ∈ N 0 1 Y x ( φ \ { x } )  (10) Let Ω() denote the pro bability distribution of the r epresentativ e cluster. Using the Ca mpbell- Me cke theo- rem [16], we ge t ˜ Ω ! 0 ( Y ) = 1 ¯ c  R 2  N 1 Y x ( φ )Ω ! x (d φ ) ¯ c F (d x ) =  R 2  N 1 Y x ( φ )Ω ! x (d φ ) f ( x )d x (11) Here N denotes the spa ce of lo cally finite and s imple po int seq uences [16 ] o n R 2 . Since the re presentativ e cluster ha s a Poisson distribution of p oints, by Slivny a k’s theorem [16] we hav e Ω ! x ( . ) = Ω( . ). Hence ˜ Ω ! 0 ( Y ) =  R 2  N 1 Y x ( φ )Ω(d φ ) f ( x )d x =  R 2 Ω( Y x ) f ( x ) d x (12) F or notational conv enience let ψ denote N 0 . L e t ψ y = ψ + y . Using (7), we have G ( v ) =  N  N Y x ∈ φ ∪ ψ v ( x ) P (d φ ) ˜ Ω ! 0 (d ψ ) =  N Y x ∈ φ v ( x ) P (d φ )  N Y x ∈ ψ v ( x ) ˜ Ω ! 0 (d ψ ) = ˜ G ( v )  N Y x ∈ ψ v ( x ) ˜ Ω ! 0 (d ψ ) (13) ( a ) = ˜ G ( v )  N Y x ∈ ψ v ( x )  R 2 Ω(d ψ y ) f ( y )d y = ˜ G ( v )  R 2  N Y x ∈ ψ v ( x )Ω(d ψ y ) f ( y )d y = ˜ G ( v )  R 2  N Y x ∈ ψ v ( x − y )Ω(d ψ ) f ( y )d y ( b ) = ˜ G ( v )  R 2 G c ( v ( · − y )) f ( y )d y ( a ) follows fro m (1 2), and ( b ) follows from the definition of G ( . ). So from the ab ove lemma, we hav e G ( v ) = exp  − λ p  R 2 h 1 − M   R 2 v ( x + y ) f ( y )d y i d x  ×  R 2 M   R 2 v ( x − y ) f ( x )d x  f ( y )d y (14) The ab ov e equa tion holds when all the integrals are finite. Since f ( x ) = f ( − x ), then  R 2 v ( x + y ) f ( y )d y =  R 2 v ( x − y ) f ( y )d y = v ∗ f , so G ( v ) = exp  − λ p  R 2 h 1 − M (( v ∗ f )( x )) i d x   R 2 M (( v ∗ f )( y )) f ( y )d y (15) DRAFT May 29, 2018 9 Likelihoo d a nd ne a rest neighbor functions of the Poisson clus ter pro cess, which inv olve simila r calc ula tions with Palm distributio ns a re provided in [21]. One can obtain the nea r est-neighbor distribution function of Thomas o r Ma tern cluster pr o cess a s D ( r ) = G (1 B ( o,r ) c ( . )). In some cases the nu mber of p oints p er cluster may b e fixed r a ther than Poisson. The conditional g enerating functional, for this c ase is given in App endix B. I I I. Interference a nd Out age Probability of Poisson Cluster Processes In this section, we first der ive the characteristics of interference in a Poisso n clustered pro cess conditioned on the existence of a transmitting no de at the origin. W e then ev alua te the outage probability for a tr ansmit- receive pair when the transmitters are distributed as a Neyman-Sc o tt cluster pro cess, with the n umber of po int s in each cluster is Poisson with mean ¯ c and density function f ( x ). A. Pr op erties of the In t erfer enc e I φ ( z ) Let L h ( s ) denote the Laplace tra nsform o f the fading ra ndom v ariable h . L emma 2: The co nditional La place tra nsform of the interference is given by L I φ ( z ) ( s ) = G ( L h ( sg ( · − z ))) (16) Pr o of: F rom (2) we hav e L I φ ( z ) ( s ) = E ! 0 exp( − s X x ∈ φ h x g ( x − z )) = E ! 0   Y x ∈ φ exp( − sh x g ( x − z ))   ( a ) = E ! 0   Y x ∈ φ L h ( sg ( x − z ))   (17) where ( a ) follows fro m the indep endence of h x and (16) follows fro m (9). W e observe from Lemma 2 and (15), that the conditio na l Laplac e transfo rm of the interference L I φ ( z ) ( s ) depe nds on the p osition z . This implies that the distribution of the in terference dep ends o n the lo cation z a t which we obs erve the interference. This is in contrast to the fact that the interference distribution is independent of the lo cation z when the trans mitters are Poisson distributed on the plane [9], [12]. This is due to the non-statio narity of the reduce d Palm mea sure of the Neyma n-Scott clus ter pro cess es. If o ne interprets I φ ( z ) as a sto chastic pro cess, it is then a non stationa ry pro cess due to the ab ov e reason. Let K n ( B ) denote the reduced n -th factoria l mo ment measure [16], [18 ] of a po int pro cess ψ , and let B = B 1 × . . . × B n − 1 , B i ∈ R 2 . K n ( B ) = E ! 0   x i 6 = x j X x 1 ,...,x n − 1 ∈ ψ 1 B ( x 1 , . . . , x n − 1 )   (18) May 29, 2018 DRAFT 10 K 2 ( B (0 , R )), for example, denotes the average n umber of p oints inside a ball of ra dius R centered ar ound the origin, given that a p oint exists at the origin. Fir s t and sec ond moments of the int erference ca n b e determined using the seco nd and third order reduced factoria l moments. The av er age interference (co nditioned o n the even t tha t there is a p oint of the pro cess at the origin) is g iven by E ! 0 [ I φ ( z )] = E ! 0   X x ∈ φ h x g ( x − z )   = E [ h ] λ  R 2 g ( x − z ) K 2 (d x ) (19) Since the pr o cess φ is statio nary , K 2 ( B ) can b e expres sed as [16], [22] K 2 ( B ) = 1 λ 2  B ρ (2) ( x )d x, where ρ (2) ( x ) is the se cond order pr o duct dens ity 1 . So w e have E ! 0 [ I φ ( z )] = E [ h ] λ  R 2 g ( x − z ) ρ (2) ( x ) d x (20) Example: Thomas Cluster Pr o c ess . In this ca s e, from [16 ] ρ (2) ( x ) λ 2 = 1 + 1 4 π λ p σ 2 exp  −k x k 2 4 σ 2  where λ = λ p ¯ c . W e obtain E ! 0 [ I φ ( z )] = E I Poi ( λ ) + ¯ cE [ h ] 4 π σ 2  R 2 g ( x − z ) exp  −k x k 2 4 σ 2  d x (21) Where E I Poi ( λ ) is the average interference seen by a receiver lo cated at z , when the no des are distributed as a P PP with intensit y λ . The ab ove expression als o shows that the mean interference 2 is indeed la rger than for the PPP . One can also get the ab ove from the co nditional Lapla c e tra nsform in Lemma 2 and using E ! 0 [ I φ ( z )] = − d ds L I φ ( z ) ( s ) | s =0 . In the following theorem we provide bounds to the tail pr obability o f the interference I φ ( z ) for any stationary distribution φ of transmitters. W e adapt the techn ique presented in [15] to derive the tail b ounds of the interference. W e deno te the tail probability (CCDF) of I φ ( z ) by ¯ F I ( y ) = P ( I φ ( z ) ≥ y ). The or em 1: When the transmitters a re distributed as a sta tionary and isotropic point pr o cess φ of in tensity λ with conditional g enerating functiona l G and s econd o rder pro duct dens ity ρ (2) , the tail probability ¯ F I ( y ) of the interference at lo ca tion z , conditioned on a transmitter pres ent a t the orig in 3 is low er b ounded by 1 In tuitiv ely , this indicates the probability th at there are t wo points separated by k x k . F or PPP , it is ρ (2) ( x ) = λ 2 independent of x . Also the second or der pro duct density is a function of tw o argumen ts i. e. , ρ (2) ( x 1 , , x 2 ). But when the pro cess φ is stationary , ρ (2) depends only on the difference of i ts arguments i .e. , ρ (2) ( x 1 , x 2 ) = ν ( x 1 − x 2 ) for al l x 1 , x 2 ∈ R 2 . F urthermore i f φ is motion-inv ar i an t, i.e., stationary and isotropic, then ν depends only on k x 1 − x 2 k [16, pg 112]. 2 Note that for g ( x ) = k x k − α , E ! 0 [ I φ ( z )] is diverging. 3 W e do not include the cont ribution of the transmitter at the origin in the interference. This is b ecause the transmitter at the origin is the inten ded transmitter which we f ocus on. DRAFT May 29, 2018 11 ¯ F l I ( y ) and upp er b ounded by ¯ F u I ( y ), where ¯ F l I ( y ) = 1 − G  F h  y g ( . − z )  (22) ¯ F u I ( y ) = 1 − (1 − ϕ ( y )) G  F h  y g ( . − z )  (23) where F h ( x ) denotes the CDF of the p ow e r fading co efficient h a nd ϕ ( y ) = 1 y λ  R 2 g ( x − z ) ρ (2) ( x )  y /g ( x − z ) 0 ν d F h ( ν )d x. Pr o of: The basic idea is to divide the transmitter set into tw o s ubs ets φ y and φ c y where, φ y = { x ∈ φ, h x g ( x − z ) > y } (24) φ c y = { x ∈ φ, h x g ( x − z ) ≤ y } (25) φ y consists of those transmitter s , who se contribution to the interference e x ceeds y . W e have I φ ( z ) = I φ y ( z ) + I φ c y ( z ), where I φ y ( z ) corr esp onds to the interference due to the transmitter set φ y and I φ c y ( z ) corr esp onds to the interference due to the trans mitter set φ c y . Hence we have ¯ F I ( y ) = P ( I φ y ( z ) + I φ c y ( z ) ≥ y ) ≥ P ( I φ y ( z ) ≥ y ) = 1 − P ( I φ y ( z ) < y ) = 1 − P ( φ y = ∅ ) . (26) W e can ev aluate the probability P ( φ y = ∅ ) that φ y is empty using the co nditional Laplace functional a s follows: P ( φ y = ∅ ) = E ! 0 Y x ∈ φ 1 h x g ( x − z ) ≤ y ( a ) = E ! 0 Y x ∈ φ E h x  1 h x g ( x − z ) ≤ y  = E ! 0 Y x ∈ φ F h  y g ( x − z )  = G  F h  y g ( · − z )  , (27) where ( a ) follows fro m the indep endence of h x . T o obtain the upp er b ound ¯ F I ( y ) = P ( I φ > y | I φ y > y ) ¯ F l I ( y ) + P ( I φ > y | I φ y ≤ y )(1 − ¯ F l I ( y )) ( a ) = 1 − G  F h  y g ( · − z )  + P ( I φ > y | I φ y ≤ y ) G  F h  y g ( · − z )  = 1 − (1 − P ( I φ > y | I φ y ≤ y )) G  F h  y g ( · − z )  (28) May 29, 2018 DRAFT 12 where ( a ) fo llows fro m the lower b ound we hav e establis he d. T o ev a lua te P ( I φ > y | I φ y ≤ y ) we use the Marko v inequa lit y (the Cheb eshev inequalit y can also b e used but is more difficult to b e ev aluated in this particular s e tting). W e hav e P ( I φ > y | I φ y ≤ y ) = P ( I φ > y | φ y = ∅ ) ( a ) ≤ E ! 0 ( I φ | φ y = ∅ ) y = 1 y E ! 0 X x ∈ φ h x g ( x − z )1 h x g ( x − z ) ≤ y = 1 y E ! 0 X x ∈ φ g ( x − z )  y /g ( x − z ) 0 ν d F h ( ν ) ( b ) = 1 y λ  R 2 g ( x − z )  y /g ( x − z ) 0 ν d F h ( ν ) ρ (2) ( x )d x (29) ( a ) follows from the Ma rko v inequality , and ( b ) follows from a pro c e dure similar to the calculation of the mean int erference in (20). In the pro o f o f Lemma 3, we show ϕ ( y ) ∼ θ 2 y − 2 /α when g ( x ) = k x k − α . This indicates the tightness of the bo unds for la r ge y . Lemma 3 shows that the interference is a heavy-tailed dis tribution with pa rameter 2 / α when the no des a re distr ibuted as a Neyman-Scott cluster pro cess. L emma 3: F or g ( x ) = k x k − α , the lower and upp er b ounds to CCDF ¯ F I ( y ) of the int erference at lo cation z , when the no des are distributed as a Neyman-Scott c lus ter pro c ess sca le as follows for y → ∞ . ¯ F l I ( y ) ∼ θ 1 y − 2 /α (30) ¯ F u I ( y ) ∼ ( θ 1 + θ 2 ) y − 2 /α (31) where θ 1 = π ¯ c [( f ∗ f )( z ) + λ p ]  ∞ 0 ν 2 /α d F h ( ν ) and θ 2 = 2 θ 1 / ( α − 2). Pr o of: See App endix A. Remarks: 1) Observe that θ 1 = π λ − 1 ρ (2) ( z )  ∞ 0 ν 2 /α d F h ( ν ). A s imilar k ind o f scaling law with θ 1 = π λ − 1 ρ (2) ( z ) E h [ ν 2 /α ] and θ 2 = 2 θ 1 / ( α − 2) can b e obtained when the transmitters a r e sca tter ed as an y “nice ” 4 stationary , isotropic p oint pro cess with intensit y λ and second order pro duct density ρ (2) ( x ) 6 = 0 at x = z . 2) A simila r heavy-tailed distribution with para meter 2 /α was obtained for Poisson interference in [1], [15]. Since 2 /α < 1, the mean and hence the v aria nce diverge. This ca n a ls o b e inferr ed fro m (21) a nd is due to the singula r ity of the channel function g ( x ) = k x k − α at the o rigin. F o r Matern cluster pro c esses 4 W e require the conditional generating functional to ha ve a series expansion with r espect to reduced n -th factorial m omen t measures of the reduced Palm distribution [22] si milar to that of the expansion of generating functional [16, p.116] and [23]. The proof of the existence and the series expansion of the conditional generating functional with resp ect to reduced n - th f actorial momen t measures, would be of more technical nature following a tec hnique used in [23]. If such an expansion exists it is straight forward to pro ve the scaling la w s f or the CCDF of int erference similar to Lemma 3, with θ 1 = π λ − 1 ρ (2) ( z ) E h [ ν 2 /α ] and θ 2 = 2 θ 1 / ( α − 2). DRAFT May 29, 2018 13 ( f ∗ f )( z ) = 0, for k z k > 2 a a nd for Thomas cluster pro ces ses ( f ∗ f )( z ) is a Gaussian with v ar iance 2 σ 2 . Hence for lar ge z , we observe that the constants θ 1 bec ome similar to that of the unconditional int erference. This is b ecause, the contribution of the cluster at origin b ecomes small as we mov e far from the or ig in. 3) When the path loss function is g ( x ) = (1 + k x k α ) − 1 , the distribution of the interference mo re s trongly depe nds on the fading mo del. Using a simila r pro of a s in Lemma 3, one can deduce an exp onential tail decay when g ( x ) = (1 + k x k α ) − 1 and Rayleigh fading. Similar ly if the p ow er fading co efficient 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y CCDF of the interference σ =0.25, λ p =2,c=3,R=0.3, α =4 1 2 4 3 Fig. 3. λ p = 2 , ¯ c = 3 , σ = 0 . 25 , α = 4 , R = 0 . 3: Compari s on of the inte rference CCDF for different path-loss mo dels and differen t fading. They w ere generated using Monte -Carlo simulation. Curves #1 and #2 corresp ond to g ( x ) = (1 + k x k α ) − 1 . Curve # 1 corresp onds to Rayleigh fading and exhibits an exp onen tial decay . Cur v e #2 for whi c h h i s distributed as generalized Pa reto with parameters k = 1 , θ = 0 , σ p = 1 (a hypothetical p ow er fading distribution which exhibits p ow er l a w decay ) exhibits a p ow er law deca y . Curves #3 (generalized Pareto ) and #4 (Rayleigh) corresp ond to g ( x ) = k x k − α and exhibit a hea vy tail for b oth fading distri butions. follows a p ow er-law distribution with exp onent k , the tail of the interference shows a p ow er -law decay . This is b ecaus e of the presence o f the term y − 2 /α  ∞ y [1 − F h ( u )]( u − y ) 2 /α − 1 d u in the pro of. So when using non-singular channel mo dels , the interference has a more intricate de p endence on the fading characteristics r a ther than a simple dep endence o n E h [ ν 2 /α ] as in the singula r case. This b ehavior is well understo o d for Poisson and unco nditio nal Poisson cluster shot noise pr o cess [24 ], [25]. The prop erties of interference for different path loss mo dels with no fading , when the no des a re unifor mly distributed ar e discussed in [26 ]. May 29, 2018 DRAFT 14 B. Suc c ess pr ob ability: P ( suc c ess ) Let the desir ed transmitter b e lo cated at the origin and the receiver at lo cation z a t distance R = k z k from the transmitter . With a slight a buse of notation we shall b e using R to denote the p oint ( R, 0). The probability o f succes s for this pair is given by P (success) = P !0  hg ( z ) W + I φ ( z ) ≥ T  (32) W e now a ssume Ra yleigh fading, i.e. , the received p ow er is exp o ne ntially distributed with mean µ . So w e hav e P (success) =  ∞ 0 e − µsT /g ( z ) d P ( W + I φ \{ 0 } ( z ) ≤ s ) = L I φ ( z ) ( µT /g ( z )) L W ( µT /g ( z )) , (33) When h x is Rayleigh we have L h ( sg ( x − z )) = µ µ + sg ( x − z ) (34) A t s = µT /g ( R ) w e obser ve that the ab ov e expression will be indep endent o f the mea n o f the expo nential distribution µ . L emma 4: [Succe s s pr o bability] The pro bability of successful transmis s ion b etw een the tra nsmitter at the origin and the receiver lo cated at z ∈ R 2 , whe n W ≡ 0 (no no ise), is given by P (success) = exp n − λ p  R 2 h 1 − exp( − ¯ cβ ( z , y )) i d y o | {z } T 1 ×  R 2 exp( − ¯ cβ ( z , y )) f ( y )d y | {z } T 2 (35) where β ( z , y ) =  R 2 g ( x − y − z ) g ( z ) T + g ( x − y − z ) f ( x )d x (36) Pr o of: F ollows from (34) a nd Le mma 2. The success proba bilit y , when the num ber of no des in each cluster is fixed is g iven in the App endix B. See Figure 4 for co mpa rison. When the fading is Nak aga mi- m , the probability of suc c ess is ev a luated in the Appendix C for integer m . R emarks: 1) The term T 1 in (35) captures the interference without the c luster at the or ig in ( i.e. , without condi- tioning); it is indep endent 5 of the p osition z since the or iginal cluster pro cess is stationary (can b e verified by change of v a r iables y 1 = y + z ). The seco nd term T 2 is the contribution of the transmitter’s 5 By this we mean the unco nditional i nterference distribution whic h leads to this term do es not dep end on the lo cation z . The term T 1 does depend on g ( z ). DRAFT May 29, 2018 15 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c P(Success) λ p =0.5 α =4,T=0.5, σ =0.25,R=0.5 Cluster points~Poi(c) Fixed number of cluster points Fig. 4. Comparison of P (success) when the num b er of p oints i n a cluster are fixed and Poisson distributed with parameter ¯ c . cluster; it is iden tical for all z with k z k = R s ince f and g are is otropic. So the s uccess pr obability itself is the same for all z at distance R . This is b ecause the Palm distribution is a lwa ys isotropic w he n the original distribution is motion-inv ariant [16 ]. Hence we shall use β ( R, y ) to denote β ( z , y ) where z = Re iθ . W e sha ll a lso use R a nd ( R , 0) interc hangeably a nd will b e clear by the context. 2) F rom the ab ov e arg ument we observe that P (succes s) depe nds only o n k z k = R and not on the angle of z . So the succe s s probability sho uld be interpreted as a n av erage ov er the circle k z k = R , i.e. , the receiver may b e unifo r mly lo c ated a nywhere o n the cir cle o f ra dius R around the or igin. F or lar ge distances R , ther e is a very high probability that the receiver is lo ca ted in an empt y spa ce and not in any clus ter. Hence for larg e R the success probability is higher than that of a PPP of the same intensit y . If the receiver is also conditioned to b e in a cluster, w e hav e to multiply (at lea st heuristica lly ) by a term that is similar to T 2 and this would sig nificantly reduce the success probability . 3) F rom L e mma 4, we hav e P (suc c ess) = E ! 0 [exp( − sI φ ( z ))] ev aluated a t s = µT /g ( z ). If µT /g ( z ) is smal l, and  g ( z ) d z < ∞ (i.e, finite av er a ge interference) then, P (success) ≤ P p ( λ ). This follows fro m (21) and the fact that E ! 0 [ I φ ( z )] is the slop e of the curve E ! 0 [exp( − sI φ ( z ))] at s = 0 . This implies that at small distances, spr ead spectr um (DS-CDMA) works b etter with a Poisson distribution of no des. (If the distance R is larg e, then the spreading gain has to increase appr oximately like g ( z ) to keep µT /g ( z ) small.) 4) Let M be the DS-CDMA spreading factor. W e have P (outage) = P ( I φ ( z ) > M T h x g ( z )). F o r g ( x ) = k x k − α , we hav e the following sca ling law for the o utage probability with resp ect to the spr eading ga in. θ 1 R 2 M − 2 /α T 2 /α E h [ ν − 2 /α ] ( a ) . P (outage) ( b ) . α α − 2 θ 1 R 2 M − 2 /α T 2 /α E h [ ν − 2 /α ] , (37) May 29, 2018 DRAFT 16 where E h [ ν − 2 /α ] =  ∞ 0 ν − 2 /α d F h 2 ( ν ). ( a ) a nd ( b ) follow from Lemma 3 . Also obser ve that these scaling bo unds a re v alid for any fading distribution for which E h [ ν − 2 /α ] < ∞ . Similar scaling la ws with the exp onent of M b eing − 2 /α can b e obtained when the transmitters are Poisson distributed o n the plane. When the fading is Rayleigh i.e. , h ∼ ex p( µ ), the low er b ound is π ¯ c [( f ∗ f )( R ) + λ p ]Γ  1 + 2 α  Γ  1 − 2 α  R 2 M − 2 /α T 2 /α and the upp er b ound is α/ ( α − 2 ) times the low er bound. Γ( z ) represents the standard Gamma function. W e now derive closed form upper and lower b ounds on P (success). L emma 5: [Lower b o und] P (success) ≥ P p ( λ ) P p (¯ c ˆ f ∗ ) (38) where P p ( λ ) denotes the success probability when φ is a PPP , ˆ f ∗ = sup y ∈ R 2 ( f ∗ f )( y ), and λ = λ p ¯ c . Pr o of: The first facto r in (35), T 1 can b e low er b o unded by the success pr o bability in the sta ndard PP P P p ( λ ), and the seco nd factor can b e low er b ounded by P p (¯ c ˆ f ∗ ). F rom (35) a nd the fact that 1 − exp( − δ x ) ≤ δ x, δ ≥ 0, we have P (success) ≥ exp  − λ p ¯ c  R 2 β ( R , y )d y  | {z } T erm1 (39) ×  R 2 exp( − ¯ cβ ( R, y )) f ( y )d y | {z } T erm2 T erm1 = exp  − λ  R 2 β ( R , y )d y  ( a ) = exp  − λ  R 2 g ( y ) g ( R ) T + g ( y ) d y  = P p ( λ ) (40) ( a ) follows fro m change o f v aria bles, interchanging integrals and using  f ( x ) = 1. T erm2 =  R 2 exp( − ¯ cβ ( R, y )) f ( y )d y Since exp( − x ) is co nv ex and f ( x ) > 0 ,  f ( x ) = 1, Using J ensen’s ineq ua lity ( E f ( x ) ≥ f ( E ( x ))) w e hav e, T erm2 ≥ exp  − ¯ c  R 2 β ( R , y ) f ( y )d y  Changing v ariables a nd us ing f ( x ) = f ( − x ),we get, T erm2 ≥ exp  − ¯ c  R 2 g ( x ) g ( R ) T + g ( x )  R 2 f ( x + z − y ) f ( y )d y d x  ≥ exp  − ¯ c  R 2 g ( x ) g ( R ) T + g ( x ) ( f ∗ f )( x + z )d x  (41) Hence T erm2 ≥ P p (¯ c ˆ f ∗ ) (42) DRAFT May 29, 2018 17 Since f ∈ L p , by Y oung’s inequality [27] we ha ve ˆ f ∗ ≤ k f k p k f k q , where 1 /p + 1 /q = 1 (conjugate exp o nents). F or a ≥ 1 / √ π (Matern) and σ ≥ 1 / √ 2 π (Thoma s), we get P (success) ≥ P p ( λ ) P p (¯ c ). In ge neral, ˆ f ∗ ≤ k f k ∞ k f k 1 , whic h is 1 /π a 2 for Matern and 1 / 2 π σ 2 for Thomas pro cesses. In the latter case, when f is Gaussian, f ∗ f is also Gaussian with v aria nce 2 σ 2 , hence ˆ f ∗ ≤ 1 / 4 π σ 2 . F rom [9], we ge t (by change o f v ariables): P p ( λ ) = exp  − λ  R 2 β ( R , y )d y  . (43) W e hav e • for g ( x ) = k x k − α , P p ( λ ) = exp( − λR 2 T 2 /α C ( α )) [9], where C ( α ) =  2 π Γ(2 /α )Γ(1 − 2 /α )  /α = 2 π 2 α csc(2 π /α ). • for g ( x ) = (1 + k x k α ) − 1 , P p ( λ ) = exp( − λT C ( α )( T + g ( R )) 2 /α − 1 g ( R ) − 2 /α ). Let β I =  R 2 β ( R , y )d y , ˆ β = sup y ∈ R 2 β ( R , y ) and ˆ f = sup y ∈ R 2 f ( y ). By H ¨ olders ineq uality we hav e ˆ β ≤ min { 1 , ˆ f β I ( R ) } . Also let κ =  R 2 β ( R , y ) f ( y )d y . L emma 6: [Upp er b ound] P (success) ≤ P p  λ 1 + ¯ c ˆ β  (44) Pr o of: Neglecting the seco nd term T 2 and using exp( − δ x ) ≤ 1 / (1 + δx ), w e have P (success) ≤ exp  − λ p  R 2 h 1 − 1 1 + ¯ c β ( R , y ) i d y  = exp  − λ p  R 2 ¯ cβ ( R , y ) 1 + ¯ c β ( R , y ) d y  ≤ exp  − λ p ¯ c 1 + ¯ c ˆ β  R 2 β ( R , y )d y  (45) F rom the ab ov e tw o lemmata, we get P p ( λ ) P p (¯ c ˆ f ∗ ) ≤ P (success) ≤ P p  λ 1 + ¯ c ˆ β  (46) from which follows P (success) → P p ( λ ) as ¯ c σ , ¯ c a → 0 as exp ected. In Lemma 6, we have neglected the contribution of the tra nsmitter’s cluster . W e der ive the following upp er b ound in the pro of of L e mma 8, P (success) ≤ P p ( λ ) exp  λβ I ν ( ¯ c ˆ β )  h 1 −  1 − ν ( ¯ c ˆ β )  ¯ cκ i (47) where ν ( x ) = (exp( − x ) − 1 + x ) /x . Substituting for ν ( x ), we hav e P (success) ≤ P p λ (1 − exp( − ¯ c ˆ β )) ¯ c ˆ β !  1 −  1 − exp( − ¯ c ˆ β )  κ ˆ β  (48) (48) is a tighter b ound than the bo und in Lemma 6, but not ea sily computable due to the presence of κ (for a g iven R , T and σ , κ a nd β ∗ are cons tants). In (48), the o utage due to the interference by the trans mitting cluster is a lso taken into ac c ount. May 29, 2018 DRAFT 18 The pro of of Lemmata 5 and 6 also indicates that it is only by conditioning on the event that there is a p oint at the or igin that the success probability o f Neyma n-Scott cluster pr o cesses can b e low e r than the Poisson pro cess o f the same intensit y . This implies that the cluster around the tra nsmitter ca uses the maximum “damage” . So as the receiver mov es aw ay from the transmitter, the Neyman-Scott c luster pr o cess has a b etter success probability than the P PP . So, it is not true in genera l that cluster pro ce s ses hav e a low er success probability than PP Ps of the s a me intensit y . F or example from Figure 5, we se e tha t for 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R P(non−outage) α =4, T=0.1, a=0.6 Cluster λ p =1, c=2 Poisson , λ =2 Cluster, λ p =1,c=2.5 Poisson, λ =2.5 Fig. 5. Comparison of success probability f or cluster and Poisson process of inten sity 2 R < 0 . 8, the PPP has a b etter success pro bability than the Matern pro cess. In Subsection II I-C we g ive a more detailed analys is, which reveals that a PPP with intensit y λ p ¯ c has a lower s uccess proba bility than a clustered pro ce s s of the s a me intensit y for lar ge t r ansmit-r e c eiver distanc es. On the o ther hand, for s mall R , the s uc c e ss pro bability o f the PP P is higher. C. Clustering Gain G ( R ) In this subs e ction we compa re the p erfor mances of a cluster ed netw or k a nd a Poisson netw ork of the same int ensity with Rayleigh fading . W e deduce how the cluster ing gain dep ends on the tr ansmitter receiver distance. W e use the fo llowing notation, P 1 ( R, ¯ c, λ p ) ∆ = exp  − λ p  R 2 1 − exp( − ¯ cβ ( R, y ))d y  (49) P 2 ( R, ¯ c ) ∆ =  R 2 exp( − ¯ cβ ( R, y )) f ( y )d y (50) So P (success) = P 1 ( R, ¯ c, λ p ) P 2 ( R, ¯ c ). P 2 is the pr obability of success due to the pres ence of the cluster at the origin near the tra ns mitter. P 1 is the probability of success in the pr e s ence o f other clusters. Interference from these o ther clusters contributes mor e to the o utage when R is lar ge. This is a lso intuit ive, sinc e as DRAFT May 29, 2018 19 the receiver mov e s aw ay from the tra nsmitting cluster , the interference fro m the o ther clus ter s s tarts to dominate. W e define the clust ering gain G ( R ) a s G ( R ) = P 1 ( R, ¯ c, λ p ) P 2 ( R, ¯ c ) P p ( λ p ¯ c ) The fluctuation of G ( R ) around unity indicates the e x istence of a cros sov er p oint R ∗ below which the PP P per forms b etter than clustered pro ces s and vice versa. The v alues of G ( R ) at the origin and infinity indica te the gain of scheduling tra nsmitters as clusters ins tead of b eing spre a d unifor mly o n the pla ne. So it is bene fic ia l to induce logic a l clustering of trans mitter s by MAC if G ( R ) > 1 . W e firs t consider G ( R ) for large R , i.e. , lim R →∞ G ( R ). By the dominated conv er gence theorem a nd (1), we hav e lim R →∞ P 2 ( R, ¯ c ) =  R 2 exp  − ¯ c  R 2 lim R →∞ f ( x ) 1 + g ( R ) T g ( x − y − R ) d x  f ( y )d y = exp  − ¯ c 1 + 1 /T  (51) Also fro m the deriv atio n of upp er b ound we hav e P 1 ( R, ¯ c, λ p ) ≤ P p  λ 1+¯ c ˆ β  . Hence fro m the definition of P p ( x ) we have, lim R →∞ P 1 ( R, ¯ c, λ p ) = 0. Hence for large R , P 1 ( R, ¯ c, λ p ) < P 2 ( R, ¯ c ) (52) So for large R , most of the damage is do ne by transmitting no des other tha n the clus ter in which the in tended transmitter lies. L emma 7: lim R →∞ P p ( λ p ¯ c ) P 1 ( R, ¯ c, λ p ) = 0 (53) Pr o of: See App endix D Hence fo r large R , P p ( λ p ¯ c ) P 1 ( R, ¯ c,λ p ) ≤ exp( − ¯ c 1+1 /T ). F ro m (5 1) w e have P p ( λ p ¯ c ) ≤ P 1 ( R, ¯ c, λ p ) P 2 ( R, ¯ c ), for large R , i.e. , G ( R ) > 1 for lar ge transmit-r eceive distance. W e hav e lim R →∞ G ( R ) = ∞ . Hence the Poisson p oint pr o c ess with int ensity λ p ¯ c , has a lower su c c ess pr ob ability than the clu s ter e d pr o c ess of the same intensity for lar ge tr ansmit r e c eiver distanc es. F or small R , G ( R ) dep ends o n the behavior of the pa th loss function, g ( x ) at k x k = 0. W e conside r the t wo cases when the channel function is singular at the orig in or no t. 1) lim k x k→ 0 g ( x ) = ∞ : In this case we observe that G (0) = 1 . But at sma ll R , G ( R ) is less than 1. W e hav e the following lemma. L emma 8: If ( f ∗ f )( x ) > k x k fo r smal l k x k and g ( x ) = k x k − α , then for sm al l R , P (success) ≤ P p ( λ p ¯ c ) (54) Pr o of: See App endix E. Note that f ( x ) for Matern and Tho mas clus ter pro c e ss have the required pr op erty . Hence when g ( x ) = k x k − α , the PPP with intensity λ p ¯ c , has a higher suc c ess pr ob ability than the cluster e d pr o c ess of t he same int en sity May 29, 2018 DRAFT 20 for smal l tr ansmit r e c eiver distanc e . Lemma 8 and the fact that G ( ∞ ) = ∞ also indicate the existence of a crossover point R ∗ betw een the succes s cur ves of the PP P a nd the cluster pr o cess. So it is no t true in gener al that the p erfo r mance of the clustered pro ces s is b etter or worse than that of the Poisson pro cess. This is bec ause, for the same intensit y , a clustered pro cess will have clusters of transmitters (where interference is high) and also v aca nt are as (where there are no tra nsmitters a nd interference is low), whereas in a Poisson pro cess, the transmitters are uniformly sprea d. 2) lim k x k→ 0 g ( x ) = ˆ g < ∞ : P 1 ( R, ¯ c, λ p ) can b e wr itten as P 1 ( R, ¯ c, λ p ) = P p ( λ p ¯ c ) exp  λ p  R 2 ∞ X n =2 ( − 1) n n ! ¯ c n β ( R , y ) n | {z } > 0 d y  Hence G ( R ) can also be wr itten a s follows G ( R ) = P 2 ( R, ¯ c ) exp  λ p ¯ c  R 2 β ( R , y ) η ( ¯ c, R, y )d y  (55) where η ( ¯ c, R, y ) = ν ( ¯ cβ ( R, y )), with ν ( x ) = (exp( − x ) − 1 + x ) /x . Observe that 0 ≤ η ( ¯ c, R , y ) ≤ 1 , ∀ x > 0. If the tota l density of the transmitters is fixed i.e. , λ = λ p ¯ c is consta n t, how do es G ( R ) b ehave with resp ect to ¯ c ? W e have the following lemma which characterizes the monotonicity o f G ( R ) with resp ect to ¯ c . L emma 9: Given λ = λ p ¯ c is c o nstant, G ( R ) is dec r easing with ¯ c , i.e. , d G ( R ) d¯ c ≤ 0 , ∀ ¯ c > 0 iff λ ≤ λ ∗ ( R, T ), where λ ∗ ( R, T ) = 2  R 2 β ( R , y ) f ( y )d y  R 2 β ( R , y ) 2 d y Pr o of: F rom (55), G ( R ) = P 2 ( R, ¯ c ) e xp  λ p ¯ c  R 2 β ( R , a ) η ( ¯ c , R, a )d a  =  R 2 exp  − ¯ cβ ( R, y ) + λ  R 2 β ( R , a ) η ( ¯ c , R, a )d a  f ( y )d y (56) W e hav e d η (¯ c,R,z ) d ¯ c | ¯ c =0 = β ( R, z ) / 2 and d η (¯ c,R,z ) d ¯ c is decreasing in ¯ c . d G ( R ) d¯ c =  R 2  − β ( R, y ) + λ  R 2 β ( R , a ) d η ( ¯ c, R, a ) d¯ c d a  exp  − ¯ cβ ( R, y ) + λ  R 2 β ( R , a ) η ( ¯ c , R, a )d a  f ( y )d y = exp[ λ  R 2 β ( R , a ) η ( ¯ c , R, a )d a ]  R 2 h − β ( R , y ) + λ  R 2 β ( R , z ) d η ( ¯ c , R, a ) d¯ c d a i exp  − ¯ cβ ( R, y )  f ( y )d y | {z } T 2 (¯ c ) Since η ′ (¯ c, R , z ) is decreasing in ¯ c , w e hav e T 2 (¯ c ) is decr easing in ¯ c . So a necessar y and sufficien t condition for dG ( R ) d ¯ c ≤ 0 ∀ ¯ c > 0 is T 2 (0) ≤ 0. W e wan t T 2 (0) =  R 2  − β ( R, y ) + λ 2  R 2 β 2 ( R, z )d z  f ( y )d y ≤ 0 ⇒ λ ≤ 2  R 2 β ( R , y ) f ( y )d y  R 2 β 2 ( R, z )d z (57) Remarks: DRAFT May 29, 2018 21 1) Since β (0 , y ) 6 = 0 , we hav e that, G (0) is incr easing with λ p (like exp( λ p )), and hence will be grea ter than 1 at so me λ p for a fix e d ¯ c . 2) W e hav e lim ¯ c → 0 G ( R ) = 1 a nd sp ecifically G (0) = 1 a t ¯ c = 0 . 3) F rom Lemma 9 and Remar k 2 we can deduce G ( R ) < 1 , ∀ ¯ c > 0 if λ < λ ∗ ( R, T ) i.e. , the gain G ( R ) decreases fr om 1 with increa sing ¯ c if the total intensit y o f transmitters is less than λ ∗ ( R, T ). 4) Since G ( R ) is co n tinuous w ith re s pe c t to R , G ( R ) is clo s e to G (0) for small R . 5) F rom Figure 7, we o bserve that G ( R ) increases mono tonically with R . In Figure 6, λ ∗ (0 , T ) is plotted ag a inst T . W e pr ovide so me heuristics as to when log ical clustering do es no t per form b etter than a unifor m distribution of p oints: • The exact v alue of R a t which G ( R ) c rosses 1 is difficult to find a nalytically due to the highly nonlinear nature of G ( R ). If s uch a cro ssov er p oint ex is ts (dep ends on the path-loss mo del) we will denote it by R ∗ . • If g ( x ) = k x k − α , it is b etter to induce logical cluster ing by the MAC scheme if the link distanc e is lar g er than R ∗ . Otherwise it is b etter to s chedule the transmissions so that they ar e s c attered uniformly on the pla ne. • If g (0) < ∞ and for a co nstant intensit y λ p ¯ c , it is a lwa ys b eneficial to induce clustering for lo ng-hop transmissions . When R is small the answer dep ends on the total int ensity λ p ¯ c . If λ p ¯ c < λ ∗ (0 , T ) then G (0) < 1 by obser v ation 3, and hence G ( R ) < 1 for sm al l R by obser v ation 4. Also when λ p ¯ c < λ ∗ (0 , T ), it is b etter to r e duce lo gical clustering by decr easing ¯ c and increasing λ p , since G (0) is a decreas ing function of ¯ c . F r om Figure 6 we observe tha t λ ∗ (0 , 0 . 5) ≈ 1 . 26 when g ( x ) = (1 + k x k 4 ) − 1 and σ = 0 . 25. In Figure 7, G ( R ) is plotted for λ p ¯ c = 0 . 7 5 , 9 for the sa me v alues of σ, α and the sa me channel function as of Figure 6. When λ p ¯ c = 9 > λ ∗ (0 , 0 . 5), we obs erve tha t the gain curve G ( R ) is approximately 10 at the orig in a nd increas es. When λ p ¯ c = 0 . 75 < λ ∗ (0 , 0 . 5), G ( R ) star ts around 0 . 2 5 and cross e s 1 at R ≈ 1 . 2 . W e also observe that G ( R ), for the non-sing ular g ( x ), s eem to increa s e mono tonically . W e also observe that the g a in function for g ( x ) = k x k − α decreases fro m 1 initially and then increases to infinity . • F or DS-CDMA, the v alue of T is sma lle r by a factor e q ual to the spre ading gain. F ro m Figure 6, we observe that the threshold λ ∗ (0 , T ) for cluster ing to b e b eneficia l a t small distance s increa s es with decreasing T . Hence for a co ns tant intensit y of transmiss io ns λ p ¯ c , the b enefit of clustering decr eases with increasing spr e a ding ga in for small link distances. So fo r DS-CDMA (for a large s preading gain) it is better to make the trans missions unifor m on the plane for smaller link distances and cluster the transmitters for lo ng-rang e communication. • F or FH-CDMA, the tota l num b er of transmis s ions λ p ¯ c is reduced by the sprea ding gain while T remains constant (see Figur e 6). Hence λ p ¯ c < λ ∗ (0 , T ) for small distances and o ne can draw similar conclusions as that of DS-CDMA. The relative gain be t ween FH-CDMA a nd DS-CDMA with clustering is more difficult to characterize analy tically . May 29, 2018 DRAFT 22 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −1 10 0 10 1 T λ * (0,T) λ * (0,T), σ =0.25, g(x)=(1+|x| α ) −1 α =4, σ =0.25 Normal operating point FH−CDMA DS−CDMA G(0) <1 λ * (0,0.5) ≈ 1.26 Fig. 6. λ ∗ (0 , T ) versus T for g ( x ) = (1 + k x k 4 ) − 1 , σ = 0 . 25. The region b elow the curv e consists of all the pairs of ( T , λ = λ p ¯ c ) suc h that G (0) < 1. “Normal op erating p oint” denotes a pair ( T , λ ) that lies ab ov e the curve ( T , λ ∗ (0 , T )). Suppose we use FH- CDMA, the total inte nsity decreases by a factor of spreading gain and hence we mov e vertically down wards into the G (0) < 1 region. If DS-CDMA is used, the threshold T decreases by a factor of spreading gain and hence we mov e hori zon tally tow ards the left into the G (0) < 1 region. IV. Transmission Cap acity o f Clustered Transmitters It is impo rtant to understand the p erfor mance of ad ho c wir eless netw orks. T ransmission capacity was int ro duced in [12], [14], [15 ] a nd is defined as the pro duct of the maximum density of s uccessful transmissions and their data ra te, g iven an outag e co ns traint. More formally , if the intensit y of the co ntending tr ansmitters is λ with an outage threshold T and a bit r ate b bits p e r seco nd p er her tz, then the transmission capacity at a fixed distance R is g iven by C ( ǫ, T ) = b (1 − ǫ ) sup λ { λ : P ( λ, T ) ≥ 1 − ǫ } (58) where P ( λ, T ) denotes the success pr obability of a given transmitter rec eiver pair. Mor e discussion ab o ut the transmissio n capa cit y and its relation to other metrics like transp ort ca pa city is provided in [1 5 ]. Note that the r e s ults proved in [12 ], [14], [15] are for Poisson a rrang e men t of tra ns mitters. In this s ection we ev aluate the transmissio n capa c ity when the transmitters ar e arrang ed as a Poisson clustered pro cess. W e pr ov e that fo r small v alues of ǫ , the transmissio n capacity o f the clustered pr o cess coincides with that o f the Poisson ar rangement of no des. W e a lso show that care should b e taken in defining transmission ca pacity for genera l distribution of no des. F o r notatio nal conv enience we shall assume b = 1. F or the clustered pro cess, P ( λ, T ) denotes the success proba bility of the cluster pro ce ss with int ensity λ = λ p ¯ c DRAFT May 29, 2018 23 0 0.5 1 1.5 10 0 10 1 R G(R) G(R), α =4, σ =0.25 λ p =0.25, c=3,g(x)=(1+|x| α ) − 1 λ p =3, c=3,g(x)=(1+|x| α ) −1 λ p =0.25,c=3, g(x)=|x| − α λ p =3, c=3, g(x)=|x| − α 1 2 3 4 Fig. 7. G ( R ) ve rsus R , α = 4 , σ = 0 . 25. Observe that the gain curves #2 and #3, which corresp ond to the singular cha nnel, start at 1 decrease and then i ncrease ab ov e unity . F or the gain curve #4, the total inte nsity of transmi tters i s 3 ∗ 0 . 25 = 0 . 75 which i s less than the threshold λ ∗ (0 , 0 . 5) ≈ 1 . 26. Hence the gain curv e for this case starts below unity at R = 0 and then increases. F or the gain curve #1 the total inte nsity is 9 > 1 . 26. By cha nce, in the present case the gain curve #1 starts around 10 and increases. and thr eshold T . Let P l ( λ, T ) , P u ( λ, T ) denote lower and upp er b ounds of the success pr obability P ( λ, T ) and the co rresp onding sets A l , A u defined by A χ := { λ : P χ ( λ, T ) ≥ 1 − ǫ } for χ ∈ { l , u } . W e then hav e A l ⊂ A ⊂ A u which implies sup A l ≤ sup A ≤ sup A u . (59) Let C l ( ǫ, T ) = sup A l and C u ( ǫ, T ) = sup A u denote low er and upper bounds to the trans mission capacity . F or a PP P we hav e from (43), P p ( λ, T ) = ex p( − λβ I ) ( β I do es not dep end on λ ). Hence the tra ns mission capacity of a P P P denoted by C p ( ǫ, T ) is given by C p ( ǫ, T ) = 1 − ǫ β I ln  1 1 − ǫ  ≈ ǫ (1 − ǫ ) β I , ǫ ≪ 1 (60) F or Neyman-Scott cluster pr o cesses, the intensit y λ = λ p ¯ c . W e first to try to consider b oth λ p and ¯ c as optimization pa rameters for the tra nsmission capa city , i.e. C ( ǫ, T ) := (1 − ǫ ) sup { λ p ¯ c : λ p > 0 , ¯ c > 0 , outage-c o nstraint } (61) without individua lly cons tr aining the par ent no de density or the av era ge n umber of no des p er clus ter. May 29, 2018 DRAFT 24 L emma 10: The transmiss ion capa cit y of Poisson clus ter ed pr o cesses is lower bo unded by the transmissio n capacity of the P PP , C ( ǫ, T ) ≥ C l ( ǫ, T ) = C p ( ǫ, T ) (62) Pr o of: F ro m Lemma 5 , we hav e P l ( λ, T ) = P p ( λ p ¯ c ) P p (¯ c ˆ f ∗ ). So to get a lower b ound, fr o m (59 ) we have to find sup n λ p ¯ c : λ p ¯ c + ¯ c ˆ f ∗ ≤ 1 β I ln  1 1 − ǫ  = C p ( ǫ, T ) 1 − ǫ o (63) This maximum v alue of λ p ¯ c is attained when, λ p → ∞ , w hile ¯ c → 0, such that ¯ cλ p = C p ( ǫ, T )(1 − ǫ ) − 1 . So we hav e C l ( ǫ, T ) = C p ( ǫ, T ). Also o bserve that λ p → ∞ a nd ¯ c → 0. This corr esp onds to the scena rio in which the clustered pro cess degenerated to a P PP . W e also hav e the following upp er b ound. L emma 11: Let ρ ( T ) = k / ˆ β with k =  β ( R , y ) f ( y )d y . F o r ǫ < 1 − e − ρ ( T ) , we hav e C ( ǫ, T ) ≤ C u ( ǫ, T ) = C p ( ǫ, T ) (64) Pr o of: See App endix F. The or em 2: F or ǫ ≤ 1 − e − ρ ( T ) we hav e C ( ǫ, T ) = C p ( ǫ, T ). Pr o of: F ollows from the Lemmata 1 0 and 11. F rom the ab ov e tw o pr o ofs, when ǫ is small, the tra nsmission capacity is equal to the Poisson pro cess of same intensit y . This ca pacity is achieved when λ p → ∞ and ¯ c → 0. This is the s cenario in which the c luster pro cess beco mes a P PP . This is due to the definitio n of the transmission capacity as C ( ǫ, T ) := sup { λ p ¯ c : λ p > 0 , ¯ c > 0 , outage- constraint } wher e we have tw o v ar iables to optimize ov er. Instead we may fix λ p as constant and find the transmiss io n capacity with r e sp e c t to ¯ c . So we define constrained tra nsmission capacity as C ∗ ( ǫ, T ) := λ p (1 − ǫ ) sup { ¯ c : ¯ c > 0 , outag e-constra in t } (65) W e hav e the following b ounds for C ∗ ( ǫ, T ) The or em 3: λ p C p ( ǫ, T ) λ p + ˆ f ∗ ≤ C ∗ ( ǫ, T ) ≤ λ p C p ( ǫ, T ) max n 0 , λ p − ˆ β β I ln  1 1 − ǫ o (66) Pr o of: F rom the low er b ound on P (s uc c e ss), we have to find sup n ¯ c : λ p ¯ c + ¯ c ˆ f ∗ ≤ 1 β I ln  1 1 − ǫ  = C p ( ǫ, T ) 1 − ǫ o (67) So we have C ∗ l ( ǫ, T ) = C p ( ǫ, T ) / ( ˆ f ∗ + λ p ). F rom the upp er b ound on P (success),w e have to find sup n ¯ c : λ p ¯ c 1 + ¯ c ˆ β ≤ 1 β I ln  1 1 − ǫ  = C p ( ǫ, T ) 1 − ǫ o (68) DRAFT May 29, 2018 25 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 α Transmission Capacity C * R=0.25, σ =0.25, ε =0.1,T=1, λ p =1 C u * (0.1,1) C l * (0.1,1) C p (0.1,1) C * (0.1,1) Upper bound PPP Actual (order) Lower bound Fig. 8. Upp er and low er b ounds of C ∗ ( ǫ, T ) versus α , g ( x ) = k x k − α , T = 1 , σ = 0 . 25 , ǫ = 0 . 1 , λ p = 1 One ca n also derive a n order approximation to the co nstrained tra nsmission capacity when ǫ is very s mall. W e hav e the following order approximation to transmission ca pacity . Pr op osition 1: When λ p is fixed, the co nstrained trans mission capacity is given by C ∗ ( ǫ, T ) = (1 − ǫ )  ǫλ p λ p β I + k + o ( ǫ )  (69) when ǫ → 0. Pr o of: Let γ ( ¯ c ) deno te the outa ge probability , i.e. , γ ( ¯ c ) = 1 − exp n − λ p  R 2 1 − exp[ − ¯ cβ ( R, y )]d y o  R 2 exp( − ¯ cβ ( R, y )) f ( y )d y (70) W e have d γ ( ¯ c ) / d ¯ c > 0 , which implies γ ( ¯ c ) is incr easing and inv ertible and hence C ∗ ( ǫ, T ) = λ p (1 − ǫ ) γ − 1 ( ǫ ). W e appr oximate γ − 1 ( ǫ ) for small ǫ by the Lagra nge inv ersion theorem. Observe that γ ( ¯ c ) is a smo oth function of ¯ c and all deriv atives e x ist. Expa nding γ − 1 ( ǫ ) a round ǫ = 0 by the Lagr ange in version theor em and using γ (0) = 0 yields γ − 1 ( ǫ ) = ∞ X n =1 d n − 1 d ¯ c n − 1  ¯ c γ ( ¯ c )  n     ¯ c =0 ǫ n n ! (71) = ¯ cǫ γ ( ¯ c ) | ¯ c =0 + o ( ǫ ) ( a ) = ǫ λ p β I + k + o ( ǫ ) where ( a ) follows by applying de L’Hˆ opital’s rule . W e hav e the following observ a tions 1) The constr ained tra nsmission capac ity increa ses (slowly) with λ p . May 29, 2018 DRAFT 26 2) W e also observe that the co nstrained trans mission capacity for the cluster pro cess is a lwa ys less than that o f a Poisson netw o rk (see Figure 8) and approa ches C p ( ǫ, T ) as λ p → ∞ . 3) When FH-CDMA with in tra -cluster frequency hopping is utilized, we hav e the cluster in tensity ¯ c reduced by a fa ctor M (spreading ga in). One ca n e a sily o btain the co nstrained tr ansmission ca pacity of this s ystem to b e C ∗ F H ( ǫ, T ) = (1 − ǫ )  ǫλ p M λ p β I + k + o ( ǫ )  When DS-CDMA is used, the constrained transmissio n capacity is C ∗ DS ( ǫ, T ) = C ∗ ( ǫ, T / M ). When the tr a nsmitters are sprea d as a Poisson p oint pro cess, we have from [28 ], [29] ln  C F H ( ǫ, T ) C DS ( ǫ, T )  = (1 − 2 /α ) ln( M ) . In Figure 9, we plot ln( C ∗ F H ( ǫ, T ) /C ∗ DS ( ǫ, T )) / ln( M ) with resp ect to spreading ga in M , when the path loss function is g ( x ) = k x k − α and ǫ = 0 . 01. F ro m the figure , we observe a similar M 1 − 2 /α gain, even in the ca s e of clustered transmitters . 0 20 40 60 80 100 120 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 M Gain in using FH−CDMA to DS−CDMA ε =0.01,T=1,R=0.25, σ =0.25, λ p =1 α =4 α =3 α =5 Fig. 9. ln( C ∗ F H ( ǫ, T ) /C ∗ D S ( ǫ, T )) / ln( M ) versus M for ǫ = 0 . 01 , λ p = 1 V. Conclusions Previous work character iz ing interference, outage, and tra nsmission capacity in large ra ndom net works exclusively fo cused on the ho mogeneous Poisson p oint pro ces s as the underlying no de distribution. In this pap er, we extend these results to cluster ed pro cess es. Clustering may b e geog raphical, i.e. , given by the spatial distribution of the no des , or it may b e induced logica lly by the MA C scheme. W e use to ols from sto chastic geo metry a nd Palm probabilities to obta in the co nditional Laplace tr ansform of the interference. Upper and lower b ounds are obtained for the CCDF of the interference, for any stationar y distribution of DRAFT May 29, 2018 27 no des and fading . W e hav e shown that the distribution of interference dep ends hea vily on the path-loss mo del consider ed. In particular , the ex istence of a singula rity in the model greatly affects the results. This conditional Laplace transform is then us ed to o btain the probability of success in a clustered netw or k with Rayleigh fading. W e s how clus tering the transmitters is alwa ys b eneficial for lar ge link distances, while the clustering gain at s maller link dis ta nces dep ends on the path-lo ss mo del. The transmissio n capacity of cluster ed netw or ks is equal to the one for homo geneous netw or ks. How ever, car e m ust b e ta ken when defining this capacity since clustered pro cesses hav e tw o parameter s to optimize ov e r. W e a lso show that the transmissio n capacity of c lus tered netw ork is equal to the Poisson distributio n of no des. W e anticipate that the ana lytical techniques us ed in this w ork will b e useful fo r other problems as w ell. In particular the conditional generating functiona ls ar e likely to find wide applicability . Ackno wledgments The s upp or t of the NSF (gra n ts CNS 0 4-478 69, CCF 05-15 0 12, and DMS 5056 24) a nd the DA RP A/IT - MANET pr ogra m (g rant W9 11NF-07- 1-002 8) is gra tefully ackno wledged. References [1] E . S. Sousa and J. A. Silv ester, “Optimum transmission ranges in a di r ect-sequence spread spectrum multihop pac ket radio net work,” IEEE Journal on Sele cte d Ar e as in Communic ations , pp. 762–771, 1990. [2] R . M athar and J. M attfeldt, “On the distri bution of cumulated i nt erference p ow er in Rayleigh fading channels,” Wir eless Networks , vol. 1, no. 1, pp. 31–36, 1995. [3] M . W estcott, “On the existence of a generalized shot-noise process,” Studies in Pr ob ability and Statistics. Pap ers in Honour of Edwin JG Pitman, North-Hol land, Amster dam , p. 7388, 1976. [4] S. Rice, “Mathematical analysis of random noise,” Sele cte d Pap ers on Noise and Sto chastic Pr o c esses , pp. 133–294, 1954. [5] J. 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H einrich, “Asymptotic behaviour of an emp erical nearest-neighbour distance function for stationary poisson cluster processes,” Math. Nachr. , vol. 136, pp. 131–148, 1988. [21] M . Baudin, “Likelihoo d and Nearest-Neigh bor Distance P r operties of Multidimensional P oisson Cluster Processes,” Journal of Applie d Pr ob ability , vol. 18, no. 4, pp. 879–888, 1981. [22] K. H. Hanisch , “Reduction of the n-th momen t measures and the special case of the third moment measure of stat ionary and isotropic planar point pro cess,” Mathematische Op era tionsforschung and Statistik Series Statistics , vol. 14, pp. 421–435, 1983. [23] M . W estcott , “The probability generating functional,” Journa l of Austr alian M athematic al So ciet y , vol. 14, pp. 448–466, 1972. [24] S. Low en and M. T eich , “P ow er-law shot noise,” Info rmation The ory, IEEE T r ansactions on , vol. 36, no. 6, pp. 1302–1318, 1990. [25] G. Samorodnitsky , “T ail b ehavior of shot noise pro cesses.” [Onl ine]. Av ailable: http://cit eseer.ist.psu.edu/603758.h tml [26] H. Inaltek in and S. Wick er, “The b eha vior of un bounded path-loss mo dels and the effect of singularity on computed netw ork int erference.” F ourth Annual IEEE Communic ations So ciety Confer enc e on Sensor, M esh and A d Ho c Communic ations and Networks SECON 2007 , June 2007. [27] G. B. F olland, R e al Analysis, Mo dern T ech niques and Their Applic ations , 2nd ed. Wiley , 1999. [28] J. Andrews, S. W eber, and M . Haenggi, “Ad ho c netw orks: T o spread or not to spread,” T o app e ar: IEEE Communic ations Magazine , 2007. [29] S. W eber, J. Andrews, X. Y ang, and G. de V eciana, “T ransmissi on capacit y of wi reless ad hoc net works with successive int erference cancellation,” IEEE T r ansactions on Information The ory, under r evi sion, submitte d in , 2005. Appendix A. Pr o of of L emma 3: Pr o of: W e first ev aluate the asymptotic b ehavior of G  F h  y g ( . − z )  . Let v ( x ) = F h  y g ( x − z )  . W e have M   R 2 v ( x + u ) f ( u )d u  = exp  − ¯ c  R 2 [1 − F h  y g ( x + u − z )  ] f ( u )d u  ( a ) ∼ 1 − ¯ c  R 2  1 − F h  y g ( x + u − z )  f ( u )d u (72) where ( a ) fo llows from the fact that exp( − x ) = 1 − x + O ( x 2 ) for x close to 0 and (1 − F h ) → 0 for large y . By a similar expansion of exp, (72) a nd the dominated conv ergence theorem, we hav e exp  − λ p  R 2 1 − M (  R 2 v ( x + u ) f ( u )d u )d x  ∼ 1 − λ p ¯ c  R 2  R 2 [1 − F h  y g ( x + u − z )  ] f ( u )d u d x = 1 − y − 2 /α λ p ¯ c  R 2 [1 − F h ( k x k α )]d x (73) DRAFT May 29, 2018 29 By change of v ar iables a nd using lim y →∞ [1 − F h ( y )] y 2 /α = 0 [27, p.198 ], we hav e  R 2 [1 − F h ( k x k α )]d x = π  ∞ 0 ν 2 /α d F h ( ν ) W e similarly hav e  R 2 M   R 2 v ( x + u ) f ( u )d u  f ( x )d x ∼ 1 − ¯ c  R 2  1 −  R 2 F h  y g ( x + u − z )  f ( u )d u  f ( x )d x = 1 − ¯ c  R 2  R 2 [1 − F h ( y k x + u − z k α )] f ( u ) f ( x )d u d x = 1 − ¯ c  R 2 [1 − F h ( y k x k α )] ( f ∗ f )( x + z )d x = 1 − ¯ cy − 2 /α  R 2 [1 − F h ( k x k α )] ( f ∗ f )  x y 1 /α + z  d x ( a ) ∼ 1 − ¯ c ( f ∗ f )( z ) y − 2 /α  R 2 [1 − F h ( k x k α )] d x (74) where ( a ) fo llows from the Lebes gue dominated co n vergence theorem ( f ∗ f is a very nice function since f is a P DF). So we hav e G  F h  y g ( . − z )  ∼ θ 1 y − 2 /α . F o r a Neyman-Scott cluster pro cess, the seco nd o rder pro duct density is g iven by [16, p.15 8], ρ (2) ( r ) = λ 2 + λµ ( r ) π ¯ c ∞ X n =2 p n n ( n − 1) where p n is the distr ibution of the num b er of p oints in the re pr esentativ e cluster . µ ( r ) /π denotes the density of the distributio n function for the distance b etw een tw o indep endent random p oints which were sc a ttered using the distribution f ( x ) of the representativ e cluster. When the num b er of p oints inside each cluster is Poisson distributed with mean ¯ c , we have P ∞ n =2 p n n ( n − 1) = ¯ c 2 . W e also hav e µ ( x ) /π = ( f ∗ f )( x ) Estimating ϕ ( y ) we hav e ϕ ( y ) = 1 y λ  R 2 g ( x − z ) ρ (2) ( x )  y /g ( x − z ) 0 ν d F h ( ν )d x = λ y  R 2 k x k − α  y k x k α 0 ν d F h ( ν ) d | {z } T 1 + ¯ c y π  R 2 g ( x − z ) µ ( x )  y /g ( x − z ) 0 ν d F h ( ν )d x | {z } T 2 (75) By change of v ar iables, we hav e T 1 = 2 π λy − 2 /α α − 2  ∞ 0 ν 2 /α d F h ( ν ) (76) May 29, 2018 DRAFT 30 F or the term T 2 , T 2 = ¯ c y π  ∞ 0 ν d F h ( ν )  R 2 k x − z k − α 1 k x − z k α >vy − 1 µ ( x )d x = ¯ c π y 2 /α  ∞ 0 ν d F h ( ν )  R 2 k x k − α 1 k x k α >v µ  x y 1 /α + z  d x ∼ µ ( z ) ¯ c π y 2 /α  ∞ 0 ν d F h ( ν )  R 2 k x k − α 1 k x k α >v d x = y − 2 /α 2 µ ( z )¯ c α − 2  ∞ 0 ν 2 /α d F h ( ν ) (77) So we hav e ϕ ( y ) ∼ θ 2 y − 2 /α . Hence from Theore m 1, we have ¯ F l I ( y ) ∼ θ 1 y − 2 /α and ¯ F u I ( y ) ∼ ( θ 1 + θ 2 ) y − 2 /α . B. Outage pr ob ability, in Poisson clus t er pr o c ess when the numb er of cluster p oints ar e fixe d. In this subs e ction we derive the conditional Laplace transform in a Poisson cluster pro c ess, when the nu mber of p o ints in ea ch clus ter are fixed to b e ¯ c ∈ N and ¯ c > 0. W e als o a ssume that each p oint is independently distributed with density f ( x ). In this ca se the moment gener ating function o f the num b er of po int s in the r epresentativ e cluster is given by M ( z ) = z ¯ c Using the s ame nota tio n as in Section I I-B, and from (1 1) and (13), we have E ! 0   Y x ∈ φ v ( x )   = ˜ G ( v )  N Y x ∈ ψ v ( x ) ˜ Ω ! 0 ( dψ ) = ˜ G ( v )  N Y x ∈ ψ v ( x )  R 2 Ω ! y ( dψ y ) f ( y )d y = ˜ G ( v )  R 2  N Y x ∈ ψ v ( x )Ω ! Y ( dψ y ) f ( y )d y ( a ) = ˜ G ( v )  R 2   R 2 v ( x − y ) f ( x )  ¯ c − 1 f ( y )d y (78) where ( a ) follows from the fact that the points are independently distributed and we a r e not coun ting the po int at the o r igin. In this case ˜ G ( v ) is given by ˜ G ( v ) = exp ( − λ p  R 2 1 −   R 2 v ( x + y ) f ( y )d y  ¯ c d x ) Hence the succ e ss pro bability (Rayleigh fading) is given by P (success) = exp  − λ p  R 2 1 − ˜ β ( R , y ) ¯ c d y   R 2 ˜ β ( R , y ) ¯ c − 1 f ( y )d y (79) where ˜ β ( R , y ) =  R 2 f ( x ) 1 + g ( R ) T g ( x − y − z ) d x DRAFT May 29, 2018 31 C. Outage pr ob ability of Nakagami-m fading Here, we derive the success probability when the fading distribution is Nak agami- m distributed. W e also assume m ∈ N and W = 0. The PDF of the p ower fading co efficient y = h x is given b y p ( y ) = 1 Γ( m )  m Ω  m y m − 1 e − my / Ω F rom (35), we hav e P (success) = P  hg ( z ) W + I φ \{ 0 } ( z ) ≥ T  P (success) = 1 −  T g ( z )  m  ∞ 0 1 Γ( m )  m Ω  m y m − 1 e − T m Ω g ( z ) y P !0 ( I φ ( z ) > y )d y (80) Using int egra tio n b y parts we g et, P (success) = 1 Γ( m )  ∞ 0 Γ( m, T m Ω g ( z ) y ) d P !0 ( I φ ( z ) ≤ y ) ( a ) = ( m − 1)! Γ( m ) m − 1 X k =0 1 k !  ∞ 0 e − T m Ω g ( z ) y y k d P !0 ( I φ ( z ) ≤ y ) ( b ) = m − 1 X k =0 ( − 1) k k ! d k ds k L I φ ( z ) ( s ) | s = T m/ Ω g ( z ) (81) where ( a ) follows from the series e xpansion of incomplete Gamma function when m is an integer and ( b ) follows from the prop er ties of the La place tra nsform and Γ( m ) = ( m − 1 )! when m is an int eger. W e als o hav e L h x ( sg ( x − z )) = 1 (1 + Ω m sg ( x − z )) m Hence fr om Lemma 2, we hav e L I φ ( z ) ( s ) = ex p  − λ p  R 2 1 − exp( − ¯ c ¯ β ( s, z , y ))d y   R 2 exp( − ¯ c ¯ β ( s, z , y )) f ( y )d y (82) where ¯ β ( s, z , y ) = 1 −  R 2 1 (1 + Ω m sg ( x − y − z )) m f ( x )d x F or integer m ≥ 1, P (succes s) c a n be ev aluated fro m (81) and (82). F or m = 1, the probability ev alua ted from (81) a nd (8 2) matches that of Lemma 4. May 29, 2018 DRAFT 32 D. Pr o of of L emma 7 Pr o of: P p ( λ p ¯ c ) P 1 ( R, ¯ c, λ p ) = ex p h − λ p ¯ c  R 2 β ( R , y )d y + λ p  R 2 (1 − exp[ − ¯ cβ ( R, y )])d y i = ex p h − λ p  R 2 { ¯ cβ ( R, y ) − 1 + exp h − ¯ cβ ( R, y ) i | {z } ν ( R,y ) } d y i (83) Since 1 − exp( − ax ) ≤ ax , we hav e that ν ( R , y ) > 0. W e also have from the dominated conv ergence theore m and (1) lim R →∞ ν ( R, y ) = ¯ c 1 + T − 1 − 1 + exp  − ¯ c 1 + T − 1  > 0 which is a co ns tant. So using F ato u’s lemma [27 ] (lim inf  f n ≥  (lim inf f n ) , f n > 0), we hav e lim R →∞ P p ( λ p ¯ c ) P 1 ( R, ¯ c, λ p ) = exp[ − λ p lim R →∞  R 2 ν ( R, y )d y ] ≤ exp[ − λ p  R 2 lim R →∞ ν ( R, y )d y ] = exp[ − λ p ∞ ] = 0 (84) E. Pr o of of L emma 8 Pr o of: F rom (55), the probability o f suc c e ss is P (success) = P p ( λ p ¯ c ) exp h λ p ¯ c  R 2 β ( R , y ) η ( ¯ c, R , y )d y i | {z } T 1 P 2 ( R, ¯ c ) | {z } T 2 (85) where η ( ¯ c , R, y ) = ν ( ¯ cβ ( R, y )) and ν ( x ) = (exp( − x ) − 1 + x ) /x a n incr easing function of x . F rom Y oung’s inequality [2 7, Sec. 8 .7 ] we have β ( R, y ) ≤ min { 1 , sup { f ( x ) } R 2 T 2 /α C ( α ) } . Hence η ( ¯ c, R, y ) ≤ ν ( ¯ c min { 1 , sup { f ( x ) } R 2 T 2 /α C ( α ) } ) With a slig ht abuse o f nota tio n, let η ( ¯ c, R ) = ν ( ¯ c min { sup { f ( x ) } R 2 T 2 /α C ( α ) , 1 } ). Hence T 1 ≤ exp[ λ p ¯ c  R 2 β ( R , y ) η ( ¯ c, R )d y ] = exp[ λ p ¯ cT 2 /α R 2 η ( ¯ c, R ) C ( α )] (86) Also observe that η ( ¯ c , R ) / R 2 . So T 1 / exp( R 4 ). T 2 =  R 2 1 − ¯ c β ( R , y ) + ¯ c β ( R , y ) ∞ X k =2 ( − 1) n n ! (¯ cβ ( R , y )) n − 1 f ( y )d y ≤  R 2 [1 − ¯ c β ( R , y ) + ¯ cβ ( R, y ) η ( ¯ c, R )] f ( y )d y = 1 − [1 − η (¯ c , R )] ¯ c  R 2 β ( R , y ) f ( y )d y (87) DRAFT May 29, 2018 33 If one co nsiders x and y a s identical a nd indep endent random v ariables with density functions f , w e then hav e  R 2 β ( R , y ) f ( y )d y = E [ 1 1+ g ( R ) T k x − y − R k α ]. Let 0 < κ < 1 be some constant. Using the Chebyshev inequality we get E " 1 1 + g ( R ) T k x − y − R k α # ≥ κP " 1 1 + g ( R ) T k x − y − R k α ≥ κ # = κP h k x − y − R k ≤ ( 1 κ − 1 ) 1 /α RT 1 /α i ( ∗∗ ) (88) The PDF of z = x − y is given by ( f ∗ f )( z ), sinc e y is rota tion-inv a r iant. Cho osing κ = T / (1 + T ) we hav e ( ∗∗ ) = T 1 + T  B ( R,R ) ( f ∗ f )( x )d x ≥ T 1 + T  B ( R,R ) k x k d x = R 3 T 1 + T  B (1 , 1) k x k d x | {z } C 2 (89) So we have P 2 ≤ 1 − [1 − η (¯ c , R )] R 3 C 2 / 1 − R 3 + R 5 (90) Also we hav e T 1 / exp( R 4 ) / 1 + 1 . 01 R 4 . So we hav e P 2 T 1 / 1 − R 3 + R 5 − 1 . 01 R 7 + 1 . 01 R 9 < 1 for sma ll R 6 = 0. Henc e for smal l R we have P ( suc c ess ) ≤ P p ( λ p ¯ c ) . F. Pr o of of L emma 11 Pr o of: W e find C u ( ǫ, T ) a nd hence upp er b ound the tra nsmission capacity . W e hav e from the deriv ation of Lemma 8 P ( λ, T ) ≤ P p ( λ p ¯ c ) exp[ λ p ¯ cβ I η ( ¯ c, R )] P 2 ( R, ¯ c ) = P u (¯ cλ p , T ) (91) where η (¯ c , R ) = (exp( − ¯ c ˆ β ) − 1 + ¯ c ˆ β ) / ¯ c ˆ β . With A u = { λ p ¯ c, P u ( λ p ¯ c, T ) ≥ 1 − ǫ } , it is sufficient to prov e sup A u ≤ C p ( ǫ, T ). Also observe that P u (¯ cλ p , T ) → 0 as ¯ c → ∞ indep endent o f λ p . Hence we ca n a ssume ¯ c is finite for the pr o of. W e pr o ceed by co nt radiction. Let sup A u > C p ( ǫ, T ). Hence there ex ists a δ > 0 , λ p ≥ 0 , ¯ c ≥ 0 such that λ p ¯ c = ( C p ( ǫ, T ) / (1 − ǫ ))+ δ ∈ A u . A t this v a lue of λ p ¯ c we have 1 − ǫ ≤ P u (¯ cλ p , T ) = (1 − ǫ ) P p ( δ ) exp[ η ( ¯ c, R ) { ln( 1 1 − ǫ ) + δ β I } ] P 2 ( R, ¯ c ) = (1 − ǫ ) 1 − η ( ¯ c ,R ) P p ( δ (1 − η ( ¯ c, R ))) | {z } T 1 P 2 ( R, ¯ c ) (92) May 29, 2018 DRAFT 34 F rom the der iv ation o f Lemma 8 , w e have P 2 ( R, ¯ c ) ≤ 1 − [1 − η (¯ c, R )]¯ c k , with equality only when ¯ c = 0 . Hence we hav e p u (¯ c λ p , T ) ≤ T 1 (1 − ǫ ) 1 − η ( ¯ c ,R ) (1 − [1 − η (¯ c , R )] ¯ ck ) (93) Since exp( − x ) − (1 − x ) x ≤ x 1+ x , we hav e η ( ¯ c, R ) ≤ ¯ c ˆ β / (1 + ¯ c ˆ β ). Using the upper bound for η ( ¯ c, R ) p u (¯ cλ p , T ) ≤ T 1 (1 − ǫ )(1 − ǫ ) − ¯ c ˆ β 1+ ¯ c ˆ β 1 − " 1 − ¯ c ˆ β 1 + ¯ c ˆ β # ¯ c ˆ β ρ ( T ) ! = T 1 (1 − ǫ ) (1 − ǫ ) − ¯ c ˆ β 1+ ¯ c ˆ β 1 − ¯ c ˆ β ρ ( T ) 1 + ¯ c ˆ β ! | {z } T 2 (94) Using the inequality 1 − ay ≤ (1 − b ) y , b ≤ 1 − e − a , y ≥ 0, substituting y = ¯ c ˆ β 1+¯ c ˆ β , b = ǫ, a = ρ ( T ), we get T 2 ≤ 1. Hence we hav e p u (¯ cλ p , T ) ≤ (1 − ǫ ) P p ( δ (1 − η ( ¯ c, R ))) (95) So if δ > 0, and ¯ c finite, we also have P p ( δ (1 − η ( ¯ c, R ))) < 1. So w e have a contradiction from (92) and (95). Hence there exists no such δ and hence sup A u ≤ C p ( ǫ, T ). W e ca n achiev e C u ( ǫ, T ) = C p ( ǫ, T ), by us ing λ p = n C p ( ǫ,T ) 1 − ǫ − 1 , ¯ c = 1 /n for n very large. As n → ∞ , P u (¯ c λ p , T ) → P p (¯ cλ p , T ) . DRAFT May 29, 2018