Transmission Capacities for Overlaid Wireless Ad Hoc Networks with Outage Constraints

T ransmission Capaciti es for Ove rlaid W irel ess Ad Hoc Networks with Outage Co nstraints Changch uan Y in, Long Gao, Ti e Liu, and Sh uguang Cu i Departmen t of E lectrical an d Com puter En gineerin g T exas A& M Univ ersity , College Station, TX 7784 3 Email: ccyin@ieee.org, {lgao, tieliu, cui}@ece. tamu.edu Abstract —W e stu dy the transmission capacities of two coex- isting wireless netw orks (a primary network vs. a secondary network) th at operate in the same geographic region and share the same spectrum. W e define transmission capacity as th e product among the d ensity of transmissions, the transmission rate, and the successful transmission probability (1 minus t he outage pro bability). The primary (PR) network has a higher priority to acce ss the spectrum without particular considerations fo r the seco ndary (S R) network, where the SR netw ork limits its interference to the PR network by carefully controlling the density of its transmitters. Assuming that the nodes are distributed according to Poisson point proc esses and the two networks use different transmission ranges, we quantify the transmission capac ities f or both of these two netw orks and discuss their tradeoff based on asymptotic analyses. Our results sh ow that if the PR network permits a small in crease of its outage probability , the su m transmission capacity of th e two networks (i.e., the o verall spectrum effi ciency per unit area) will be boosted significantly over that of a si ngle network. I . I N T RO D U C T I O N Initiated by the seminal work of Gupta and Kumar [1], the studies for understanding the capacities of wireless ad hoc networks have mad e great prog resses. Considering n nodes that are ran domly distributed in a unit area and grou ped indepen dently into one- to-one source-de stination (S-D) pairs, Gupta and Kumar [1] showed that a ty pical time-slotted multi-hop arch itecture with a com mon transmission r ange and ad jacent-neig hbor commun ication can a chieve a sum throug hput that scales a s Θ  p n/ log n  . By using p ercola- tion theor y , Francesch etti et al. [2] sho wed that th e Θ ( √ n ) sum throu ghpu t scaling is ach iev able. In [3], Grossglauser and Tse showed th at by allowing th e nodes to move ind epende ntly and unifo rmly , a constant th rough put scaling Θ (1) per S- D p air can be achieved. In [4], Bacce lli et al. pr oposed a multi-hop spatial reuse ALOHA protoco l. By op timizing the product betwee n the nu mber of simultaneo us successful transmissions per unit area and the a verage transmission range, they showed that the tr ansport capacity is p ropor tional to the square root of the no de density , which achieves the up per bound of Gupta an d Kumar [ 1]. W eber et al in [5] d eriv ed the upper an d lower boun ds on transmission c apacity o f spread - spectrum wireless ad h oc networks, where the tran smission capacity is defined as th e p roduct between the max imum density of successful transmissions and the correspo nding data rate, u nder a co nstraint on th e ou tage probab ility . All the above results focus on the capacity o f a single ad hoc wireless n etwork. In recent years, due to the scarc ity and poor utilization of spectrum , the regulation bo dies are beginning to co nsider the po ssibility of permitting second ary (SR) networks to coexist with licensed prima ry (PR) networks, which is the main dri v ing force b ehind the cognitiv e radio technolog y [6]. In cognitive radio networks, the PR users h av e a higher priority to access the spectru m and the SR users need to o perate c onservati vely suc h that their interferen ce to the PR users is limited below an “acceptable lev el”. In this overlaid regime, the capacity or throughpu t scaling laws for both of th e PR and SR networks are interesting problems. Recently , some preliminary works along this line ap peared. In [7], V u et al. considered the th rough put scaling law for a single-ho p cognitive r adio network, wh ere a linear scaling law is obtained f or the SR n etwork with an outage con straint for the PR network. In [8], Jeon et al . considere d a m ulti- hop cognitive network on top o f a PR n etwork and assum ed that the SR no des k now the locatio n of each PR node . W ith an elegant tr ansmission scheme, they showed that by defining a pr eservation region around each PR node, both networks can achieve the same throug hput scaling law as a stand-alo ne wireless network, while the SR network may suf fer from a finite outage probab ility . In [9 ], Y in e t al . assumed that the SR nodes o nly kon w th e locations o f PR transmitters (TXs) and propo sed a tran smission scheme to show that both networks can achieve the same throug hput scaling law as a stand-alo ne wireless network, with zero outa ge. In this paper, we study the c oexistence of two ad h oc networks with different tr ansmission scales (power and/or transmission range) based on the transmission capacity define d in [5]. W e extend the definition o f transmission capacity fr om a single network to two overlaid network s. D ifferent fr om th e approa ches in [7], [8], and [9], we resort to stochastic geome- try tools to q uantify the transmission capacities f or both the PR and SR ne tworks without defining an y pre servation regions. By considerin g the mutua l interfer ences from the two networks, we discuss the tradeoff of the transmission cap acities between them. The results show that if we permit a slight inc rease over the outage probability o f the PR network, the sum transmission capacity (i.e., the ov erall spectru m ef ficien cy p er unit area) o f the overlaid networks will be b oosted sign ificantly over th at of a sing le ne twork. The r est of the paper is organ ized as follows. The network model and symbo l notations are descr ibed in Sectio n II . The transmission capacity for a single ne twork case is an alyzed in Section III . The transmission capacities for the PR and SR ne tworks and their trad eoff ar e discussed in Sectio n IV . The num erical results and observations are given in Section V . Finally , Section VI summarizes ou r co nclusions. I I . N E T W O R K M O D E L A N D S Y S T E M S E T U P Consider the scenar io where a network of PR nodes and a network o f SR nodes co exist in the sam e g eograp hic region, and assume that the PR n etwork is the legacy ne twork, wh ich has a hig her prio rity to access the spec trum. Th e pr erequisite condition for in troducin g a ne w SR n etwork into the terr itory of the PR network is that the outage proba bility increment of the PR network is uppe r-limited by a target constrain t △ ǫ , where △ ǫ u sually takes a very small value. W e assume that at a certain tim e instance the d istribution of PR TXs fo llows a h omogen eous Poisson poin t pro cess (PPP) Π 0 of density λ 0 , and the distribution of SR TXs follows another inde penden t homog eneous PPP Π 1 of density λ 1 . Our goal is to ev aluate the outag e pr obability of th e PR network, P 0 , and that of the SR network, P 1 , wh ich are fu nctions o f the TX node densities λ 0 and λ 1 . The specific definitions of outage pro bability will be giv e n in Section III and Section IV . Similar to that in [5], in orde r to e valuate the ou tage probab ilities, we con dition on a typical PR (or SR) RX at the origin, which yields the Palm d istribution for PR (or SR) TXs. Follo wing the Slivnyak’ s theo ry in stoch astic geom etry [1 0], these conditional d istributions also follow homo geneou s PPPs with the corresp onding densities (i.e., λ 0 and λ 1 , respectively). Let  X i ∈ R 2 , i ∈ Π 0  and  Y j ∈ R 2 , j ∈ Π 1  denote the locations of th e PR TXs and the SR TXs, respectively , | X i | and | Y j | d enote the distances from PR TX i an d SR TX j to the origin, r espectively . An attemp ted transmission is successful if the received signal-to-in terference -plus-no ise ratio (SINR) at the r eference RX is ab ove a thresh old, β ; otherwise, the transmission f ails, i.e ., an outage occurs. W e use β 0 and β 1 to represent the SINR thr esholds for the PR network and the SR network, respe ctiv e ly . For simplicity , we limit our discussion to sing le-hop tr ans- missions, and assume th at all PR TXs use th e same transmis- sion power ρ 0 , and all PR transmissions ar e over the same distance r 0 . Similarly , all SR TXs use the same transmission power ρ 1 over the same tran smission distance r 1 . For th e wireless channel, we o nly co nsider the large-scale path-loss, and ignore the ef f ects of shadowing an d small-scale multipath fading. As such, the n ormalized ch annel power gain g ( d ) is giv en as g ( d ) = A d α , (1) where A is a system-depen dent constant, d is the distance between the TX and the co rrespon ding RX, and α > 2 denotes the path -loss expon ent. In the fo llowing discussion, we norm alize A to be unity for simplicity . Th e ambient noise is assum ed to be additi ve wh ite Gaussian noise ( A WGN) with an average power η . W e assume that a ll the PR T Xs and the SR TXs use the sam e spectrum with ba ndwidth n ormalized to be u nity . As in [5], we de fine tran smission cap acity a s f ollows. Definition 1: T ransmission capacity C ǫ of a rando mly- deployed wire less network is defined as the pro duct among the maximum density λ ǫ of transmissions, the common trans- mission data rate R , an d (1 − ǫ ) with ǫ an asympto tically small outage p robability . The refore, we have C ǫ = Rλ ǫ (1 − ǫ ) . (2) As no ted in [5], C ǫ also r epresents the u nit-area spectral efficiency of the successful transmissions. I I I . A S Y M P T OT I C A N A L Y S I S O F T H E T R A N S M I S S I O N C A PAC I T Y : S I N G L E N E T W O R K C A S E In this section, we derive th e asympto tic result (asymptotic over vanishingly-sm all o utage probability v alues) for the trans- mission cap acity of the PR network wh en the SR network is absent. As an example, we fo cus on the c ase when the p ath- loss exponent α = 4 , ov er which we build an asympto tic analysis framew o rk that is useful fo r the futu re stud y ov er the cases o f g eneral α values. When the SR network is ab sent, deno te th e target ou tage probab ility of the PR network ov er per-link SINR as ǫ 0 . Then we have P 0 = Prob     ρ 0 r − α 0 η + X i ∈ Π 0 ρ 0 | X i | − α ≤ β 0     = ǫ 0 . (3) Re write (3) as Prob ( X ≥ T 0 ) = ǫ 0 , (4) where X = P i ∈ Π 0 ρ 0 | X i | − α and T 0 = ρ 0 r − α 0 β 0 − η . The moment g enerating fu nction ( MGF) of X is given by [11] Φ X ( s ) = exp  − π λ 0 ρ 2 α 0 s 2 α Γ  1 − 2 α  . (5) When α = 4 , we have Φ X ( s ) = exp h − π 3 2 λ 0 ρ 1 2 0 s 1 2 i . (6) V ia the inverse Laplac e transfor m, we obtain the proba bility density function (PDF) of X as f X ( x ) = π 2 λ 0 √ ρ 0 x − 3 2 exp  − π 3 4 x λ 2 0 ρ 0  , (7) and the co rrespon ding cum ulativ e density fu nction (CDF) o f X as F X ( x ) = 2 Q π 3 2 λ 0 √ ρ 0 √ 2 x ! . (8) From (8), we have Prob ( X ≥ T 0 ) = 1 − 2 Q π 3 2 λ 0 √ ρ 0 √ 2 T 0 ! . ( 9) Combined (4) and (9), it is clear that the fo llowing condition has to b e satisfied: Q π 3 2 λ 0 √ ρ 0 √ 2 T 0 ! = 1 − ǫ 0 2 . (10) When ǫ 0 → 0 such that π 3 2 λ 0 √ ρ 0 √ 2 T 0 → 0 , with T aylor series expansion, we o btain the max imum a llow ab le value (via the monoto nicity of the Q f unction) of λ 0 asymptotically for α = 4 as λ ǫ 0 0 = ǫ 0 π  T 0 ρ 0  1 2 = ǫ 0 π  r − 4 0 β 0 − η ρ 0  1 2 . (11) As we can see f rom (11) that when the ou tage pro bability ǫ 0 is very small, the de nsity of TXs is a linear function of ǫ 0 . Th erefore, the tr ansmission capacity of the PR network is giv en by C ǫ 0 0 = R 0 λ ǫ 0 0 (1 − ǫ 0 ) , (12) where R 0 is the data rate wh en th e tran smission b etween th e TX and its associated RX is successful, which is set to be same for all the link s. I V . A S Y M P T OT I C A N A L Y S I S O F T H E T R A N S M I S S I O N C A PAC I T Y : O V E R L A I D N E T W O R K C A S E A. T ransmission Capac ity of the PR Network When th e SR n etwork is presen t, it intro duces interf erence to the PR network and the outage pro bability of the PR net- work will be incr eased. I f we set the target outage probability increment of th e PR ne twork as △ ǫ , we have P 0 = Prob     ρ 0 r − α 0 η + X i ∈ Π 0 ρ 0 | X i | − α + X j ∈ Π 1 ρ 1 | Y j | − α ≤ β 0     = ǫ 0 + △ ǫ. (13) W ith Y = P j ∈ Π 1 ρ 1 | Y j | − α , (13) can be r ewritten as Prob ( X + Y ≥ T 0 ) = ǫ 0 + △ ǫ. (14) The M GF of Y is given by Φ Y ( s ) = exp  − π λ 1 ρ 2 α 1 s 2 α Γ  1 − 2 α  . (15) Define Z = X + Y such th at the MGF of Z is given by Φ Z ( s ) = Φ X ( s )Φ Y ( s ) = exp  − π s 2 α Γ  1 − 2 α   λ 0 ρ 2 α 0 + λ 1 ρ 2 α 1   . For α = 4 , we have Φ Z ( s ) = exp h − π 3 2 s 1 2 ( λ 0 √ ρ 0 + λ 1 √ ρ 1 ) i , (16) and the PDF of Z is given by f Z ( z ) = π 2 ( λ 0 √ ρ 0 + λ 1 √ ρ 1 ) z − 3 2 × exp  − π 3 4 z ( λ 0 √ ρ 0 + λ 1 √ ρ 1 ) 2  . (17) Applying (17) in (14), w e hav e 1 − 2 Q π 3 2  λ 0 √ ρ 0 + λ 1 √ ρ 1  √ 2 T 0 ! = ǫ 0 + ∆ ǫ, (18) i.e., Q π 3 2  λ 0 √ ρ 0 + λ 1 √ ρ 1  √ 2 T 0 ! = 1 − ǫ 0 − △ ǫ 2 . (19) When ǫ 0 → 0 and ∆ ǫ → 0 , with bivariate T aylor series expansion, we obtain 1 2 − π λ 0 √ ρ 0 2 √ T 0 − π λ 1 √ ρ 1 2 √ T 0 = 1 − ǫ 0 − △ ǫ 2 . (20) If we choose λ 0 = λ ǫ 0 0 as in (11), the maximu m allowable value o f λ 1 correspo nding to a target outage p robab ility increment ∆ ǫ is given by λ ∆ ǫ 1 = 1 π  T 0 ρ 1  1 2 ∆ ǫ = 1 π  ρ 0 ρ 1 · r − 4 0 β 0 − η ρ 1  1 2 ∆ ǫ, (21) and the tran smission cap acity of th e PR network is given by C ǫ 0 = R 0 λ ǫ 0 0 (1 − ǫ 0 − ∆ ǫ ) . (22) As sho wn in (20), wh en the SR n etwork is p resented, the outage probability of the PR network can b e approximated by an affine fu nction o f λ 0 and λ 1 over asym ptotically small ǫ 0 ’ s and ∆ ǫ 0 ’ s. B. T ransmission Capacity of the SR Network Denote the ou tage probab ility of the SR network as ǫ 1 , the outage p robability of th e SR n etwork is given by P 1 = Prob     ρ 1 r − α 1 η + X i ∈ Π 0 ρ 0 | X i | − α + X j ∈ Π 1 ρ 1 | Y j | − α ≤ β 1     = ǫ 1 . (23) Re write (23) as Prob  Z ≥ ρ 1 r − α 1 β 1 − η  = ǫ 1 . (24) Define T 1 = ρ 1 r − α 1 β 1 − η , and we have Prob ( Z ≥ T 1 ) = ǫ 1 . (25) Similar to ( 19), we o btain Q π 3 2  λ 0 √ ρ 0 + λ 1 √ ρ 1  √ 2 T 1 ! = 1 − ǫ 1 2 . (26) When ǫ 1 → 0 , with bivariate T aylor series expansion, we have 1 2 − π λ 0 √ ρ 0 2 √ T 1 − π λ 1 √ ρ 1 2 √ T 1 = 1 − ǫ 1 2 . (27) Therefo re, th e outag e proba bility o f the SR network is given by ǫ 1 = π √ T 1 ( λ 0 √ ρ 0 + λ 1 √ ρ 1 ) , (28) and the tran smission cap acity of th e SR network is given by C ǫ 1 = R 1 λ ǫ 1 (1 − ǫ 1 ) , (29) where R 1 is the data rate ad opted b y su ccessful SR lin ks. On the other hand, if we set the target ou tage pro bability of the PR n etwork to b e ǫ 0 + ∆ ǫ , and set the target outag e probab ility of th e SR network to be ǫ 1 simultaneou sly , we could choose th e value of λ ǫ 1 in (29) a s follows λ ǫ 1 = min  λ ∆ ǫ 1 , λ ǫ 1 1  , (30) where λ ǫ 1 1 is g iv e n by (via (28)) λ ǫ 1 1 = ǫ 1 π  r − α 1 β 1 − η ρ 1  1 2 − λ ǫ 0 0 r ρ 0 ρ 1 . (31) C. Sum T ransmission Capacity of the Ov erlaid Network When th e SR network is pre sent, based o n the above anal- yses, th e sum tran smission capacity of the overlaid network s is g iv e n by C ǫ s = C ǫ 0 + C ǫ 1 = R 0 λ ǫ 0 0 (1 − ǫ 0 − ∆ ǫ ) + R 1 λ ǫ 1 (1 − ǫ 1 ) = R 0 λ ǫ 0 0  1 − π √ T 0 ( λ ǫ 0 0 √ ρ 0 + λ ǫ 1 √ ρ 1 )  + R 1 λ ǫ 1  1 − π √ T 1 ( λ ǫ 0 0 √ ρ 0 + λ ǫ 1 √ ρ 1 )  = ( R 0 λ ǫ 0 0 + R 1 λ ǫ 1 ) − π ( λ ǫ 0 0 √ ρ 0 + λ ǫ 1 √ ρ 1 ) ×  R 0 √ T 0 λ ǫ 0 0 + R 1 √ T 1 λ ǫ 1  . (32) Compared to the single n etwork case, th e gain of the trans- mission capacity (i.e., th e overall spectr um efficiency) of the overlaid networks over that of a single network is given by K g = C ǫ s C ǫ 0 0 ≈ 1 + C ǫ 1 C ǫ 0 . (33) D. T radeoff o f th e T ransmission Capacities Here we conside r two setups to study the trad eoff between the transmission capacities of the PR n etwork and the SR network. The first setup is that we change the value of ∆ ǫ only , and fix o ther p arameters ( ρ 0 , ρ 1 , r 0 , r 1 , β 0 , β 1 , η , and ǫ 0 ). The second setup is that we chan ge th e value of ρ 1 , and let oth er param eters ( ρ 0 , r 0 , r 1 , β 0 , β 1 , η , ǫ 0 , an d λ 1 ) b e fixed. Let us co nsider the first setup. When ǫ 0 is fixed, λ 0 is also fixed, see (11). Fr om ( 22), we can see th at C ǫ 0 is a linear function of ∆ ǫ . As such, when ∆ ǫ is increased, C ǫ 0 is r educed. Re write (29) as C ǫ 1 = R 1 π s T 0 ρ 1 ∆ ǫ 1 − r T 0 T 1 ǫ 0 − r T 0 T 1 ∆ ǫ ! . (34) From (3 4), we can easily verify that when p T 1 /T 0 > ǫ 0 , C ǫ 1 is a con vex fun ction of ∆ ǫ , and wh en ∆ ǫ < 1 2 ( p T 1 /T 0 − ǫ 0 ) , C ǫ 1 increases m onoton ically over ∆ ǫ . T able I N E T W O R K P A R A M E T E R S . Symbol Descript ion V alue ρ 0 Tra nsmission po wer of PR TXs 20 W ρ 1 Tra nsmission po wer of SR TXs 0.1 W r 0 Tra nsmission range of PR TXs 20 m r 1 Tra nsmission range of SR TXs 5 m η A verag e power of ambient noise 10 − 6 W β 0 T arge t SINR for PR network 10 dB β 1 T arge t SINR for SR network 10 dB Now , we con sider th e second setup. Rewrite (22) and (29) as f ollows, C ǫ 0 = R 0 λ ǫ 0 0 (1 − ǫ 0 − π √ T 0 λ ǫ 1 √ ρ 1 ) (35) and C ǫ 1 = R 1 λ ǫ 1   1 − π λ ǫ 0 0 q ρ 1 ρ 0 r − α 1 β 1 − η ρ 0 + π λ ǫ 1 q r − α 1 β 1 − η ρ 1   . (36) W e can easily sho w that when ρ 1 increases, C ǫ 0 decreases and C ǫ 1 increases. V . N U M E R I C A L R E S U LT S A N D I N T E R P R E TA T I O N S In this section, we present some n umerical results based on o ur pre v ious analyses and give some interpretations. W e set the values of the network parameters as in T able I un less otherwise specified. A. Single Network Case In Fig. 1, we show the normalized transmission capacity C ǫ 0 0 /R 0 as a function of the o utage prob ability ǫ 0 , as well as the d ensity o f PR TXs λ 0 vs. the o utage pr obability ǫ 0 . Note that these are exact r esults (not asymptotic ones) by using (2) and (10). W e could see from this figure that when ǫ 0 is about 0.55, C ǫ 0 0 is maximiz ed, and when ǫ 0 < 0 . 4 , λ 0 is nearly a linear function o f ǫ 0 , which verifies the asymp totic result in (11). 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 x 10 −5 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 x 10 −4 C ǫ 0 0 /R 0 Outage probability ǫ 0 Normalized transmission capacity C ǫ 0 0 /R 0 Density of PR TXs λ 0 λ 0 Figure 1. Normalize d transmission capaci ty/density of PR TXs vs. outage probabil ity for th e PR network when the SR network is absent. In Fig. 2, we show the normalized asymptotic transmission capacity C ǫ 0 0 /R 0 as a fun ction of the o utage prob ability ǫ 0 , and the up per and lower bou nds o f the transmission cap acity based on the results derived in [ 5], which verifies the tigh tness of th e upper bound . 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −5 Upper bound Lower bound Asymptotic result Outage probability ǫ 0 Normalized transmission capacity C ǫ 0 0 /R 0 Figure 2. Normalize d transmission capac ity vs. outage probability for the PR netw ork whe n SR network is absent. B. Overlaid Network Case The normalize d transmission capacity of the PR network C ǫ 0 /R 0 vs. the in crement of the ou tage probability ∆ ǫ of the PR network is shown in Fig. 3. As expected, C ǫ 0 /R 0 is in versely p ropor tional to ∆ ǫ . On th e other h and, since C ǫ 0 is a co n vex fun ction of ǫ 0 ; and when ǫ 0 < 1 − ∆ ǫ 2 , C ǫ 0 increases over ǫ 0 monoto nically for a fixed ∆ ǫ . 0 0.05 0.1 0.15 0.2 0.25 0.3 0.5 1 1.5 2 2.5 3 3.5 x 10 −5 Normalized transmission capacity C ǫ 0 /R 0 Increment of the outage probability of the PR network ∆ ǫ ǫ 0 = 0 . 15 ǫ 0 = 0 . 05 ǫ 0 = 0 . 10 Figure 3. Normalize d transmission capacity of the PR netw ork vs. increment of the outage probabili ty of the PR netw ork. In Fig. 4, we show the normalized transmission capac ity of the SR n etwork C ǫ 1 /R 1 as a function of ∆ ǫ , see ( 29). As shown in the fig ure, we see that C ǫ 1 increases mono tonically over ∆ ǫ , since the larger ∆ ǫ is, the larger the values of λ ǫ 1 and ǫ 1 are, but th e effect of λ ǫ 1 on C ǫ 1 is dom inant wh en ǫ 1 is small. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 x 10 −4 Normalized transmission capacity of the SR network C ǫ 1 /R 1 Increment of the outage probability of the PR network ∆ ǫ ǫ 0 = 0 . 05 ǫ 0 = 0 . 1 5 ǫ 0 = 0 . 1 0 Figure 4. Normalize d transmission capacity of the SR netw ork vs. increment of the outage probabili ty of the PR netw ork. Assuming that R 0 = R 1 , the capacity gain K g of the overlaid networks (i.e., the sum transmission capacity) over that of a single network is shown in Fig. 5, see (33). W e see that K g increases over ∆ ǫ since the extra capacity contribution from th e secondary ne twork incr eases over ∆ ǫ . 0 0.05 0.1 0.15 0.2 0.25 0.3 0 20 40 60 80 100 Increment of the outage probability of the PR network ∆ ǫ ǫ 0 = 0 . 05 ǫ 0 = 0 . 15 ǫ 0 = 0 . 10 Gain of the transmission capacity K g Figure 5. Gain of the transmission capaci ty of the ov erlaid network over that of the PR network. In Fig . 6, we show the tradeoff between the normalize d transmission cap acity of the PR n etwork C ǫ 0 /R 0 and tha t of the SR network C ǫ 1 /R 1 when ∆ ǫ chan ges as a n intermedia te variable. W e see th at C ǫ 0 decreases over C ǫ 1 , which verifies the result in Sectio n I V . V I . C O N C L U S I O N S In this p aper, we extended th e co ncept o f tr ansmission capacity defined for the single network case to overlaid network c ase. By co nsidering th e mutual in terferenc e effect across two ov erlaid n etworks, i.e., the PR network vs. the SR network, we der i ved the tran smission capacities for the se two n etworks a nd studied their tradeoffs. Different f rom th e previous approach f or the single network case, we resorted to obtain the asy mptotic solutions for these capacities. The results 0 1 2 3 4 5 6 x 10 −4 0.5 1 1.5 2 2.5 3 3.5 x 10 −5 Normalized transmission capacity of PR network C ǫ 0 /R 0 Normalized transmission capacity of SR network C ǫ 1 /R 1 ǫ 0 = 0 . 15 ǫ 0 = 0 . 10 ǫ 0 = 0 . 05 Figure 6. Tra deof f of the transmission capaci ties of the PR and the SR netw orks when the v alue of ∆ ǫ i s changed. showed that by letting a SR network co exist with a le g acy PR network, the spectrum efficienc y per unit area could be increased significantly . 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