A Polynomial Rooting Approach for Analysis of Competition among Secondary Users in Cognitive Radio Networks

Transcription

1 A Polynomial Rooting Approach for Analysis of Competition among Seconary Users in Cognitive Raio Networs Xiaohua Li, Chengyu Xiong an Wenel Caeau Department of Electrical an Computer Engineering State University of New Yor at Binghamton Binghamton, NY Abstract The competitive coexistence of a large number of seconary users in heterogeneous cognitive raio networs (CRN) is a challenging issue. In this paper, we aress this issue by eveloping an efficient approach to analyze the competition among the seconary users. Base on the arov moel ban that we evelope recently, we can change the evaluation of the seconary user throughput into stuying a special root of a polynomial that is much more simplifie. Uner this framewor, we stuy the competition among the seconary users in three special cases: homogeneous CRN with ientical seconary users, heterogeneous CRN with a group of ientical seconary users plus an outlier, an heterogeneous CRN with two ifferent groups of seconary users. In each case, we erive the polynomial, prove the uniqueness of the root, an erive the throughput of each seconary user base on the root. Furthermore, we erive close-form approximate solutions to the root, base on which constraine optimization can be formulate to stuy the optimal competition strategies for the seconary users. The optimization results are in water-filling principles an can be erive in closeform expressions. Simulations are conucte to verify the analysis results. Keywors cognitive raio networ, arov chain, throughput, ynamic spectrum access, polynomial root I. INTRODUCTION Cognitive raio networs (CRN) allow seconary users (SU) to reuse the spectrum currently occupie exclusively by primary users (PU) with ynamic spectrum access (DSA). The SUs conuct spectrum sensing, an access the spectrum that is not occupie by the PUs. The SUs must vacate the spectrum whenever the PUs become active. Due to the inherent flexibility of cognitive raios, ifferent SUs may aopt ifferent transmission parameters an ifferent spectrum sensing an spectrum access strategies. For example, since the PU signals are ifferent temporarily an geographically, the SUs may choose to aopt either more conservative or more aggressive polices towar sensing errors. Some SUs may exploit hanshaing or control channels more than others. In particular, there are many ifferent spectrum access strategies for the SUs to aopt. This not only maes the CRN heterogeneous, but also raises the issue of competition among the (possibly selfish) seconary users when they are trying to gain the access of the limite spectrum resource. This wor was supporte in part by National Science Founation uner Grant CNS Although there have been extensive research publishe in CRN, incluing spectrum sensing, Physical-layer moulation, AC/Networ layer protocols, harware/testbe evelopment, etc, very limite stuy has been conucte on the competitive coexistence of a large number of ifferent SUs in heterogeneous CRNs. There is a class of wor on the CRN transmission power/capacity analysis base on the physical-layer interference analysis, but they are conucte uner certain simplifie communication moels []. There is another class of wor on CRN AC protocol esign an analysis, such as the performance analysis of the CSA-type protocols [2]- [4]. Nevertheless, the homogeneity assumption overloos most of the complex competitions. Uner the selfish an ecentralize operations assumption, various game-theoretic spectrum sensing an spectrum access strategies have been evelope to aress the competition. However, the performance was usually too complicate to analysis. ore importantly, all the seconary users are still assume to have the same strategy space, which inherently maes the CRN homogeneous rather than heterogeneous. As a result, the equilibrium of the CRN in these wors may still be very ifferent from the competitive coexistence of heterogeneous CRNs. The large number of issimilar users maes the moeling an analysis of heterogeneous CRN challenging. ost of the existing wors on CRN analysis that consier more or less the heterogeneous nature of CRN have to mae substantial simplifications [5]-[2]. The SU s competition for spectrum access is sippe in [5] an [6] because only simultaneous transmissions are consiere, where spectrum access becomes just the constant arrival/eparting rate. In aition, most of these wors focus on the simple scenario of two or three seconary users only to simplify the arov chain moeling. The general case becomes prohibitively complex ue to the rapily increase number of arov states an transitional probabilities. In [3], we evelope an innovative arov oel Ban (B) for moeling an analyzing the heterogeneous CRN with multiple issimilar SUs an multiple issimilar channels. We focuse on evaluating the SU throughput uner certain simple spectrum access strategies. Our basic iea was to ecompose the complex arov moel into a ban of relatively separate an thus simplifie arov chains, one for each SU. This maes the arov chains easy to analyze. In aition, B allows us to exploit spatial, channel an user

2 e-correlations to rastically simplify the interference analysis, which maes it tractable to analyze large heterogeneous CRNs. In this paper, we focus on analyzing the competing behavior of the SUs using ifferent transmission parameters an ifferent spectrum access strategies. We change the performance evaluation of the SUs into stuying special roots of some polynomials. By stuying the properties an by eriving approximate close-form solutions to the roots, the competition among the seconary users an their throughput can be conveniently analyze. The organization of this paper is as follows. In Section II, we give the system moel an the B analysis framewor. In Section III, we aopt appropriate networ ecomposition techniques to change the networ throughput evaluation into polynomial rooting. Then in Section IV, we stuy the competition among the seconary users uner three typical CRN scenarios. Simulations are conucte in Section V, an conclusions are given in Section VI. II. SYSTE ODEL The CRN we consier in this paper consists of a set of N ranomly istribute SUs an some PUs. Each SU is a transmitting-receiving pair of noes, where the transmitting noe transmits to the receiving noe. While transmitting to the receiving noe, the transmitter creates interference to other users. We assume there are channels available for the N SUs to choose. For each SU, we consier a cognitive raio transmission moel that inclues four states: spectrum sensing, ata transmission, waiting an channel switching. If the spectrum sensing inicates the channel is available for seconary access, then the cognitive raio transmits a ata pacet, an the moel shifts into the ata transmission state. If the spectrum sensing inicates the channel is not available (ue to the PU activity, interference from other SUs activity, or the lac of offere loa from this SU), then the cognitive raio enters the waiting state, which will be followe by the channel switching state. We assume that the cognitive raio always shifts bac to the spectrum sensing state after either the channel switching state or the ata transmission state. We assume that the SU can successfully etect the activity of the PUs. Therefore, we can enote the activity of the PUs in each channel by a ranom variable θ, which enotes the probability that the channel is not use by any PU. This maes us to focus on analyzing the coexistence of SUs via their mutual interference. In the spectrum sensing state, the signal receive by the SU i s receiver in the channel is N u i, (n) = f j, Pj, h ji, s j (n)+v i, (n) () j=,j i where P j, is the transmission power that the SU j spens in the channel, s j (n) is the signal transmitte by the SU j, h ji, is the flat faing channel from the SU j s transmitter to the SU i s receiver, an v i, (n) is the AWGN with zero-mean an power σ 2 i,.thevariablef j, =or 0 enotes whether the SU j is currently using the channel. The instantaneous π i, q i, πw i, πs i, q i, channel πi, s q i, z i, q z i, π π i, i, c i channel π w i, z i, πw i, π i, π s q i, i, q i, channel Fig.. arov oel Ban (B) for multi-user heterogenous CRN, where only the arov chain for the SU i is shown. Each of the SUs has a similar arov chain. Each arov chain has 3 + states: 3 states in each channel, plus a single channel switching state for all the channels. The transitional probabilities q i, an z i, are etermine by the spectrum sensing an the spectrum access strategies. signal-to-noise ratio (SNR) of the SU i in the channel can thus be obtaine from () as N γi, s = j=,j i f j,p j, h ji, 2 σi, 2 (2) If SNR γi, s is larger than the etection threshol Γ i,, then the SU i maes a ecision that the channel is not available for transmission. It turns to the waiting state. Note that ue to the asynchronous an ecentralize operation of the competing seconary users in heterogeneous CRNs, we o not consier specifically any common AC-layer protocol where the seconary users compete for channel access in a slotte signaling structure. In our case, ifferent seconary users may use ifferent slot/pacet urations, an such transmission parameters are competition factors. As [3], we can use a arov oel Ban (B) to moel the competitive spectrum access among the N CRN SUs. The B consists of a arov chain for each SU, as shown in Fig., where πi, s,π i,,πw i, are the probabilities of the SU i staying in the channel sensing, ata pacet transmission an waiting state in the channel. The probability πi c refers to the probability that the SU i staying in the channel switching state. The transitional probability q i, enotes the probability that the channel can be use by the SU i to transmit a ata pacet. It is a composite result of PU etection, SU carrier sensing, as well as the offere loa. The transitional probability z i, enotes the probability for the SU i to choose the channel to sense an access. It is etermine by the channel access strategy. We assume 0 z i, an z i, =, i =,,N, (3) = which means that a SU just uses one channel each time. Note that although ifferent SUs arov chains loo separate from each other, their transitional probabilities q i, are inter-relate. In aition, epening on the strategies use in channel selection, the transitional probabilities z i, for ifferent seconary users may also be inter-relate. As a matter

3 of fact, the B approach simplifies the arov moel, but increases the complexity of q i, an possibly z i,. Fortunately, B can bring some useful networ ecomposition properties to reuce the complexity. For each SU i, the urations of the spectrum sensing, ata transmission, waiting, an channel switching are Ti, s, T i,, Ti, w an T i c, respectively. Note that we have efine the SU s offere loa as a ranom variable α i, but we will not use the SU s exiting probability. Such exiting probability, if neee, can be integrate into the ata pacet/slot length Ti,.The pacet/slot lengths are variables to be stuie in this paper. From [3], the steay state probabilities of the arov moel of Fig. is given as an π s i, = z i, ( +2 ) z i,l l= q i,l ( q i, ) π i, = q i, π s i,, π w i, =( q i, )π s i,, π c i = q i, z i, π s i,. (5) In this paper, we focus on analyzing the throughput of each seconary user. Each SU i has a throughput R i, obtaine in the channel, wherer i, is efine as the fraction of time when the SU i is using the channel for ata pacet transmissions. To simplify analysis, we use the throughput efinition that is more simplifie than those use in [3]. Specifically, we o not consier the transmission channel capacity in the throughput efinition. Note that the ata transmission can be assume free of collisions ue to the perfect PU an SU sensing assumption. Base on each single SU s throughput R i,, we can erive the overall throughput for a group of seconary users or even the entire CRN. We can also efine the channel usage as the probability of a channel being occupie by all the SUs, which can also be erive from R i,. III. NETWORK DECOPOSITION AND THE POLYNOIAL ROOTING APPROACH The most complex tas is perhaps to analyze q i, which requires us to analyze the mutual interference among all the SUs. The probability q i, is closely relate to the SU s spectrum sensing strategy. In our case, it actually epens on the PUs activity, the other SUs activity, as well as this SU s offere loa. We efine [ ] f q i, = θ α i P j, P j, h ji, 2 < Γ i,, j =,,N,j i, σ 2 i, (6) which means that the SU i can use the channel to transmit a ata pacet only if there is no PU activity in the channel, thesui has ata pacet to transmit, an the interference from any other SU using the channel simultaneously is low enough. The probability that another SU j is also using the channel can be moele as a Bernoulli ranom variable with (4) probability β j, = P[f j, =] (7) πj, = T j, πj ct j c + ( ). l= πj,l s T j,l s + π j,l T j,l + πw j,l T j,l w From (4) we can easily see π s i, πi,l s = z i,( q i,l ) z i,l ( q i, ). (8) Applying (4) an (8), we can erive from (7) β j, = z j,q j, Tj, q j, Tj c+. (9) [T s l= j,l +q j,lt j,l +( q j,l)tj,l] w z j,l q j,l In other wors, f j, is a Bernoulli ranom variable with probability β j,. Accoring to (6), q i, =0whenever there is a SU j with f j, =an P j, h ji, 2 /σi, 2 Γ i,. In this paper, we assume that the SUs use very low etection SNR threshols Γ i, so that (6) can be simplifie to where q i, = θ α i N j=,j i c i,j = 2+ T s ( c i,j β j, ). (0) i, T j,. () Note that the probability that the SU j is transmitting ata pacets while the SU i is conucting spectrum sensing is πj, (2T j, + T i, s ) πj ct j c + ( ) l= πj,l s T j,l s + π j,l T j,l + πw j,l T j,l w because when the SU i begins spectrum sensing, it can etect the SU j s transmission when the latter is transmitting from Tj, secons before the sensing to T j, + T i, s after. Following (9) we can erive the probability as c i,j β j,. The essential iea of (0) is that we set the threshol Γ i, small enough so as to forbi simultaneous transmissions among the seconary users. This is similar to the practical AC-layer protocols such as Carrier Sense ultiple Access (CSA) that prevent collisions (i.e., simultaneous transmissions) among the users. This simplification is reasonable in practice when the SUs have relatively strong interference among each other, such as in closely istribute networs. It was calle spatial e-correlation in [3] because it effectively subivies a large CRN into many smaller un-correlate clusters. In our case, it maes us focus better on the competition analysis. Generally speaing, wea mutual interference has negligible competition effects, an can thus be omitte. In this paper, the throughput of the SU i achieve in the channel is efine as R i, = β i,, (2) which is the uty-cycle when the SU i is staying in the ata transmission state. Due to the assumption of perfect PU

4 sensing an SU sensing, ata transmission can be consiere as free of collisions. Therefore, (2) is the probability of the successful ata transmission in the channel. The equations (9)-(2) are general expressions that can be use to erive numerically the throughput performance of SUs in general heterogeneous CRNs. However, they are not convenient for more insightful analysis ue to their complexity an the interconnection among all the SUs an all the channels. For analysis purpose, we nee to aopt some reasonable assumptions to further simplify the expressions. First, in (9), we fin that there is a special connection between Ti,l an T i,l w which are the slot lengths of the ata transmission slot an the waiting slot, respectively. Without loss of generality, we assume that Ti, = T i, w, i.e., the uration of waiting state is the same as the ata transmission state. In practice, when a channel is etecte as unavailable, the SUs usually wait for just one ata pacet transmission slot uration before trying another channel. Therefore, this assumption is highly practical an reasonable. Then (9) can be reuce to β j, = z j, q j, T j, ( q j, )[T c j + l= (T s j,l + T j,l ) z j,l q j,l ]. (3) To further simplify the expressions, we assume that each SU uses the same slot length parameters when accessing all the channels. Therefore, T s j, = T s j, T j, = T j, T w j, = T w j, for all =,,. In aition, we assume that T c j =0 because channel selection is just a ecision maing without the nee of any transmission or receiving. The information for the ecision maing can be gathere an processe uring the waiting state. Then, we can simplify (3) into β j, = z j, q j, Tj ( q j, )(Tj s + T j ) z i,l. (4) l= q i,l In (4), each β j, still epens explicitly on the same SU s activities in all the channels. Fortunately, as shown in [3], we can tae channel e-correlation to remove the explicit mutual epenence among all the channels l. The channel ecorrelation is conucte by replacing z i, with x i, where x i, = z i, ( q i, ) z i,l. (5) l= q i,l From the properties 0 q i, an = z i, =, we can see that 0 x i,, x i, =. (6) = In aition, given x i,, we can uniquely etermine z i, via z i, = x i, ( q i, ) l= x i,l( q i,l ). (7) Note that z i, is the channel selection probability, while x i, is a new ranom variable escribing the same channel selection probability, i.e., the probability for the SU i to select the channel. For example, with uniformly-istribute ranom channel selection strategy, we have both z i, = an x i, =. Nevertheless, x i, an z i, are ifferent in general. With the channel e-correlation, we can reuce the CRN throughput evaluation expressions (9)-(2) into { Ri, = b i x i, q i,, N q i, = a i, j=,j i ( c (8) i,jb j x j, q j, ), where b i = Ti /(T i s + Ti ) is the transmission protocol parameter an a i, = α i θ inclues both the offere loa an the PU channel activity. Note that the throughput in channel, i.e., R i,, relies on the parameters in the channel only. Base on (8), we can calculate the throughput of each SU numerically by first fining N probabilities q i,, i =,,N, which are the roots of N nonlinear equations. However, in general, it is possible to change it into a simpler form, i.e., fining a special root of one polynomial, as outline below. Consiering two SUs i an j. From (8), we fin that a i, q i,( c i,i b i x i, q i, )=a j, q j,( c j,j b j x j, q j, ). (9) Define D = a i, q i,( c i,i b i x i, q i, ), (20) which is a special parameter share by all the SUs. The value D satisfies some equations such as N D = ( c i,j b j x j, q j, ), (2) j= Conitione on D, ifferent SU s q i, an R i, can be evaluate separately, which is calle user e-correlation in [3]. Since D is a common parameter for all the SUs, it is possible to compare ifferent SUs performance even without nowing the value of D. This provies a promising way for efficient an incisive CRN performance evaluation an analysis. We can formulate some polynomials of D in orer to fin its values or the properties. Specifically, from (20), q i, has solution q i, = ± 4c i,i b i x i, a i, D. (22) 2c i,i b i x i, We usually just nee the minus sign in (22) because x i, is small. If x i, is small enough, we can even use approximately q i, a i, D.ThenD is the root to a polynomial obtaine by substituting (22) bac into (2). In summary, for CRN performance analysis, we can first fin the polynomial roots as D, an then use the value D to calculate q i, as well as the throughput R i,. This is the general outline of the polynomial rooting approach for CRN performance analysis. The etails of this approach may change when consieringspecific CRN scenarios, which will be shown in Section IV. Given each SU s throughput R i,, the overall throughput of a group of SUs obtaine in a set of channels can be efine as R + = R i,. (23) i Unfortunately, such trivial aitions may accumulate the errors in R i,, especially the inevitable systematic moeling errors (in aition to the numerical an noise-inuce errors). Even though the systematic error in each R i, is negligibly small,

5 the accumulate error over a large number of SUs an channels may become significant. To avoi this problem, for large CRN, we may use the following throughput efinition R = ( R i, ), (24) i which can suppress systematic errors more effectively. For example, if the estimate throughput for each SU is ˆR i, = R+ɛ, where ɛ is the systematic error that are ientical for all the SUs, then (23) gives ˆR + = NR+Nɛ, where the error increases linearly with N. In contrast, (24) gives ˆR = i ( R ɛ) N ( R) N + Nɛ( R) N,wherethe accumulate error is attenuate by the factor ( R) N which becomes very small for large N. Nevertheless, since (23) is easier to analyze an is accurate in CRN with moerate size, we still prefer it for analysis. Another performance metric base on R i, is the probability that a channel is use by all the SUs. It measures how the SUs changes the spectral environment, which may be calle spectral mutation. This probability can be efine as U + N = N R i,, or U = ( R i, ). (25) i= There are some interesting questions with respect to (25), such as whether a group of SUs can change arbitrarily U +,or, for an arbitrary U + whether a group of SUs can achieve it by ajusting their transmission parameters an strategies. All these are highly relative to the competition stuy of the SUs. i= IV. COPETITION ANALYSIS OF THREE SPECIAL CRNS The framewor outline in Section III moels the general CRN where each seconary user can aopt transmission parameters or strategies that are ifferent from others. This obviously maes the expressions an analysis unnecessarily complex. In many practical situations, we may have a group of SUs that aopt similar transmission parameters an strategies. We call them homogeneous SUs. In aition, there are some other SUs that aopt another set of transmission parameters an strategies, whom we call outliers. These two groups of SUs may compete with each other within the same CRN. In this section, we focus on analyzing their competition. We consier three cases: CRN with N homogeneous SUs, CRN with N homogeneous SUs plus one outlier, an CRN with two ifferent groups of homogeneous SUs. In each case, we will evelop in etails the polynomial rooting approach. A. CRN with N homogeneous SUs We first consier the CRN where all the N SUs have the same transmission parameters an spectrum access strategies. Specifically, in (8) we have c i,j = c, b i = b, a i, = a,an x i, = x,fori =,,N. Using these parameters in (8), we can see that the SUs have the same q i, an R i,. Denote q i, = q an R i, = R. To fin the throughput R is to solve { R = bx q, q = a ( cbx q ) N (26), Note that each SU s throughput in the channel can be evaluate inepenently from other channels, given the probability x of channel selection. Proposition. Consiering the constraint 0 q, the equation (26) has one an only one vali solution R = bx y N an q = y N,wherey is the root of the polynomial cbx y N + a y =0, (27) that satisfies 0 y. Proof. From the secon equation in (26), we have q = a ( cbx q ). (28) Define y = q, we can change (28) into (27). Obviously, for 0 q, wehave0 y. In aition, q = y N. The solutions to (26) can be obtaine from the roots of the polynomial f(y) =cbx y N + a y. Wefinthat f(0) = < 0, anf() = cbx + a > 0 because 0 a. Note that a = θ α where α is the offere loa for the SU which is assume ientical among all the SU. In aition, f(y)/ y =(N )cbx y N 2 + a > 0 since all the variables are non-negative. This means that f(y) is monotone increasing over 0 y. Therefore, the function f(y) has one an only one root that satisfies 0 y. This proves the existence of a unique solution to (26). Therefore, each SU can achieve a unique throughput R in the channel. The throughput is ientical for all the SUs. The throughput epens on the transmission parameters b, a an the channel selection probability x. It can be calculate from the unique vali root of (27). One of the interesting yet challenging issue is how the SUs can maximize their throughput by selecting the optimal transmission parameters b an channel selection probabilities x. From the throughput expression an (27), we have R = bx y N = c ( a y). Theoretically, the throughput can be optimize by looing for the smallest root min y, (29) {b,x } with the constraints = x =, 0 y, an (27). Unfortunately, this optimization is ifficult to solve, mainly because the root of the high-orer polynomial (27) is elusive. We turn to an approximately but informative analysis instea. Notice from (27) that y (a /c) for large N an. Therefore, we assume y =(a /c) + ɛ, where ɛ. Substituting y into (27) an applying the approximation ((a /c) + ɛ) N a /c +(N )(a /c) N 2 ɛ, we can get an approximate solution to (27) as y (a /c) a ba x + c +(N )bc a x. (30) Substituting (30) bac into R =( a y)/c we can obtain R c c N c ba x + c N c +. (3) +(N )bc a x

6 From (3) we can see that larger x or b gives higher throughput. Each SU get approximately /(N ) throughput from a channel. For a single SU, it can choose x =to maximize its throughput in the channel. Let us consier the problem of looing for the optimal channel selection strategy {x } to maximize an SU s throughput in all channels. Note that it also maximizes the overall CRN throughput. Using (3), the optimization problem max {x } R, = s.t., x = (32) = becomes a special case outline in the Appenix, where the optimal solution follows the water-filling principle. Specifically, we have α = ( ) N 2 c N c, β =(N )bc a (33) Then we can follow (58) to fin λ with all the j =,, that satisfies (57). Then the optimal channel selection probabilities can be foun from (59). In particular, consier the special case with a =an b, i.e., channels are all available, an sensing slot uration is negligibly small compare with the ata transmission uration. Then we can fin that x =/, which means uniformly istribute ranom channel selection is optimal. In this special case with x =/, from (3), we fin R 2(N ) + 2 ( 2 ) per SU per channel, which is between /(2N) to /N. Note that /N is the ieal throughput. Therefore, the approximation metho tens to uner-estimate the throughput in large CRN. From the water-filling principle (59), we can reaily see that each SU shoul increase the probability x on channels with the stronger a (i.e., larger θ ). In other wors, SUs shoul utilize more the channels that are use less by the PUs. In aition, throughput increases with larger b an a. From the x erive from (59), we can fin q,anthen calculate the practical channel selection probabilities (that is actually use irectly by the SUs) z accoring to (7). In particular, for ientical a = a, it is easy to see that the optimal uniformly istribute ranom channel selection means x = z =. B. CRN with N homogeneous SUs an one outlier If there is one outlier SU that aopts transmission parameters an spectrum access strategies that are ifferent from the rest of the N SUs, we woul lie to see how this outlier can compete with the rest of the SUs. Denote the parameters for each of the N homogeneous SUs as b, a, q, x an R, while the corresponing parameters for the outlier are b 2, a, q, x an R. Because Ti, s is usually much less than Tj,, without loss of generality, we use a constant c for all the SUs. The equation (8) becomes R = b x q R = b 2 x q q = a ( cb q x ) N 2 (34) ( cb 2 q x ) q = a ( cb q x ) N Similar to Section IV-A, we can formulate the polynomial rooting approach for (34), an fin the solution as below. Proposition 2. The equation (34) has one an only one vali solution which can be expresse as { q = a y cb x, (35) q = y N, where y is the root of the polynomial c 2 b x b 2 x a y 2N 3 cb x a y N 2 a N 3 y+a N 2 =0, (36) that satisfies 0 y. Proof. From the secon equation in (34), we have cb q x = q. (37) a which gives q = q a. (38) cb x Substituting it into the first equation in (34) we obtain q 2, q2, cb x a a = a q a ( cb 2 q x ). (39) Define y = q, then we can revise (39) into (36). Equations in (35) can also be obtaine from (38). Next, to show the existence an the uniqueness of the root, consier the constraint 0 y for the equation (36). Following similarly Proposition, at y = 0, the left han sie of (36) is larger than 0. At y =, the left han sie of (36) is less than 0. From the erivative of the left han sie of (36) with respect to y, we can show that the erivative is always negative. Therefore, the left han sie polynomial is monotone ecreasing. There is one an only one solution for 0 y. For the CRN with N homogeneous SUs an one outlier, every homogeneous SU has the same throughput R in the channel as other homogeneous SUs. However, the outlier may have ifferent R given the ifferent transmission parameters b 2, a an the channel selection probability x. Their throughputs can be calculate after fining the unique vali root of (36). From (34) an (35), the throughputs of each homogeneous SU an the outlier are connecte as (b 2 a x ) R + cr =. (40) If counting the overall throughput of the homogeneous SUs, we have (b 2 a x ) R + c N [(N )R ] =. This shows the throughputs can be ajuste by the outlier s transmission parameters. The throughputs R = c ( a y), R = b 2 x y N (4)

7 can be stuie or optimize from the root y of (36). Obviously, R ecreases with y while R increases with y. Whenb x are etermine, the outlier can choose b 2 an x to ajust the root y in (36). any questions can be investigate, such as how the outlier can maximize its throughput, or how it can reuce the throughput of the homogeneous SUs. Nevertheless, ue to the lac of close-form solution to (36), the analysis is not easy. In the sequel, we rive an approximate close-form solution to (36) to give more incisive analysis of the competition among the homogeneous SUs an the outlier. The proceure is similar to the erivation of the approximate solution in Section IV-A. Since the expressions are more complex, we assume that a = a =to simplify the notation. The proceure will be similar for the general case though. This assumption means that the channel has no PU activity an the SU s offere loa is. Then, let y =+ɛ, where ɛ. Applying the approximation y K +Kɛ to (36), we can obtain c 2 b x b 2 x 2cb x ɛ = (2N 3)c 2. (42) b x b 2 x +2(N 2)cb x Base on (42), we have the approximate solutions { R c ( y) = c ɛ R = b 2 x ( + ɛ) N (43) b 2 x [ + (N )ɛ] As a special example, in case c =2, b = b 2 =,an x =/ (which is the optimal channel selection probability for the homogeneous SUs), we have x ɛ. (44) /4+N 2 (2N 3)x The outlier can use its x to ajust both R an R.For example, it can maximize = R R over x,where the Appenix suggests some water-filling results. In general, the homogeneous SUs can optimize their own throughput R via optimal channel selection probabilities x. This leas to the constraine optimization max R, s.t., x =. (45) = = This optimization can be formulate into the form (54), an a water-filling principle on the solution x can be obtaine. In this case, the optimal x is a function of x. For the outlier to enhance its competition, it may select its own x to either solely maximize its own throughput R or to minimize the homogeneous SU s throughput R.Thefirst approach is the optimization max R, s.t., x =, (46) = = an the secon approach is the optimization min R, s.t., x =. (47) = = Both the optimization can be revise into the form (54) an fin the optimal water-filling principle on x. Details are sippe to save space. C. CRN with two groups of homogeneous SUs Now consier the case that there are two groups of SUs competing for spectrum access. There are N SUs in the first group that have the parameters b, a, x, q an R.The rest N 2 = N N SUs form the secon group that have the parameters b 2, a, x, q an R. They share the same constant c. From (8) we have R = b x q R = b 2 x q q = a ( cb q x ) N ( cb 2 q x ) N2 q = a ( cb q x ) N ( cb 2 q x ) N2. (48) We can construct a polynomial that can be use to stuy (48). Proposition 3. The equation (48) has one an only one vali solution that can be written as { q = ± 4cb x a a y( cb2x y) 2cb x, (49) q = y, where y is the root of the polynomial cb x a a N N y N N ( cb 2 x y) N +a N y N ( cb 2 x y) N 2 N =0. (50) that satisfies 0 y. Note that only one q in (49) satisfies 0 q. Proof. From (48) we have a q ( cb x q )=a q ( cb 2 x q ) (5) Define y = q 2,, then we have (49). From the secon equation in (48) we can get cb x q = a N y N ( cb 2 x y) N 2 N, (52) which gives ± 4cb x a a y( cb 2x y) =2a N y N ( cb 2 x y) N 2 N. (53) Taing square of both han-sies, we can get the polynomial equation (50). To show the solution is unique, similar to Proposition, the left han sie of (50) is less than 0 at y =0, an is larger than 0 at y =. From the erivative of the left han sie of (50) with respect to y, we can show that the erivative is always positive. Therefore, the left han sie of (50) is monotone increasing. There is one an only one solution satisfies 0 y. Even though q, has two possible values in (49), only one is vali, an the other oes not satisfy 0 y. This can be explaine if we form an equation in terms of q rather than q in (50). In this case we can show that there is only one vali solution to q. Obviously, R increases with y, while R ecreases with y. The analysis of the competition between the two groups of SUs can be conucte similarly as Section IV-B. To save space, however, we o not repeat it since the expressions are substantially longer. We stuy some of their properties in simulations instea.

8 R + : overall CRN throughput Poly Root etho Approximate etho Numerical: Root Numerical: General Simulation N: number of seconary users R *: CRN SU Throughput Analysis: Homo SU Analysis: Outlier Simulation: Homo SU Simulation: Outlier N: number of seconary users Fig. 4. Throughput of each seconary user achieve in all channels in the CRN with various number of homogeneous seconary users an one outlier. Fig. 2. Overall throughput of all the seconary users in all channels for CRN with various number of homogeneous seconary users. R : throughput per SU Poly Root etho Approximate etho Numerical: Root Numerical: General Simulation R *: CRN SU Throughput Analysis:Group Analysis:Group 2 Simulation:Group Simulation:Group N : number of SUs in Group N: number of seconary users Fig. 3. Throughput per seconary user per channel in CRN with various number of homogeneous seconary users. V. SIULATIONS In this section, we report our simulation of the numerical evaluation of the CRN throughput base on the analysis results erive in Sections III an IV. We verify them by the onte- Carlo simulation of a real ranom CRN with N seconary users an channels ( Simulation ). The noes positions were ranomly generate within a square of meters. The SNRs were calculate as ij where ij is the propagation istance. We evaluate the overall CRN throughput R + in aition to the throughput per SU. First, we simulate the CRN with homogeneous SUs only for =5, b =0/, a =0.5, c =2.. We numerically calculate the throughputs by using the equations (9)-(2) ( Numerical:General ), the equation (26) ( Numerical:Root ), the Proposition ( Poly Root etho ), an the equation (3) ( Approximate etho ). We use x =/ =/5 since it was optimal. Simulation results were shown in Fig. 2 for the overall throughput R + aninfig.3forpersuper channel throughput R. It can be clearly seen that the analysis results fit well with the onte-carlo simulation results, which verifie that our analysis was vali. Fig. 5. Throughput of each seconary user achieve in all channels in the CRN with various number of homogeneous seconary users in two ifferent groups. Next, we simulate the CRN with homogeneous SUs plus one outlier. Parameters are similar to the first experiment, except that the outlier has a =0.9. Simulation results were shown in Fig. 4, which shows the average throughputs of each homogeneous SU an the outlier achieve in all the =5channels. The analysis results fit well with the onte- Carlo simulation results, an the outlier can gain much more throughput by increasing its offere loa. Finally, we simulate the CRN with N homogeneous SUs in Group an N 2 homogeneous SUs in Group 2. The total number of SUs were N =52. Parameters are similar to the first experiment, except that the Group 2 SUs have a =0.9. Simulation results were shown in Fig. 5, which shows the average throughputs of each homogeneous SU in the two groups achieve in all the = 5 channels. The analysis results fit well with the onte-carlo simulation results. VI. CONCLUSION We evelope an efficient approach to evaluate the competition of heterogeneous CRN with a large number of ifferent users. Base on the arov oel Ban, we applie some networ ecorrelation techniques to reuce the complex problem into the problem of looing for the roots of some

9 special polynomials. This polynomial rooting approach provies feasible ways to analyze the throughput an competition among the seconary users. We consiere three ifferent CRN cases with homogeneous CRN plus some outliers. We set up the polynomials, stuie their properties, an evelope approximate close-form solutions that were convenient for various optimizations. Water-filling principles were obtaine for optimal performance of the CRN an for the competition among the seconary users. APPENDIX A CLASS OF CONSTRAINED OPTIIZATIONS WITH WATER-FILLING SOLUTIONS For the constraine optimization problem max J = = a + b x c + x, s.t., x =,x 0 (54) = we can first change J into the following form b a c J = + b +. (55) = = c x Since the first summation in (55) is a constant, the optimization (55) is equivalent to the following stanar optimization max J 2 = = α +β x, s.t., x =,x 0. (56) = With the Lagrange multiplier metho, efine J 3 = = ( α +β x + λx ) λ. From J3 x = 0, we get λ = α β (+β x ), from which we can get x 2 = α λ β β.since x are probabilities, we have constraints [3] Y. H. Bae, A. S. alfa an B. D. Choi, Performance analysis of moifie IEEE 802.-base cognitive raio networs, IEEE Commun. Lett., vol. 4, no. 0, pp , Oct [4] D. B. Zhu an B. D. Choi, Performance analysis of CSA in an unslotte cognitive raio networ with license channels an unlicense channels, EURASIP J. on Wirel. Commun. an Networing, 202:2. oi:0.86/ [5] Y. Xing, R. Chanramouli, S. angol an S. Shanar, Dynamic spectrum access in open access wireless networs, IEEE J. Sel. Areas in Commun., vol. 24, no. 3, pp , ar [6] B. Wang, Z. Ji, K. J. R. Liu an T. C. Clancy, Primary-prioritize marov approach for ynamic spectrum allocation, IEEE Trans. Wirel. Commun., vol. 8, no. 4, pp , Apr [7] C. Ghosh, C. Coreiro, D. P. Agrawal an. B. Rao, arov chan existence an hien arov moels in spectrum sensing, IEEE Int. Conf. on Pervasive Computing an Communi., arch [8] P. K. Tang an Y. H. Chew, On the moeling an performance of three opportunistic spectrum access schemes, IEEE Trans. Veh. Tech., vol. 59, no. 8, pp , Oct [9] A. S. Zahmati, X. Fernano an A. Grami, Steay-state arov chain analysis for heterogeneous cognitive raio networs, 200 IEEE Sarnoff Symposium, Princeton, NJ, Apr [0] U. ir, L. erghem-boulahia,. Esseghir an D. Gaiti, A continuous time arov moel for unlicense spectrum access, IEEE 7th Int. Conf. WiOb, pp , 20. [] H. P. Ngallemo, W. Ajib an H. Elbiaze, Dynamic spectrum access analysis in a multi-user cognitive raio networ using arov chains, 202 Int. Conf. on Computing, Networing an Commun. (ICNC), pp. 3-7, aui, HI, Jan [2] T.. N. Ngatche, S. Dong, A. S. Alfa, J. Cai, Performance analysis of cognitive raio networs with channel assembling an imperfect sensing, IEEE Int. Conf. Commun. (ICC 202), Ottawa, ON, Canaa, June 202. [3] X. Li an C. Xiong, arov moel ban for heterogenous cognitive raio networs with multiple issimilar users an channels, 204 Int. Conf. on Computing, Networing an Commun. (ICNC), Honolulu, HI, Feb. 3-6, 204. α β ( + β ) 2 λ α β. (57) From = x =,wehave α 2 λ = β = +. (58) = β Note that λ shoul satisfy (57). In other wors, we can loo iteratively for λ by selecting the channels with the strongest α β an by applying (58) until both (57) an (58) are satisfie. The optimal variables are then + α j= β j β x = α j β, if (57) is satisfie j= β j (59) 0, otherwise REFERENCES [] J. Hwu, J. Chen an X. Li, Sum transmission power of multiple cooperative seconary transmitters in ynamic spectrum access networs, the 42n Annual Conference on Information Sciences an Systems (CISS 2008), Princeton University, ar. 9-2, [2] E. W.. Wong an C. H. Foh, Analysis of cognitive raio spectrum access with finite user population, IEEE Commun. Lett., vol. 3, no. 5, pp , ay 2009.