Renewal-Theoretical Dynamic Spectrum Access in Cognitive Radio ...
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Renewal-Theoretical Dynamic Spectrum Access in Cognitive Radio ...

410 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013Fig. 4.RenewalPointI(t,X,Y)=Y+I(t-X-Y)I(t,X,Y)=t-XI(t,X,Y)=0OFF StateXON StateIllustration of function I(t).YRenewalPointRenewalPointPrimary ON StateSUs Packet Waiting TimeTransmissionTtFig. 5. SUs’ waiting time T w.TwiTONRenewalPointSUs PacketTransmissionSince X and Y are independent, their joint distributionf XY (x, y) =f OFF (x)f ON (y). In such a case, I(t) can be rewrittenas follows∫∫I(t) = I(t|x, y)f XY (x, y)dxdy,xy∫∫= (t − x)f XY (x, y)dxdy +x≤t≤x+y=∫∫x+y≤t∫ t0[y + I(t − x − y)]fXY (x, y)dxdy,(t − x)f OFF (x)dx +∫∫I(t − x − y)f OFF (x)f ON (y)dxdy −x+y≤t∫∫x+y≤t(t − x − y)f OFF (x)f ON (y)dxdy,= I 1 (t)+I 2 (t) − I 3 (t), (6)where I 1 (t), I 2 (t) and I 3 (t) represent those three terms in thesecond equality, respectively. By taking Laplace transforms onthe both sides of (6), we haveI(s) =I 1 (s)+I 2 (s) − I 3 (s), (7)where I 1 (s), I 2 (s), I 3 (s) are the Laplace transforms of I 1 (t),I 2 (t), I 3 (t), respectively.According to the expression of I 1 (t) in (6), we haveI 1 (t) =∫ t0(t − x)f OFF (x)dx = t ∗ f OFF (t). (8)Thus, the Laplace transform of I 1 (t), I 1 (s) isI 1 (s) = 1 s 2 F OFF(s), (9)where F OFF (s) = 1λ is the Laplace transform of f 0s+1 OFF(t).With the expression of I 2 (t) in (6), we have∫∫I 2 (t) = I(t − x − y)f OFF (x)f ON (y)dxdyx+y≤t= I(t) ∗ f ON (t) ∗ f OFF (t)= I(t) ∗ f p (t), (10)where the last step is according to (1). Thus, the Laplacetransform of I 2 (t), I 2 (s) isI 2 (s) =I(s)F p (s), (11)1where I(s) and F p (s) =(λ 1s+1)(λ 0s+1)are Laplace transformsof I(t) and f p (t), respectively.Similar to (10), we can re-written I 3 (t) as I 3 (t) =t∗f p (t).Thus, the Laplace transform of I 3 (t), I 3 (s) isI 3 (s) = 1 s 2 F p(s). (12)By substituting (9), (11) and (12) into (7), we haveI(s) =1 s 2 F OFF(s)+I(s)F p (s) − 1 s 2 F p(s)F p (s)= λ 1 + I(s)F p (s). (13)sThen by taking the inverse Laplace transform on the both sidesof (13), we have∫ t∫ tI(t) = λ 1 f p (w)dw + I(t − w)f p (w)dw0= λ 1 F p (t)+∫ t00I(t − w)f p (w)dw. (14)This completes the proof of the theorem.Theorem 2 illustrates the renewal characteristic of I(t). By1substituting F p (s) =(λ 1s+1)(λ 0s+1)into (13), the Laplacetransform of I(t) can be calculated byλ 1 F p (s)λI(s) = ( ) 1=s 1 − F p (s) s 2 (λ 0 λ 1 s + λ 0 + λ 1 ) . (15)Then, by taking inverse Laplace transform on (15), we canobtain the closed-form expression for I(t) asI(t) = λ 1 λ 0 λ 2 ()1t −λ 0 + λ 1 (λ 0 + λ 1 ) 2 1 − e − λ 0 +λ 1λ 0 λ t 1 . (16)2) Expected waiting time E(T w ): AsshowninFig.3,onone hand, if the transmission time T t ends in the OFF state,the following waiting time T w will be 0; on the other hand, ifT t ends in the ON state, the length of T w will depend on whenthis ON state terminates, which can be specifically illustratedin Fig. 5. In the second case, according to the Renewal Theory[19], T w is equivalent to the forward recurrence time of theON state, ̂T ON , the distribution of which is only related to thatof the ON state. Thus, we can summarize T w as follows{ 0 Tt ends in the OFF state,T w =(17)T t ends in the ON state.̂T ONTo compute the closed-form expression for T w , we introducea new function defined as follows.Definition 3: P ON (t) is the average probability that a periodof time t begins at the OFF state and ends at the ON state.

JIANG et al.: RENEWAL-THEORETICAL DYNAMIC SPECTRUM ACCESS IN COGNITIVE RADIO NETWORK WITH UNKNOWN PRIMARY BEHAVIOR 411No SU in theprimary channelWaitingTime InterferencePrimary ON-OFF ChannelSUs PacketTransmissionNo SU in theprimary channel RenewalPointTb1Tb(N-1) TbNTb1TI1 T B1 TI2 TB2TC1TC2RenewalPointTbNRenewalPointFig. 6. Illustration of the SUs’ idle-busy behavior in the primary channel when λ s ≠0.According to Definition 3 and (17), the SUs’ averagewaiting time E(T w ) can be written byTwTtTwTtTwTtE(T w )=P ON (T t )E( ̂T ON ). (18)Similar to the analysis of I(t) in Section IV-B1, P ON (t)can also be obtained through solving the following renewalequationP ON (t) =λ 1 f p (t)+∫ t0P ON (t − w)f p (w)dw. (19)By solving (19), we can obtain the closed-form expression ofP ON (t) asP ON (t) = λ ()11 − e − λ 0 +λ 1λ 0 λ t 1 . (20)λ 0 + λ 1Note that (20) can also be derived by the theory of continuousMarkov chains [29].The ̂T ON is the forward recurrence time of the primarychannel’s ON state. Since all ON sates follow a Poissonprocess. According to Renewal Theory [19], we havêT ON ∼ 1 λ 1e −t/λ1 , E( ̂T ON )=λ 1 . (21)By combining (20) and (21), the SUs’ average waiting timeE(T w ) can be obtained as follows(E(T w )= λ2 11 − e − λ 0 +λ 1λ 0 λ T t 1). (22)λ 0 + λ 1Finally, by substituting (16) and (22) into (3), we can obtainthe quantity of interference Q I1 as follows()(λ 0 + λ 1 )T t − λ 0 λ 1 1 − e − λ 0 +λ 1λ 0 λ T t 1Q I1 =(λ 0 + λ 1 )T t + λ 2 1(1 − e − λ 0 +λ 1λ 0 λ 1T t) . (23)V. INTERFERENCE CAUSED BY SUS WITHNON-ZEROARRIVAL INTERVALIn this section, we will discuss the case when the SUs’requests arrive by a Poisson process with average arrivalinterval λ s ≠ 0. Under such a scenario, the buffer at thecoordinator may be empty during some periods of time.Similar to the analysis in Section IV, we will start withanalyzing the SUs’ communication behavior, and then quantifythe interference to the PU.E0E1E l El+1 EN-1 ENFig. 7. Illustration of buffer status E l when n = N.A. SUs’ Communication Behavior AnalysisCompared with the SUs’ behavior when λ s =0, anotherstate that may occur when λ s ≠0is there is no SUs’ requestin the coordinator’s buffer. We call this new state as an idlestate of the SUs’ behavior, while the opposite busy state refersto the scenario when the coordinator’s buffer is not empty.The length of the idle state and busy state are denoted by T Iand T B , respectively. As shown in Fig. 6, the SUs’ behaviorswitches between the idle state and busy state, which is similarto the PU’s ON-OFF model. In the following, we prove thatthe SUs’ such idle-busy switching is also a renewal process.Theorem 3: When the SUs’ transmission requests arrive byPoisson process with constant rate λ −1s , the SUs’ communicationbehavior is a renewal process in the primary channel.Proof: In Fig. 6, we use T c to denote one cycle of theSUs’ idle and busy state, i.e., T c = T I + T B . For the idlestate, since the SUs’ requests arrive by Poisson process, T I ∼1λ se −t/λs and hence the lengths of all idle states are i.i.d.∑For the busy state, T B =n T bi as shown in Fig. 6, wherei=1n is the number of SUs’ transmitting-waiting times duringone busy state. Since all T bi are i.i.d as proved in Theorem1, T B1 , T B2 , ...will also be i.i.d if we can prove that then of all busy states are i.i.d. It is obvious that the n of allbusy states are independent since the SUs’ requests arrive bya Poisson process. In the following, we will focus on provingits property of identical distribution.In Fig. 7, we illustrate the case when there are Ntransmitting-waiting times during one busy state, i.e., n = N.E l represents the number of requests waiting in the coordinator’sbuffer at the end of the lth T t , i.e., the time rightafter the transmission of the SUs’ lth packet. We can see thatE l (0 ≤ l ≤ N) should satisfy the following condition⎧E 0 =1, 1 ≤ E 1 ≤ N − 1,⎪⎨...,1 ≤ E l ≤ N − l,(24)...,⎪⎩E N−1 =1, E n =0.According to the queuing theory [30], the sequence E 1 , E 2 ,

412 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013...,E N is an embedded Markov process. Thus, the probabilityP (n = N) can be written asP (n = N) =P (E 1 |E 0 )P (E 2 |E 1 ) ...P(E N |E N−1 ), (25)where P (E l+1 |E l ) denotes the probability that the last T t endswith E l requests in the coordinator’s buffer and current T tends with E l+1 requests. Suppose E l+1 = j and E l = i,we can denote P (E l+1 |E l ) simply as P ij , which representsthe probability that there are j − i +1 requests arriving at thecoordinator during the period T w +T t . Since the SUs’ requestsarrive by a Poisson process with arrival interval T s , P ij canbe calculated byP ij = P∫ +∞==( j−i+1 ∑T sk ≤ (T w + T t ) ≤k=1(t/λ s ) j−i+1T t∫ +∞0j−i+2∑k=1T sk)(j − i +1)! e−t/λs P (T w + T t = t)dt) j−i+1((t + T t )/λ s(j − i +1)!e −(t+T t )λs P (T w = t)dt,(26)where T sk is the SUs’ kth arrival interval satisfying theexponential distribution with parameter λ s ,thefirst equality isbecause j−i+1 ∑T sk and j−i+2 ∑T sk satisfy Erlang distribution.k=1k=1According to (17) and (21), the probability distribution ofT w , P (T w = t), can be written as followsP (T w = t) ={POFF (T t ) t =0,P ON (T t)λ 1e −t/λ1 t>0,where P OFF (T t )=1− P ON (T t )=λ01 − e − λ 0 +λ 1λ 0 λ T t 1).By substituting (27) into (26) , we can re-write P ij asλ 0+λ 1(P ij = P OFF (T t ) (T t/λ s ) j−i+1(j − i +1)! e−Tt/λs +(∫ +∞0 +(t + T t )/λ s) j−i+1(j − i +1)!P ON (T t )λ 1(e − λ1 +λsλ 1 λs(27))t+ T tλs dt. (28)According to (28), we can see that there are (N − 1)!possible combinations of (E 0 , ..., E l , ..., E N ). We denoteeach case as C(a), where1 ≤ a ≤ (N − 1)!. For each case,the probability is the product of N terms P ij(C(a),b),where1 ≤ b ≤ N. Thus, P (n = N) can be expressed as followsP (n = N) =(N−1)!∑a=1N∏ ( )P ij C(a),b . (29)b=1From (29), we can see that n of all busy states are identicaldistributed, and hence i.i.d.Up to now, we have come to the conclusion that T I of allidle states are i.i.d, aswellasT B of all busy states. SinceT I and T B are independent with each other, the sequence ofall cycles’ lengths T c1 , T c2 , ...are i.i.d. Therefore, the SUs’communication behavior is a renewal process.B. Interference Quantity AnalysisAccording to Definition 1 and Theorem 3, the interferencequantity Q I2 can be calculated byE(T B)Q I2 = μ B Q I1 , (30)where μ B =E(T I)+E(T B)is the occurrence probability of theSUs’ busy state.Our system can be treated as an M/G/1 queuing system,where the customers are the SUs’ data packets and the serveris the primary channel. The service time S of one SU is thesum of its transmission time T t and the waiting time of thenext SU T w . In such a case, the expected service time isE(S) =T t + E(T w ). According to the queuing theory [30],the load of the server is ρ = E(S)/λ, whereλ is the averagearrival interval of the customers. By Little’s law [30], ρ isequivalent to the expected number of customers in the server.In our system, there can be at most one customer (SUs’ onepacket) in the server, which means the expected number ofcustomers is equal to the probability that there is a customerin the server. Therefore, ρ is equal to the proportion of timethat the coordinator is busy, i.e.,ρ = T t + E(T w )λ s= μ B =E(T B )E(T I )+E(T B ) . (31)Thus, combining (23), (30) and (31), the closed-form expressionof Q I2 can be obtained as followsQ I2 =()(λ 0 + λ 1 )T t − λ 0 λ 1 1 − e − λ 0 +λ 1λ 0 λ T t 1λ s (λ 0 + λ 1 ). (32)VI. OPTIMIZING SECONDARY USERS’ COMMUNICATIONPERFORMANCEIn this section, we will discuss how to optimize the SUs’communication performance while maintaining the PU’s communicationQoS and the stability of the secondary network. Inour system, the SUs’ communication performance is directlydependent on the expected arrival interval of their packetsλ 1 s and the length of the transmission time T t .Thesetwoimportant parameters should be appropriately chosen so as tominimize the interference caused by the SUs’ dynamic accessand also to maintain a stable secondary network.We consider two constraints for optimizing the SUs’ λ s andT t as follows• the PU’s average data rate should be at least Rp ↓,whichis the PU’s lowest data rate,• the stability condition of the secondary network shouldbe satisfied.In the following, we will first derive the expressions for thesetwo constraints based on the analysis in Section IV and V.Then we formulate the problem of finding the optimal λ ∗ s andTt∗ as an optimization problem to maximize the SUs’ averagedata rate.1 To evaluate the stability condition, we only consider the scenario whenλ s ≠0.

JIANG et al.: RENEWAL-THEORETICAL DYNAMIC SPECTRUM ACCESS IN COGNITIVE RADIO NETWORK WITH UNKNOWN PRIMARY BEHAVIOR 413A. The Constraints1) PU’s Average Data Rate: If there is no interference fromthe SUs, the PU’s instantaneous rate is log(1 + SNR p ),whereSNR p denotes the Signal-to-Noise Ratio of primary signalat the PU’s receiver. On the other hand, if the ( interference )occurs, the PU’s instantaneous rate will be log 1+INR SNRp ,p+1where INR p is the Interference-to-Noise Ratio of secondarysignal received by the PU. According to Definition 1, Q I2represents the ratio of the interference periods to the PU’soverall communication time. Thus, the PU’s average data rateR p can be calculated byR p = ( ) ((1−Q I2 log 1+SNR p)+Q I2 log 1+ SNR )p. (33)INR p +12) SUs’ Stability Condition: In our system, the secondarynetwork and the primary channel can be modeled as a singleserverqueuing system. According to the queuing theory [30],the stability condition for a single-server queue with Poissonarrivals is that the load of the server should satisfy ρ 0. (35)λ 0 + λ 1B. Objective Function: SUs’ Average Data RateIf a SU encounters the PU’s recurrence, i.e., the ON stateof the primary channel, during its transmission time T t , itscommunication is also interfered by the PU’s ( signal. In such )a case, the SU’s instantaneous rate is log 1+INR SNRs ,s+1where SNR s is the SU’s Signal-to-Noise Ratio and INR s is theInterference-to-Noise Ratio of primary signal received by theSU. According to Theorem 1 and Theorem 3, the occurrenceI(Tprobability of such a phenomenon is μ t)B T = I(Tt)t+E(T w) λ s.Onthe other hand, if no PU appears during the SU’s transmission,its instantaneous rate will be log(1 + SNR s ) and the correspondingoccurrence probability is μ t−I(T t) TB T = Tt−I(Tt)t+E(T w) λ s.Thus, the SU’s average data rate R s isR s = T t−I(T t ))log(1+SNR sλ s+ I(T t)λ s(log 1+ SNR )s. (36)INR s +1C. Optimizing SUs’ Communication PerformanceBased on the analysis of constraints and objective function,the problem of finding optimal Tt∗ and λ ∗ s for the SUs can beformulated by (37) below.Proposition 1: The SUs’ average data rate R s (T t ,λ s ) is astrictly increasing function in terms of the their transmissiontime T t and a strictly decreasing function in terms of theiraverage arrival interval λ s , i.e.,∂R s∂T t> 0,∂R s∂λ s< 0. (38)The PU’s average data rate R p (T t ,λ s ) is a strictly decreasingfunction in terms of T t and a strictly increasing functionin terms of λ s , i.e.,∂R p∂T t< 0,∂R p∂λ s> 0. (39)The stability condition function S(T t ,λ s ) is a strictlydecreasing function in terms of T t and a strictly increasingfunction in terms of λ s , i.e.,∂S ∂S< 0, > 0. (40)∂T t ∂λ s(Proof: For simplification, we use R s0 to express log 1+)( )SNR s and R s1 to express log 1+ SNRs . According to(36) and (16), ∂Rs∂T t∂R s∂T t= R s0λ s=∂R s∂λ s= − 1 λ 2 sand ∂Rs∂λ sINR s+1can be calculated as follows− R s0 − R s1· ∂I(T t)λ s ∂T t1λ s (λ 0 + λ 1 )(λ 0 R s0 + λ 1 R s1 +)λ 1 (R s0 − R s1 )(1 ) − e − λ 0 +λ 1λ 0 λ T t 1 , (41)( ( )T t − I(T t ))R s0 + I(T t )R s1 . (42)Since R s0 >R s1 , e − λ 0 +λ 1λ 0 λ T t 1 < 1, andT t ≥ I(T t ),wehave∂R s ∂R s> 0, < 0. (43)∂T t ∂λ s)(1+SNR pSimilarly, we use R p0 to express log(to express log 1+INR SNRpp+1∂R p∂λ s). According to (33),and R p1∂R p∂T tandcan be calculated as follows∂R p∂T t= − ∂Q I 2(R s0 − R s1 ),∂T t(44)∂R p∂λ s= − ∂Q I 2(R s0 − R s1 ).∂λ s(45)max(T t,λ s)R s (T t ,λ s )= T t − I(T t ))log(1+SNR sλ s+ I(T t)λ ss.t. R p (T t ,λ s )= ( 1 − Q I2)log(1+SNR p)+ Q I2 log(log(S(T t ,λ s )=λ s − T t − λ2 1λ 0 + λ 1(1 − e − λ 0 +λ 1λ 0 λ 1T t)> 0.1+ SNR )s,INR s +1)≥ Rp, ↓ (37)1+ SNR pINR p +1

414 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013Fig. 8. Interference quantity Q I1 with λ s =0s.Fig. 9. Interference quantity Q I2 with λ s =1.3 s.According to (32), we have∂Q I2= 1 − λ 0 +λ 1e− λ 0 λ 1 ∂Q I2> 0,∂T t λ s ∂λ s< 0. (46)Thus, combining (44), (45) and (46), we have∂R p< 0,∂T tAccording to (35), ∂S∂T t(∂S= − 1+ λ 1e − λ 0 +λ 1∂T t λ 0T t∂R p> 0. (47)∂λ sand ∂S∂λ scan be calculated as followsλ 0 λ 1T t)< 0,∂S∂λ s=1> 0. (48)This completes the proof of the theorem.From Proposition 1, we can see that the objective functionand the constraints are all monotonous functions in terms ofT t and λ s . Thus, the solution to the optimization problem (37)can be found using gradient descent method [32].VII. SIMULATION RESULTSIn this section, we conduct simulations to verify the effectivenessof our analysis. The parameters of primary ON-OFFchannel are set to be λ 0 =2.6 sandλ 1 =3.6 s. Accordingto Fig. 2, we build a queuing system using Matlab to simulatethe PU’s and SUs’ behaviors.A. Interference Quantity Q IIn Fig. 8 and Fig. 9, we illustrate the theoretic and simulatedresults of Q I1 and Q I2 , respectively. The theoretic Q I1 andQ I2 are computed according to (23) and (32) with differentvalues of the SUs’ transmission time T t . The average arrivalinterval of the SUs’ packets λ s is set to be 1.3 s whencalculating theoretic Q I2 . For the simulated results, once theinterference occurs, we calculate and record the ratio of theaccumulated interference periods to the accumulated periodsof the ON states. We perform 2000 times simulation runs andaverage all of them to obtain the final simulation results.From Fig. 8 and Fig. 9, we can see that all the simulatedresults of Q I1 and Q I2 are accord with the correspondingtheoretic results, which means that the closed-form expressionsin (23) and (32) are correct and can be used to calculatethe interference caused by the SUs in the practical cognitiveradio system. We also denote the standard deviation of Q Iat several simulation time points when T t = 0.6 sandtheresults show that the standard deviation converges to 0 alongwith the increasing of the simulation time, i.e., the systemgradually tends to steady state. Moreover, we can also seethat the interference increases as the SUs’ transmission timeT t increases. Such a phenomenon is because the interferenceto the PU can only occur during T t and the increase of T tenlarges the occurrence probability of T t . Finally, we find thatdue to the existence of the idle state when λ s ≠0, Q I2 is lessthan Q I1 under the same condition.B. Stability of The Secondary NetworkSince we have modeled the secondary network as a queuingsystem shown in Fig. 2, the stability of the network is reflectedby the status of the coordinator’s buffer. A stable networkmeans that the requests waiting in the coordinator’s buffer donot explode as time goes to infinite, while the requests in thebuffer of an unstable network will eventually go to infinite. InSection VI-A2, we have shown the stability condition of thesecondary network in (35). On one hand, if the SUs’ accesstime T t is given in advance, the SUs’ minimal average arrivalinterval λ s can be computed by (35). On the other hand, if λ sis given, the maximal T t can be obtained to restrict the SUs’transmission time.In this simulation, we set T t =0.6 s, and thus λ s shouldbe larger than 1.25 s to ensure the SUs’ stability according to(35). In Fig. 9, we show the queuing length, i.e., the number ofrequests in the coordinator’s buffer, versus the time. The blacklines shows the queuing length of a stable network, in whichλ s =1.3 s is larger than the threshold 1.25 s. It can be seenthat the requests dynamically vary between 0 and 60. However,if we set λ s =1.2 s, which is smaller than the lower limit,from Fig. 9, we can see that the queuing length will finally goto infinite, which represents an unstable network. Therefore,the stability condition in (35) should be satisfied to maintaina stable secondary network.

JIANG et al.: RENEWAL-THEORETICAL DYNAMIC SPECTRUM ACCESS IN COGNITIVE RADIO NETWORK WITH UNKNOWN PRIMARY BEHAVIOR 415Fig. 10.Queuing length under stable and unstable conditions.Fig. 12.SUs’ average data rate.the SUs’ average data rate can achieve around 0.6 bps/Hzaccording to Fig. 12. For any fixed Rp ↓ , the optimal valuesof Tt ∗ and λ ∗ s are determined by the channel parameters λ 0and λ 1 . Therefore, the SUs should dynamically adjust theircommunication behaviors according to the channel parameters.Fig. 11.PU’s average data rate.C. PU’s and SUs’ Average Data RateThe simulation results of the PU’s average data rate R pversus the SUs’ transmission time T t and arrival interval λ sare shown in Fig. 11, where we set SNR p =SNR s =5dB andINR p = INR s =3dB. We can see that R p is a decreasingfunction in terms of T t given a certain λ s , and an increasingfunction in terms of λ s for any fixed T t , which is in accordancewith Proposition 1. Such a phenomenon is because an increaseof T t or a decrease of λ s will cause more interference tothe PU and thus degrade its average data rate. In Fig. 12, weillustrate the simulation results of the SUs’ average data rateR s versus T t and λ s . Different from R p , R s is an increasingfunction in terms of T t given a certain λ s , and a decreasingfunction in terms of λ s for any fixed T t , which also verifiesthe correctness of Proposition 1.Suppose that the PU’s data rate should be at least2.0 bps/Hz, i.e., Rp ↓ = 2.0 bps/Hz. Then, according to theconstraints in (37), T t should be no larger than the locationof those three colored vertical lines in Fig. 11 correspondingto λ s =1.3 s, 1.5 s, 2.0 s, respectively. For example, whenλ s =1.3 s, the optimal Tt∗ should be around 400 ms to satisfyboth the Rp ↓ and stability condition constraints. In such a case,VIII. CONCLUSIONIn this paper, we analyzed the interference caused by theSUs confronted with unknown primary behavior. Based onthe Renewal Theory, we showed that the SUs’ communicationbehaviors in the ON-OFF primary channel is a renewal processand derived the closed-form for the interference quantity.We further discussed how to optimize the SUs’ arrival rateand transmission time to control the level of interference tothe PU and maintain the stability of the secondary network.Simulation results are shown to validate our closed-formexpressions for the interference quantity. In the practicalcognitive radio networks, these expressions can be used toevaluate the interference from the SUs when configuring thesecondary network. In the future work, we will study howto concretely coordinate the primary spectrum sharing amongmultiple SUs.REFERENCES[1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,”IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, 2005.[2] K.J.R.LiuandB.Wang,Cognitive Radio Networking and Security:A Game Theoretical View. Cambridge University Press, 2010.[3] B. Wang and K. J. R. Liu, “Advances in cognitive radios: A survey,”IEEE J. Sel. Topics Signal Process., vol. 5, no. 1, pp. 5–23, 2011.[4] B. Wang, Z. Ji, K. J. R. Liu, and T. C. Clancy, “Primary-prioritizedmarkov approach for efficient and fair dynamic spectrum allocation,”IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 1854–1865, 2009.[5] Z. Chen, C. Wang, X. Hong, J. Thompson, S. A. Vorobyov, and X. Ge,“Interference modeling for cognitive radio networks with power orcontention control,” in Proc. IEEE WCNC, 2010.[6] G. L. Stuber, S. M. Almalfouh, and D. Sale, “Interference analysis ofTV band whitespace,” Proc. IEEE, vol. 97, no. 4, pp. 741–754, 2009.[7] M. Vu, D. Natasha, and T. Vahid, “On the primary exclusive regions incognitive networks,” IEEE Trans. Wireless Commun., vol. 8, no. 7, pp.3380–3385, 2008.

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Prob., vol. 25, no. 3, pp. 690–701, 1993.[32] D. P. Bertsekas, Nonlinear Programming. Athena Scientific, 1999.Chunxiao Jiang (S’09) received his B.S. degree ininformation engineering from Beijing University ofAeronautics and Astronautics (Beihang University)in 2008 with the highest honor as a candidateof Beihang Golden Medal Award. He is currentlya Ph.D. candidate in wireless communication andnetworking at Department of Electronic Engineeringof Tsinghua University. During 2011-2012, hevisited the Signals and Information Group (SIG) atDepartment of Electrical & Computer Engineering(ECE) of University of Maryland (UMD), supportedby China Scholarship Council (CSC) for one year. His research interestsinclude cognitive radio networks, dynamic/opportunistic spectrum access andAd Hoc networks.Yan Chen (S’06-M’11) received the Bachelor’s degreefrom University of Science and Technology ofChina in 2004, the M. Phil degree from Hong KongUniversity of Science and Technology (HKUST)in 2007, and the Ph.D. degree from University ofMaryland College Park in 2011.He is currently a research associate in the Departmentof Electrical and Computer Engineering atUniversity of Maryland College Park. His currentresearch interests are in social learning and networking,smart grid, cloud computing, crowdsourcing,network economics, multimedia signal processing and communication.Dr. Chen received the University of Maryland Future Faculty Fellowshipin 2010, Chinese Government Award for outstanding students abroad in2011, University of Maryland ECE Distinguished Dissertation FellowshipHonorable Mention in 2011, and was the Finalist of A. James Clark Schoolof Engineering Dean’s Doctoral Research Award in 2011.K. J. Ray Liu (F’03) was named a DistinguishedScholar-Teacher of University of Maryland, CollegePark, in 2007, where he is Christine Kim EminentProfessor of Information Technology. He leads theMaryland Signals and Information Group conductingresearch encompassing broad areas of signalprocessing and communications with recent focuson cooperative communications, cognitive networking,social learning and networks, and informationforensics and security.Dr. Liu is the recipient of numerous honors andawards including IEEE Signal Processing Society Technical AchievementAward and Distinguished Lecturer. He also received various teaching andresearch recognitions from University of Maryland including university-levelInvention of the Year Award; and Poole and Kent Senior Faculty TeachingAward and Outstanding Faculty Research Award, both from A. James ClarkSchool of Engineering. An ISI Highly Cited Author, Dr. Liu is a Fellow ofIEEE and AAAS.Dr. Liu is President of IEEE Signal Processing Society where he has servedas Vice President – Publications and Board of Governor. He was the Editor-in-Chief of IEEE Signal Processing Magazine and the founding Editor-in-Chiefof EURASIP Journal on Advances in Signal Processing.Yong Ren received his B.S, M.S and Ph.D. degreesin electronic engineering from Harbin Institute ofTechnology, China, in 1984, 1987, and 1994, respectively.He worked as a post doctor at Department ofElectronics Engineering, Tsinghua University, Chinafrom 1995 to 1997. Now he is a professor of Departmentof Electronics Engineering and the directorof the Complexity Engineered Systems Lab (CESL)in Tsinghua University. He holds 12 patents, andhas authored or co-authored more than 100 technicalpapers in the behavior of computer network, P2Pnetwork and cognitive networks. He has serves as a reviewer of IEICETransactions on Communications, Digital Signal Processing, Chinese PhysicsLetters, Chinese Journal of Electronics, Chinese Journal of Computer Science& Technology, Chinese Journal of Aeronautics and so on. His currentresearch interests include complex systems theory and its applications to theoptimization and information sharing of the Internet, Internet of Things andubiquitous network, cognitive networks and Cyber-Physical Systems.

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