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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.**Renewal**Po**in**tI(t,X,Y)=Y+I(t-X-Y)I(t,X,Y)=t-XI(t,X,Y)=0OFF StateXON StateIllustration of function I(t).Y**Renewal**Po**in**t**Renewal**Po**in**tPrimary ON StateSUs Packet Wait**in**g TimeTransmissionTtFig. 5. SUs’ wait**in**g time T w.TwiTON**Renewal**Po**in**tSUs PacketTransmissionS**in**ce X and Y are **in**dependent, their jo**in**t 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 tak**in**g 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.Accord**in**g 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 accord**in**g 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 substitut**in**g (9), (11) and (12) **in**to (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 tak**in**g the **in**verse 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). By1substitut**in**g F p (s) =(λ 1s+1)(λ 0s+1)**in**to (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 tak**in**g **in**verse Laplace transform on (15), we canobta**in** 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 wait**in**g time E(T w ): Asshown**in**Fig.3,onone hand, if the transmission time T t ends **in** the OFF state,the follow**in**g wait**in**g 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 term**in**ates, which can be specifically illustrated**in** Fig. 5. In the second case, accord**in**g 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 **in**troducea new function def**in**ed as follows.Def**in**ition 3: P ON (t) is the average probability that a periodof time t beg**in**s 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 channelWait**in**gTime InterferencePrimary ON-OFF ChannelSUs PacketTransmissionNo SU **in** theprimary channel **Renewal**Po**in**tTb1Tb(N-1) TbNTb1TI1 T B1 TI2 TB2TC1TC2**Renewal**Po**in**tTbN**Renewal**Po**in**tFig. 6. Illustration of the SUs’ idle-busy behavior **in** the primary channel when λ s ≠0.Accord**in**g to Def**in**ition 3 and (17), the SUs’ averagewait**in**g 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 obta**in**ed through solv**in**g the follow**in**g renewalequationP ON (t) =λ 1 f p (t)+∫ t0P ON (t − w)f p (w)dw. (19)By solv**in**g (19), we can obta**in** 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 cont**in**uousMarkov cha**in**s [29].The ̂T ON is the forward recurrence time of the primarychannel’s ON state. S**in**ce all ON sates follow a Poissonprocess. Accord**in**g to **Renewal** Theory [19], we havêT ON ∼ 1 λ 1e −t/λ1 , E( ̂T ON )=λ 1 . (21)By comb**in****in**g (20) and (21), the SUs’ average wait**in**g timeE(T w ) can be obta**in**ed as follows(E(T w )= λ2 11 − e − λ 0 +λ 1λ 0 λ T t 1). (22)λ 0 + λ 1F**in**ally, by substitut**in**g (16) and (22) **in**to (3), we can obta**in**the quantity of **in**terference 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 arrival**in**terval λ s ≠ 0. Under such a scenario, the buffer at thecoord**in**ator may be empty dur**in**g some periods of time.Similar to the analysis **in** Section IV, we will start withanalyz**in**g the SUs’ communication behavior, and then quantifythe **in**terference 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’ request**in** the coord**in**ator’s buffer. We call this new state as an idlestate of the SUs’ behavior, while the opposite busy state refersto the scenario when the coord**in**ator’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 follow**in**g, we prove thatthe SUs’ such idle-busy switch**in**g 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, s**in**ce 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’ transmitt**in**g-wait**in**g times dur**in**gone busy state. S**in**ce 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 **in**dependent s**in**ce the SUs’ requests arrive bya Poisson process. In the follow**in**g, we will focus on prov**in**gits property of identical distribution.In Fig. 7, we illustrate the case when there are Ntransmitt**in**g-wait**in**g times dur**in**g one busy state, i.e., n = N.E l represents the number of requests wait**in**g **in** the coord**in**ator’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 follow**in**g condition⎧E 0 =1, 1 ≤ E 1 ≤ N − 1,⎪⎨...,1 ≤ E l ≤ N − l,(24)...,⎪⎩E N−1 =1, E n =0.Accord**in**g to the queu**in**g 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 coord**in**ator’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 arriv**in**g at thecoord**in**ator dur**in**g the period T w +T t . S**in**ce the SUs’ requestsarrive by a Poisson process with arrival **in**terval 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 **in**terval satisfy**in**g 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=1Accord**in**g 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 substitut**in**g (27) **in**to (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)Accord**in**g to (28), we can see that there are (N − 1)!possible comb**in**ations 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. S**in**ceT I and T B are **in**dependent 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 AnalysisAccord**in**g to Def**in**ition 1 and Theorem 3, the **in**terferencequantity 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 queu**in**g 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 wait**in**g time of thenext SU T w . In such a case, the expected service time isE(S) =T t + E(T w ). Accord**in**g to the queu**in**g theory [30],the load of the server is ρ = E(S)/λ, whereλ is the averagearrival **in**terval 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 customer**in** the server. Therefore, ρ is equal to the proportion of timethat the coord**in**ator 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, comb**in****in**g (23), (30) and (31), the closed-form expressionof Q I2 can be obta**in**ed 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 ma**in**ta**in****in**g the PU’s communicationQoS and the stability of the secondary network. Inour system, the SUs’ communication performance is directlydependent on the expected arrival **in**terval of their packetsλ 1 s and the length of the transmission time T t .Thesetwoimportant parameters should be appropriately chosen so as tom**in**imize the **in**terference caused by the SUs’ dynamic accessand also to ma**in**ta**in** a stable secondary network.We consider two constra**in**ts for optimiz**in**g 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 follow**in**g, we will first derive the expressions for thesetwo constra**in**ts based on the analysis **in** Section IV and V.Then we formulate the problem of f**in**d**in**g 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 Constra**in**ts1) PU’s Average Data Rate: If there is no **in**terference fromthe SUs, the PU’s **in**stantaneous 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 ( **in**terference )occurs, the PU’s **in**stantaneous rate will be log 1+INR SNRp ,p+1where INR p is the Interference-to-Noise Ratio of secondarysignal received by the PU. Accord**in**g to Def**in**ition 1, Q I2represents the ratio of the **in**terference 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 s**in**gleserverqueu**in**g system. Accord**in**g to the queu**in**g theory [30],the stability condition for a s**in**gle-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, dur**in**g its transmission time T t , itscommunication is also **in**terfered by the PU’s ( signal. In such )a case, the SU’s **in**stantaneous 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. Accord**in**g 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 dur**in**g the SU’s transmission,its **in**stantaneous rate will be log(1 + SNR s ) and the correspond**in**goccurrence 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. Optimiz**in**g SUs’ Communication PerformanceBased on the analysis of constra**in**ts and objective function,the problem of f**in**d**in**g 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 **in**creas**in**g function **in** terms of the their transmissiontime T t and a strictly decreas**in**g function **in** terms of theiraverage arrival **in**terval 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 decreas**in**gfunction **in** terms of T t and a strictly **in**creas**in**g function**in** 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 strictlydecreas**in**g function **in** terms of T t and a strictly **in**creas**in**gfunction **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 . Accord**in**g 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)S**in**ce 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). Accord**in**g 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.Accord**in**g 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, comb**in****in**g (44), (45) and (46), we have∂R p< 0,∂T tAccord**in**g 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 constra**in**ts are all monotonous functions **in** terms ofT t and λ s . Thus, the solution to the optimization problem (37)can be found us**in**g 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. Accord**in**gto Fig. 2, we build a queu**in**g system us**in**g 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 accord**in**g to (23) and (32) with differentvalues of the SUs’ transmission time T t . The average arrival**in**terval of the SUs’ packets λ s is set to be 1.3 s whencalculat**in**g theoretic Q I2 . For the simulated results, once the**in**terference occurs, we calculate and record the ratio of theaccumulated **in**terference periods to the accumulated periodsof the ON states. We perform 2000 times simulation runs andaverage all of them to obta**in** the f**in**al 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 correspond**in**gtheoretic results, which means that the closed-form expressions**in** (23) and (32) are correct and can be used to calculatethe **in**terference caused by the SUs **in** the practical cognitiveradio system. We also denote the standard deviation of Q Iat several simulation time po**in**ts when T t = 0.6 sandtheresults show that the standard deviation converges to 0 alongwith the **in**creas**in**g of the simulation time, i.e., the systemgradually tends to steady state. Moreover, we can also seethat the **in**terference **in**creases as the SUs’ transmission timeT t **in**creases. Such a phenomenon is because the **in**terferenceto the PU can only occur dur**in**g T t and the **in**crease of T tenlarges the occurrence probability of T t . F**in**ally, we f**in**d 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 NetworkS**in**ce we have modeled the secondary network as a queu**in**gsystem shown **in** Fig. 2, the stability of the network is reflectedby the status of the coord**in**ator’s buffer. A stable networkmeans that the requests wait**in**g **in** the coord**in**ator’s buffer donot explode as time goes to **in**f**in**ite, while the requests **in** thebuffer of an unstable network will eventually go to **in**f**in**ite. 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’ m**in**imal average arrival**in**terval λ s can be computed by (35). On the other hand, if λ sis given, the maximal T t can be obta**in**ed 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 accord**in**g to(35). In Fig. 9, we show the queu**in**g length, i.e., the number ofrequests **in** the coord**in**ator’s buffer, versus the time. The blackl**in**es shows the queu**in**g 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 queu**in**g length will f**in**ally goto **in**f**in**ite, which represents an unstable network. Therefore,the stability condition **in** (35) should be satisfied to ma**in**ta**in**a stable secondary network.

JIANG et al.: RENEWAL-THEORETICAL DYNAMIC SPECTRUM ACCESS IN COGNITIVE RADIO NETWORK WITH UNKNOWN PRIMARY BEHAVIOR 415Fig. 10.Queu**in**g length under stable and unstable conditions.Fig. 12.SUs’ average data rate.the SUs’ average data rate can achieve around 0.6 bps/Hzaccord**in**g to Fig. 12. For any fixed Rp ↓ , the optimal valuesof Tt ∗ and λ ∗ s are determ**in**ed by the channel parameters λ 0and λ 1 . Therefore, the SUs should dynamically adjust theircommunication behaviors accord**in**g 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 **in**terval 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 decreas**in**gfunction **in** terms of T t given a certa**in** λ s , and an **in**creas**in**gfunction **in** terms of λ s for any fixed T t , which is **in** accordancewith Proposition 1. Such a phenomenon is because an **in**creaseof T t or a decrease of λ s will cause more **in**terference 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 **in**creas**in**gfunction **in** terms of T t given a certa**in** λ s , and a decreas**in**gfunction **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, accord**in**g to theconstra**in**ts **in** (37), T t should be no larger than the locationof those three colored vertical l**in**es **in** Fig. 11 correspond**in**gto 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 constra**in**ts. In such a case,VIII. CONCLUSIONIn this paper, we analyzed the **in**terference 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 **in**terference quantity.We further discussed how to optimize the SUs’ arrival rateand transmission time to control the level of **in**terference tothe PU and ma**in**ta**in** the stability of the secondary network.Simulation results are shown to validate our closed-formexpressions for the **in**terference quantity. In the practicalcognitive radio networks, these expressions can be used toevaluate the **in**terference from the SUs when configur**in**g thesecondary network. In the future work, we will study howto concretely coord**in**ate the primary spectrum shar**in**g amongmultiple SUs.REFERENCES[1] S. Hayk**in**, “**Cognitive** radio: bra**in**-empowered wireless communications,”IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, 2005.[2] K.J.R.LiuandB.Wang,**Cognitive** **Radio** Network**in**g 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 model**in**g 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 **in**cognitive networks,” IEEE Trans. Wireless Commun., vol. 8, no. 7, pp.3380–3385, 2008.

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Kulkarni, “Stability condition for a s**in**gle-serverretrial queue,” Adv. Appl. Prob., vol. 25, no. 3, pp. 690–701, 1993.[32] D. P. Bertsekas, Nonl**in**ear Programm**in**g. Athena Scientific, 1999.Chunxiao Jiang (S’09) received his B.S. degree **in****in**formation eng**in**eer**in**g from Beij**in**g 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 andnetwork**in**g at Department of Electronic Eng**in**eer**in**gof Ts**in**ghua University. Dur**in**g 2011-2012, hevisited the Signals and Information Group (SIG) atDepartment of Electrical & Computer Eng**in**eer**in**g(ECE) of University of Maryland (UMD), supportedby Ch**in**a Scholarship Council (CSC) for one year. His research **in**terests**in**clude 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 ofCh**in**a **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 Eng**in**eer**in**g atUniversity of Maryland College Park. His currentresearch **in**terests are **in** social learn**in**g and network**in**g,smart grid, cloud comput**in**g, crowdsourc**in**g,network economics, multimedia signal process**in**g and communication.Dr. Chen received the University of Maryland Future Faculty Fellowship**in** 2010, Ch**in**ese Government Award for outstand**in**g students abroad **in**2011, University of Maryland ECE Dist**in**guished Dissertation FellowshipHonorable Mention **in** 2011, and was the F**in**alist of A. James Clark Schoolof Eng**in**eer**in**g Dean’s Doctoral Research Award **in** 2011.K. J. Ray Liu (F’03) was named a Dist**in**guishedScholar-Teacher of University of Maryland, CollegePark, **in** 2007, where he is Christ**in**e Kim Em**in**entProfessor of Information Technology. He leads theMaryland Signals and Information Group conduct**in**gresearch encompass**in**g broad areas of signalprocess**in**g and communications with recent focuson cooperative communications, cognitive network**in**g,social learn**in**g and networks, and **in**formationforensics and security.Dr. Liu is the recipient of numerous honors andawards **in**clud**in**g IEEE Signal Process**in**g Society Technical AchievementAward and Dist**in**guished Lecturer. He also received various teach**in**g andresearch recognitions from University of Maryland **in**clud**in**g university-levelInvention of the Year Award; and Poole and Kent Senior Faculty Teach**in**gAward and Outstand**in**g Faculty Research Award, both from A. James ClarkSchool of Eng**in**eer**in**g. An ISI Highly Cited Author, Dr. Liu is a Fellow ofIEEE and AAAS.Dr. Liu is President of IEEE Signal Process**in**g Society where he has servedas Vice President – Publications and Board of Governor. He was the Editor-**in**-Chief of IEEE Signal Process**in**g Magaz**in**e and the found**in**g Editor-**in**-Chiefof EURASIP Journal on Advances **in** Signal Process**in**g.Yong Ren received his B.S, M.S and Ph.D. degrees**in** electronic eng**in**eer**in**g from Harb**in** Institute ofTechnology, Ch**in**a, **in** 1984, 1987, and 1994, respectively.He worked as a post doctor at Department ofElectronics Eng**in**eer**in**g, Ts**in**ghua University, Ch**in**afrom 1995 to 1997. Now he is a professor of Departmentof Electronics Eng**in**eer**in**g and the directorof the Complexity Eng**in**eered Systems Lab (CESL)**in** Ts**in**ghua 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 Process**in**g, Ch**in**ese PhysicsLetters, Ch**in**ese Journal of Electronics, Ch**in**ese Journal of Computer Science& Technology, Ch**in**ese Journal of Aeronautics and so on. His currentresearch **in**terests **in**clude complex systems theory and its applications to theoptimization and **in**formation shar**in**g of the Internet, Internet of Th**in**gs andubiquitous network, cognitive networks and Cyber-Physical Systems.