Stability Analyi in a Cognitive Radio Sytem with Cooperative Beamforming Mohammed Karmooe, Ahmed Sultan, Moutafa Youef Department of Electrical Engineering, Alexandria Univerity, Alexandria, Egypt Wirele Reearch Center, Egypt-Japan Univ. of Sc. & Tech. E-JUST and Alex. Univ., Alexandria, Egypt Abtract We conider a cognitive radio etting in which a relay-aited econdary link employ cooperative beamforming to enhance it throughput and to provide protection to the primary receiver from interference. We aume the preence of infinite buffer at both the primary and econdary tranmitter and characterize the maximum table throughput region exactly uing the dominant ytem approach. Numerical example are provided to give inight into the impact of power control on the tability region. Index Term Cognitive radio, cooperative beamforming, relay, queue tability I. INTRODUCTION Cognitive radio CR technology ha been propoed a a potential olution to the problem of pectrum underutilization. The CR-enabled unlicened or econdary device may ue the pectrum if they do not dirupt the operation of the licened or primary uer. In the opportunitic pectrum acce OSA model, a econdary uer utilize the pectrum whenever a primary uer i not, and vacate the pectrum upon the preence of the primary uer. Thi coexitence trategy enure high level of protection to the primary uer while allowing the econdary uer to make ue of the ilent period of the primary uer. In thi work we invetigate buffered primary and econdary uer and, hence, include a queueing analyi of the propoed model. Such analyi ha been conidered in the tudy of cognitive radio network CRN under different cenario. One of the main objective i to characterize the achievable arrival rate of the network uer which guarantee tability of all ytem queue. For intance, in [], an overlay CRN coniting of a primary and econdary pair i conidered, and maximum econdary table throughput i characterized for fixed primary throughput. The model i extended to include the inertion of a econdary relay to ait the primary tranmiion. In [2], the model i further extended to accommodate everal econdary uer which coexit with a primary link through a colliion channel. Secondary uer are required to comply to interference and QoS contraint of the primary uer. However, the major drawback in the aumed model i that econdary uer are only allowed to tranmit if the primary uer i not detected in the pectrum. Thi dratically reduce the available pectrum opportunitie for the econdary network and therefore limit the achievable ervice rate of the econdary network. We conider in thi paper a relay-aited econdary link. Relay-baed cooperative beamforming ha been widely conidered in CRN a a mean to increae the available pectrum opportunitie for ingle-antenna uer [3], [4]. By providing econdary uer with a et of aiting relay, the econdary uer can utilize the patial diverity of the relay by forming a virtual antenna array VAA. Moreover, uing the appropriate beamforming vector, econdary tranmiion can be nulled out at the primary receiver allowing econdary operation even when the primary uer i ened to be active. In [5], [6], a et of relay equipped with finite-ized buffer are aumed to help the econdary tranmiion in a cooperative beamforming manner. Scheduling i tudied to manage the two phae of econdary tranmiion: from econdary ource to relay, and from relay to detination, while minimizing the delay in econdary uer tranmiion. Spectrum ening i aumed to be perfect, thereby rendering the primary receiver completely protected from interference. In thi paper, we conider buffered uer and a relay-aited econdary link, but take the pectrum ening error into account. The econdary uer i able to ue the pectrum if the pectrum i vacant, or imultaneouly with the primary uer by applying beamforming technique to utilize unued patial dimenion. We aim at characterizing the maximum table throughput region of the primary and econdary uer. Furthermore, we aume that econdary tranmiion i equipped with power control capabilitie in order to optimize ytem performance. Specifically, optimal power allocation i obtained to maximize the table region achieved by the primary and econdary uer. In order to arrive at the provided reult, we reort to the concept of dominant ytem to decouple the interacting queue and how that it provide the exact maximum table throughput region. To the bet of our knowledge, thi i the firt tudy of the propoed cognitive etting from a queueing theory point of view, while avoiding unrealitic aumption uch a perfect pectrum ening. The ret of the paper i organized a follow. Section II introduce the aumed ytem model. In Section III we characterize the mean ervice rate of both uer. In Section IV we ue the concept of dominant ytem to perform the queueing tability analyi. We provide numerical example to provide inight into the obtained problem formulation in Section V, and we conclude the paper in Section VI.
PU - Tx SU - Tx H p H p R R 2 R K K Relaying Node H p H H 2 H p2 H pk H K PU - Rx SU - Rx Fig.. The ytem model conit of a primary link compoed of primary channel with gain H p connecting the primary tranmitter PU-Tx and primary receiver PU-Rx. The econdary tranmitter SU-Tx i linked to the econdary receiver SU-Rx via K relay, where R k i the kth relay. The tranmiion between SU-Tx and the relay i aumed to be perfect and unaffected by the interference from PU-Tx. Channel H k i the channel between R k and SU-Rx. SU-Tx overhear PU-Tx tranmiion and can ene it activity. Channel H p i the interference channel from PU-Tx to SU-Rx, wherea channel H pk i the interference channel from R k to PU-Rx. II. SYSTEM MODEL We aume the preence of a ingle econdary tranmitter trying to communicate with it repective receiver opportunitically in the preence of a primary network that conit of a tranmitter-receiver pair. The tranmit power ued by the primary tranmitter i P p. The primary link operate in a timelotted fahion. Each of the uer conidered are equipped with ingle antenna. The primary and econdary tranmitter are both equipped with infinite buffer, Q p and Q, repectively, to tore their data packet. The arrival at the primary and econdary queue are independent and identically ditributed i.i.d Bernoulli random variable from lot to lot with mean λ p and, repectively. Arrival procee at the primary and econdary buffer are tatitically independent of one another. Becaue of the ignificant pathlo to the econdary detination, direct tranmiion from the econdary tranmitter i undecodable at the detination. Hence, the econdary ourcedetination communication i aited by a relay network of K relay which receive the packet from the econdary tranmitter and operate in a decode-and-forward fahion. When the relay operate, their total tranmit power i. Power can be changed in order to control the level of interference that may be inflicted on the primary receiver, and uch that P max. The relay are aumed to be in the vicinity of the econdary tranmitter. Thi allow low-power tranmiion by the econdary tranmitter, thereby reducing ignificantly the interference inflicted on the primary link. Moreover, due to the mall pathlo from the econdary ource to the relay, the communication between them i almot errorfree. The thermal noie at each of the receiver in the ytem follow a complex Gauian ditribution CN,. We aume that all the channel are i.i.d and follow a complex Gauian ditribution CN,. We adopt a low fading model where the channel gain remain contant over ten of time lot. Channel etimation occur during a tiny fraction of the time lot via overhearing the tranmiion by the econdary and primary receiver. The econdary receiver may tranmit dedicated ymbol for channel etimation at the relay, where the primary receiver i aumed to tranmit automatic repeat requet ARQ feedback to the primary tranmitter in the form of acknowledgment ACK and negative acknowledgment NACK packet. Due to the broadcat nature of wirele communication, thee packet can be overheard by the relay and their received ignal trength can be ued to etimate the gain of the channel between themelve and the primary receiver. We aume that channel etimation i perfect and that the relay forward the etimated channel to the econdary tranmitter. The primary uer tranmit the packet at the head of it queue tarting at the beginning of the time lot provided that it queue i nonempty. The econdary tranmitter ene the channel at the beginning of the time lot in order to determine the tate of primary activity. Baed on the ening outcome, and given it knowledge of the channel gain between the relay and the primary and econdary receiver, the econdary tranmitter compute the precoding or beamforming vector. It end a data packet over a fraction of the lot duration to the relay. The beamforming vector i incorporated within the packet. 2 Due to the aumption of relay vicinity to the econdary tranmitter, the packet i received correctly by the relay. There are two poibilitie for the beamforming vector depending on the pectrum ening outcome. a The primary uer i ened to be idle: In thi cae, the econdary relay employ conventional tranmit beamforming to enhance the ignal-to-interference-plu-noie ratio SINR at the econdary receiver. The beamforming vector, w a, i given by: H w a = H. where H i a column vector of length K repreenting the channel between the econdary relay and the econdary receiver, and. repreent the L2-norm of a vector. b The primary uer i ened to be active: In thi cae, the econdary tranmiion i till allowed provided that the relay ue a beamforming vector deigned to null out the ignal at the direction of the primary receiver while maximizing SINR at the econdary receiver. Aume that H p i a column If the channel vary more frequently, then the reduction in throughput due to the time needed for channel etimation will be coniderable. 2 The econdary packet i aumed to have a different ize from the primary packet to accommodate the fact that it i ent via a two-hop link within a ingle time lot.
vector of length K repreenting the channel between the relay and the primary receiver. Thu, the beamforming vector, w p, i given by: w p = I ΦH 2 H H I Φ H where upercript H denote vector Hermitian, Φ = HpHH p H p 2 i the projection matrix onto vector H p, and I i the K K identity matrix. The pectrum ening proce i not perfect. The probability of midetecting the activity of the primary tranmitter i given by p md, wherea the fale alarm probability, which i the probability of ening the primary tranmitter to be buy while it i idle, i given by p fa. III. QUEUE SERVICE RATES Here, we derive the achievable ervice rate for both primary and econdary queue. A. Primary Uer Service Rate We can now enumerate the poible ituation that can occur in cae the primary uer ha a packet to tranmit. Secondary uer queue i empty: We denote the probability of Q being empty a Pr{Q = }. In thi cae, the channel i ued olely by the primary uer. Given that the noie variance i unity, the probability of outage, defined a the event when the receive SINR fall below a certain threhold, i given by p out,p = Pr{P p H p 2 < β p } = exp β p P p 3 where H p i the primary link complex channel gain, and β p i the SINR threhold for correct reception of primary tranmiion. The primary ervice rate in thi cae i p out,p. 2 Secondary uer queue i nonempty and the primary uer i detected: Thi ituation happen with a probability equal to p md Pr{Q }, where Pr{Q } = Pr{Q = }. Given that the relay ue the beamforming vector given in 2, the interference caued by econdary tranmiion i eliminated at the primary receiver. Therefore, the ervice rate i alo given by p out,p. 3 Secondary uer queue i nonempty and the primary uer i midetected: Thi ituation happen with a probability equal to p md Pr{Q }. Since the econdary uer midetect the activity of the primary tranmitter, the relay employ beamforming vector w a given in. We denote the outage probability in thi cae by p md out,p, where the upercript md denote a ituation of midetection with the econdary tranmiion cauing interference at the primary receiver. Outage probability p md out,p i given by: { p md Pp H p 2 } out,p = Pr Hpw H a 2 + < β p The ervice rate i then equal to p md out,p. 4 Baed on the preceding enumeration, the primary mean ervice rate, µ p, can be written a: µ p = p out,p Pr{Q = } + p md Pr{Q } + p md out,pp md Pr{Q } B. Secondary Uer Service Rate We enumerate the poible ituation that a econdary tranmitter can find when trying to tranmit. Primary queue i empty and the econdary uer detect the channel to be vacant: Denoting the probability of the primary queue being empty by Pr{Q p = }, thi cae happen with a probability p fa Pr{Q p = }. The beamforming vector w a i ued and the outage probability of the econdary tranmiion i then equal to 5 p out, = Pr{ H 2 < β } 6 where β i the SINR threhold for correct reception of econdary tranmiion. 2 Primary queue i empty and the econdary uer find the channel buy: Thi cae happen with a probability p fa Pr{Q p = }. In thi cae, the econdary uer i falely led to ue the beamforming vector of the form 2. The outage probability of the econdary link in thi cae, denoted by p fa out,, i given by: p fa out, = Pr{ H H w p 2 < β } 7 3 Primary queue i nonempty and the econdary uer detect primary activity: Thi cae happen with a probability p md Pr{Q p }, where Pr{Q p } i the probability of the primary queue being nonempty. The outage probability in thi cae i p d out, = Pr { H H w p 2 } P p H p 2 + < β. 8 where H p i the complex gain of the channel between the primary tranmitter and the econdary receiver. 4 Primary queue i nonempty and the econdary uer mie primary activity: Thi cae happen with a probability p md Pr{Q p }. The econdary uer will now ue the beamforming vector of the form and the outage probability become { p md P H 2 } out, = Pr P p H p 2 + < β. 9 Baed on the preceding enumeration, the econdary mean ervice rate, µ, i given by: µ = p out, p fa + p fa out, p fa Pr{Q p = } + p d out, p md + p md out, p md Pr{Q p } Our main objective in thi paper i to characterize the tability region defined a the et of arrival rate pair λ p, uch that the ytem queue are table. A i evident from 5 and, Q p and Q are interacting and their direct analyi i intractable. Hence, we reort to the concept of dominant ytem a explained in the next ection.
IV. STABILITY ANALYSIS USING DOMINANT SYSTEMS An important performance meaure of a communication network i the tability of the queue. Stability can be defined rigorouly a follow. Denote by Q t the length of queue Q at the beginning of time lot t. Queue Q i aid to be table if [7], [8] lim lim x t Pr{Qt < x} = In a multiqueue ytem, the ytem i table when all queue are table. We can apply Loyne theorem to check the tability of a queue [9]. Thi theorem tate that if the arrival proce and the ervice proce of a queue are trictly tationary, and the mean ervice rate i greater than the mean arrival rate of the queue, then the queue i table, otherwie it i untable. In order to analyze the interacting queue, we employ the concept of dominant ytem introduced in []. In a dominant ytem a uer tranmit dummy packet if it queue i empty. Since we have two uer, we can contruct two dominant ytem, one with the primary tranmitter ending dummy packet when Q p i empty and the other with the econdary tranmitter ending dummy packet when Q i empty. We explain below the relation between the tability region of both the original and dominant ytem. A. Firt dominant ytem In the firt dominant ytem, the primary tranmitter end dummy packet when it queue i empty, wherea the econdary tranmitter behave a it would in the original ytem. Thi effectively mean that Pr{Q p = } =. By plugging thi probability into, the mean ervice rate for the econdary uer in thi ytem become = p d out, p md + p md out, p md 2 where the upercript pd indicate that thi econdary mean ervice rate correpond to the dominant ytem where the primary queue i made to tranmit dummy packet when Q p i empty. Note that in the firt dominant ytem, the econdary mean ervice rate no longer depend on the tate of occupancy of Q p. Provided that the econdary queue i table, i.e., <, the probability of the econdary queue being empty i given by: Pr{Q = } = 3 Now, by ubtituting with 3 and 2 in 5, we obtain the mean ervice rate of the primary queue a p = p out,p p md p md out,p p out,p 4 Baed on the contruction of the firt dominant ytem it can be noted that the queue of the dominant ytem are never le in length than thoe of the original ytem, provided that they are both initialized identically. Thi i becaue the primary tranmitter end dummy packet even if it doe not have any packet of it own, and therefore the tranmiion opportunitie for the econdary tranmitter are reduced. The econdary mean ervice rate i thu reduced in the dominant ytem and Q i emptied le frequently. Thi in turn lower the occurrence of the event where Q i empty and the primary tranmitter operate freely without any interference. Thi reduce the primary mean ervice rate. Given thi, if the queue are table in the dominant ytem then they are table in the original ytem. That i, the tability condition of the dominant ytem are ufficient for the tability of the original ytem. Now if Q p aturate in the dominant ytem, the primary tranmitter will not tranmit dummy packet a it alway ha it own packet to end. For <, thi make the behavior of the dominant ytem identical to that of the original ytem and both ytem are inditinguihable at the boundary point. The tability condition of the dominant ytem are thu both ufficient and neceary for the tability of the original ytem given that <. The tability region baed on the firt dominant ytem i given by the cloure of the rate pair λ p, contrained by tability of the queue. One method to characterize thi cloure i to olve a contrained optimization problem to find the maximum feaible λ p correponding to each feaible [8], []. For a fixed, the maximum table arrival rate of the primary tranmitter i given by the following optimization problem: max. λ p = p.t. <, P max 5 We comment now on how the tranmitted power of the econdary network affect the ervice rate of the primary and econdary uer. From a econdary uer point of view, it directly enhance the received SINR of the econdary uer and thu increae. However, increaing the econdary tranmitted power can have two contradicting influence, with relatively different impact depending on the value of and. On one hand, increaing the tranmitted power of the econdary uer can degrade the ervice rate of the primary uer; in the cae where the primary uer i tranmitting and the econdary uer midetect the preence of the primary uer, increaing the tranmitted power introduce extra interference level at the primary receiver and thu degrade p. On the other hand, increaing the tranmitted power can help the econdary uer to empty it queue fater and evacuate the wirele link, thu enhancing p. B. Second dominant ytem In the econd dominant ytem, the econdary tranmitter end dummy packet when it queue i empty, wherea the primary tranmitter behave a it would in the original ytem. Thi mean that Pr{Q = } = from the primary uer point of view. Uing thi value in 5 give the mean ervice rate for the primary uer a µ d p = p out,p p md + p md out,pp md 6 where upercript d refer to the dominant ytem where the econdary uer i made to tranmit dummy packet when Q i
.9.8.7.6 pd µ.5.4.3.2. t Dominant Sytem range for Q tability =.35.37.368.366.364 pd µ p.362.36.358.356 t Dominant Sytem = =. =.35 2.8.6.4.2.8.6.4.2 t Dominant Sytem.5.5 2.354.5.5 2.2.4.6.8 Fig. 2. Mean ervice rate for the econdary uer veru econdary tranmit power in the firt dominant ytem. Fig. 3. Mean ervice rate for the primary uer veru econdary tranmit power in the firt dominant ytem. Fig. 4. Optimal econdary tranmit power veru for the firt dominant ytem of Figure 2 and 3. empty. Given that the primary queue i table, i.e., λ p < µ d p, the probability of the primary queue being empty i given by: µ d = λ p µ d p Pr{Q p = } = λ p. 7 µ d p The econdary mean ervice rate can thu be written a follow. p d out, p md + p md p fa out, p fa + out, p md p out, p fa p out, p fa + p fa out, p fa 8 Following the argument provided in the previou ubection, the econd dominant ytem and the original ytem are inditinguihable at the boundary point given that λ p < µ d p. The tability region can be obtained by olving the following optimization problem for each λ p < µ d p : max. =.t. λ p < p, P max 9 The tability region of the original ytem i the union of that of both dominant ytem. V. NUMERICAL SIMULATION We conider a primary uer with a maximum tranmiion power of P p = and the detection threhold for the primary receiver i β p =. The econdary tranmitter i aided by K = 4 relay. It ha a probability of midetection and fale alarm equal to p md =. and p fa =., repectively. The threhold for detection at the econdary receiver i β =. We obtain the outage probabilitie defined in 4, 6, 7, 8 and 9 numerically by averaging over 5, channel realization. Due to the non-convexity of the optimization problem in 5 and 9, we olve the problem via numerical earch over the optimal value of. In order to olve 5 for a certain, the contraint on the tability of the econdary queue poe a lower limit on, which ha an upper limit of P max. Figure 2 how veru in the firt dominant ytem. Power.5 to atify the condition on the tability of Q when =.35. Generally peaking, the contraint of 5 define the range of econdary power for which the problem i feaible. We earch for the maximum of p over thi range. Figure 3 how the variation of p with at three different value for. The figure how the range of for which the problem i olved. Note that the range i reduced with the lower limit on increaing due to the demand for a higher power to atify the tability of Q. The optimal power i provided in Figure 4. Power i zero when no value le than or equal to P max can atify the tability condition of Q. We dicu now the olution of 9. The contraint on the tability of Q p impoe a limit on the econdary power to be ued. Since any increae in produce more interference on the primary uer in the cae of midetection, then the primary mean ervice rate i reduced. To atify the tability contraint, hould be le than ome value, which depend on λ p. Thi i demontrated in Figure 5 where hould not exceed about.85 for λ p =.35. The econdary mean ervice rate increae with a hown in Figure 6. Neverthele, the range over which we earch for the optimal power value i reduced in order to atify Q p tability. Figure 7 how the optimal for each poible λ p. Note that the region where = mark the value of λ p at which the primary queue cannot be made table. Figure 8 how the tability region of both dominant ytem together with their union which i the tability region of the original ytem under invetigation. Figure 9 compare the table region achieved by two ytem with P p = and P p =. A expected, increaing P p enhance the primary arrival rate that can be operated at while preerving the tability of the ytem queue. Thi come at the expene of. VI. CONCLUSION A CRN i conidered which conit of a ingle primary and a ingle econdary link. The econdary tranmitter utilize a et of dedicated relay by applying beamforming technique to null out econdary tranmiion at the primary receiver to allow for concurrent tranmiion with the primary uer. Primary and econdary tranmitter are aumed to be equipped with
.37.365.36 d µ p.355 2nd Dominant Sytem.9.8.7.6 d µ.5 2nd Dominant Sytem λ = p λ p =. λ p =.35 λ =.36 p 2.5 2nd Dominant Sytem.4.35.345 range for Q p tability λ p =.35.3.2..5.34.5.5 2.5.5 2.2.4.6.8 λ p Fig. 5. Mean ervice rate for the primary uer veru econdary tranmit power in the econd dominant ytem. Fig. 6. Mean ervice rate for the econdary uer veru econdary tranmit power in the econd dominant ytem. Fig. 7. Optimal econdary tranmit power veru λ p for the econd dominant ytem of Figure 5 and 6. λ p.5 Stability region of the 2nd dominant ytem Stability region of the t dominant ytem.9.8 Stability region of the ytem with two interacting queue P p = Stability region of the ytem with two interacting queue P p = λ p.5.2.4.6.8 Stability region of the ytem with two interacting queue.2.4.6.8.7.6 λ.5 p.4.3.2..2.4.6.8 Fig. 8. Stable region of the two interacting queue ytem. Fig. 9. Stable region of the two interacting queue ytem for P p =,. finite buffer, and we tudied the table region of both queue. Sening error are taken into account, and their effect i hown on the achievable table region. We reorted to the concept of dominant ytem in order to decouple the interacting queue and arrive at the provided reult, and we proved that thi approach provide the exact maximum table region. Numerical evaluation are provided to give ueful inight on the obtained reult. VII. ACKNOWLEDGEMENT Thi work ha been upported in part by a grant from the Egyptian National Telecommunication Regulatory Authority NTRA. REFERENCES [] O. Simeone, Y. Bar-Ne, and U. Spagnolini, Stable throughput of cognitive radio with and without relaying capability, Communication, IEEE Tranaction on, vol. 55, no. 2, pp. 235 236, 27. [2] J. Gambini, O. Simeone, U. Spagnolini, Y. Bar-Ne, and Y. Kim, Stability analyi of a cognitive multiple acce channel with primary qo contraint, in Signal, Sytem and Computer, 27. ACSSC 27. Conference Record of the Forty-Firt Ailomar Conference on. IEEE, 27, pp. 787 79. [3] G. Zheng, S. Ma, K.K. Wong, and T.S. Ng, Robut beamforming in cognitive radio, Wirele Communication, IEEE Tranaction on, vol. 9, no. 2, pp. 57 576, 2. [4] M.A. Beigi and S.M. Razavizadeh, Cooperative beamforming in cognitive radio network, in Wirele Day WD, 29 2nd IFIP. IEEE, 29, pp. 5. [5] J. Liu, W. Chen, Z. Cao, and Y.J. Zhang, Delay optimal cheduling for cognitive radio with cooperative beamforming: A tructured matrixgeometric method, Mobile Computing, IEEE Tranaction on, vol., no. 8, pp. 42 423, 22. [6] J. Liu, W. Chen, Z. Cao, and Y.J.A. Zhang, Cooperative beamforming for cognitive radio network: A cro-layer deign, Communication, IEEE Tranaction on, vol. 6, no. 5, pp. 42 43, 22. [7] W. Szpankowki, Stability condition for ome ditributed ytem: buffered random acce ytem, Advance in Applied Probability, pp. 498 55, 994. [8] A.K. Sadek, K.J.R. Liu, and A. Ephremide, Cognitive multiple acce via cooperation: protocol deign and performance analyi, IEEE Tranaction on Information Theory, vol. 53, no., pp. 3677 3696, Oct. 27. [9] R.M. Loyne, The tability of a queue with non-independent interarrival and ervice time, in Proc. Cambridge Philo. Soc. Cambridge Univerity Pre, 962, vol. 58, pp. 497 52. [] R.R. Rao and A. Ephremide, On the tability of interacting queue in a multiple-acce ytem, IEEE Tranaction on Information Theory, vol. 34, no. 5, pp. 98 93, Sep. 988. [] S. Kompella, G.D. Nguyen, J.E. Wieelthier, and A. Ephremide, Stable throughput tradeoff in cognitive hared channel with cooperative relaying, in Proceeding IEEE INFOCOM, Apr. 2, pp. 96 969.