EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X 1 Traic-Aware Channel Sensing Order in Dynamic Spectrum Access Networks Chun-Hao Liu, Jason A. Tran, Student Member, EEE, Przemysław Pawełczak, Member, EEE, and Danijela Cabric Member, EEE Abstract n this paper we present new results on the problem o inding the best channel sensing order or multi-channel Dynamic Spectrum Access DSA) networks. We start with the general assumption that all Secondary Users SUs) cooperatively sense each Primary User PU) channel at one time. Then, the SU sensing results are reported to a DSA base station that schedules SU transmissions in order to maximize DSA network throughput. We then assume that PU traic parameters are not perectly known to DSA network and change over time, and propose a novel PU channel sensing order scheme based on the quality o PU traic estimation. We adopt a maximum likelihood estimator to estimate the traic statistics o PU channels and derive the Cramér-Rao CR) bounds or the PU traic estimation perormance. Based on the CR bound and its Gaussian approximation, we analyze the impact o the estimation error on the DSA network throughput by computing a new metric called sensing order conidence, i.e., the probability that the best selected sensing order is not aected by PU traic estimation errors. Finally, we ormulate a convex optimization problem to determine the minimum number o PU channel state samples required or estimating PU traic parameters ater determining a certain constraint on the sensing order conidence metric to achieve the best sensing order. ndex Terms Dynamic spectrum access networks, channel sensing order, throughput maximization, traic estimation, sensing order conidence, convex optimization.. NTRODUCTON THE use o radio requency channels assigned to primary licensed) users PUs) is not always high. n turn, there is vacant spectrum that can be exploited by secondary unlicensed) users SUs) in both time and the requency domains on a Dynamic Spectrum Access DSA) basis. Thus, inding the unused PU channels becomes an important problem in DSA networks [1]. Vacant spectrum can be ound by sensing licensed channels to determine whether PU is present or not. the PU is present, the SU should switch, within a short time, to another PU channel and sense again. Otherwise, the SU can Manuscript received November 18, 01; revised April 4, 013; accepted May 30, 013. Date o publication X X, 01X; date o current version May 30, 013. The associate editor coordinating the review o this paper and approving it or publication was Dr. Y.-C. Liang. Chun-Hao Liu, Jason A. Tran, and Danijela Cabric are with the Department o Electrical Engineering, University o Caliornia, Los Angeles, 56-15B Engineering V Building, Los Angeles, CA 90095-1594, USA email: liuch37, danijela}@ee.ucla.edu, jasonat@ucla.edu). Przemysław Pawełczak is with the Department o Electrical Engineering, Mathematics and Computer Science, Delt University o Technology, Mekelweg 4, 600 GA Delt, The Netherlands email: p.pawelczak@tudelt.nl). This work has been supported by the National Science Foundation under CNS grant 1149981 and by the Dutch Technology Foundation STW under contract 1491. Digital Object dentiier XX.XXXX/JSAC.013.XXXXXX transmit its own data and exploit the spectrum vacancy. n the event o multiple SUs and PU channels, it is important to have a well-designed PU channel selection protocol and channel sensing order scheme, i.e. the process o ordering PU channels to sense to optimize a predeined objective throughput, delay, etc.) o a DSA network []. For instance, in a centralized DSA network, PU channel selection and sensing order scheduling can be done by the base station or a coordinator). The DSA base station would then decide on the best sensing policy based on the PU channel statistics and would schedule SU transmissions accordingly when a channel is sensed to be vacant. n order to acquire the PU traic statistics o each channel which vary over time, making exact a priori knowledge o PU traic impossible [3]), a highly accurate PU traic estimation algorithm is needed. A. Related Work Techniques o estimating the PU traic parameters were studied in, e.g., [3], [4] and the reerences therein), in isolation rom the channel sensing order problem. Considering channel sensing order, [5] was the irst work to investigate this aspect to the best o the authors knowledge) and introduced the concept o semi-markov chain to model the availability o PUs assuming perect knowledge o the PU traic statistics). Therein, it was proposed to sense the channels in descending order o the PU idle probability to ind the spectrum opportunities with a small delay. n [6], the authors improved the sensing order results o [5]. Their objective was to determine the optimal sensing sequence that minimizes the average delay in inding idle channels with the constraint o the total capacity o the system being above a certain threshold. The approach was not only based on the idle probability o the PU channel again, assumed to be known perectly), but also on sensing time and channel capacity. n [7], the authors proposed a simple channel sensing order or SUs by sensing the channels in descending order o their achievable rates. Therein they proved that the SUs should stop sensing at the irst sensed-ree channel or maximum throughput without requiring a priori knowledge o PU traic. This work considered maximization o the throughput or one SU to transmit in one channel while neglecting the transmission opportunities in other channels. The authors o [8] investigated the optimal sensing order with opportunistic transmissions by considering both adaptive and non-adaptive modulations or SUs to exploit changes in PU channel physical layer parameters. The problem o PU traic parameter estimations was not considered in this work.
EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X Finally, [9] considered inding an optimal channel sensing order through a constrained Markov decision process) taking collision probability between PU and DSA network into consideration. As in all above reerred works, the authors also assumed that the arrival and departure rates or each PU channel were known perectly in advance. B. Our Contribution Unlike [5], [6] which aimed at minimizing the channel selection delay, our goal is to maximize the throughput or all SUs per time slot. n contrast to [7], [8] which considered the throughput or one SU, our system targets the problem o inding opportunities or the whole DSA network transmitting on any PU channel, which is a more general problem to tackle. Most importantly, in the aorementioned works, the authors proposed the sensing order simply using either PU activities or only parts o physical layer parameters. these are considered simultaneously, then a DSA network will be able to maximize its throughput even urther. Thereore, this paper ocuses on maximizing the throughput or the whole DSA network through a sensing order that combines the inormation on PU traic statistic estimates, channel capacities, and sensing times. The contribution o this work is threeold: a) PU Channel Sensing Order Algorithm and ts Derivation: A sensing order algorithm maximizing the throughput or SUs is proposed, assuming irst a perect knowledge o PU traic statistics and PU channel physical layer parameters; b) PU Traic Estimation Modeling and Analysis: For discrete PU traic models, parameters o PU traic are estimated using Maximum Likelihood ML) estimation. Moreover, the Cramér-Rao CR) bound is derived or the adopted estimators by inding their Fisher inormation matrix; c) Analysis o Throughput or the Proposed Channel Sensing Order Algorithm Considering the Traic Estimation Process: The proposed sensing order algorithm is analyzed as a unction o PU traic estimation errors. We derive the throughput loss in terms o a new sensing metric called sensing order conidence, which is derived rom the CR bound and Gaussian approximation or the proposed PU traic estimators errors. We orm a convex optimization problem to determine the minimum number o PU traic samples needed to obtain a certain level o sensing order conidence on the chosen sensing order. n addition, traic estimation and our sensing order algorithm are discussed when the traic is nonstationary. The rest o the paper is organized as ollows. The system model is presented in Section. n Section, we propose a sensing order algorithm which maximizes the system throughput. The algorithm is analyzed in terms o traic estimation error, which is quantiied by the CR bound. n addition, the relation between traic estimation and the proposed sensing order algorithm is discussed in terms o non-stationary traic. Section V shows numerical results comparing dierent sensing order algorithms, discusses the estimation and conidence perormance, and gives design examples to understand the connection between the proposed sensing order algorithm and the traic estimator. Finally, Section V concludes this paper.. SYSTEM MODEL Following the model and assumptions o [5] [8], in this paper, we consider a DSA network consisting o M PU channels o known capacity C i and K SUs, i M, where M = 1,,, M} and M K. Each SU can be reconigured to be a transceiver or a sensor, i.e., it can either transmit on or sense one PU channel at a time [5] [7]. The transmission structure or each PU is assumed to ollow a synchronous, slotted rame within slot time T, implying the activity or the PU is constant during one slot time [10], [11]. To enhance the detectability o the PU channels and to minimize the time spent on sensing, the SUs jointly sense one channel at a time via a cooperative sensing scheme [1], [13]. n turn, the sensing results are orwarded to a DSA base station which makes a collective decision on whether or not there is a PU present on a channel. To simpliy analysis, this paper considers the case where the DSA controller sensing decision is ree o spectrum sensing errors, unless explicitly speciied otherwise. Note that spectrum sensing ree decision is a well justiied assumption as under the cooperative sensing with large K sensing errors are amply eradicated. a PU vacancy in channel i is detected within sensing time T i) < T, the DSA base station schedules one SU to switch to a transceiver and transmit on that channel. The remaining SUs continue to sense the rest o the channels in an order governed by a predeined algorithm. Our task is to design such algorithm, i.e., a sequential sensing order A or SUs to sense all PU channels, which maximizes the overall DSA network throughput on a slot basis. PU traic is modeled as a Markov process [14], [15], where s represents the current state o the PU. States s = 0 and s = 1 indicate the PU absence and presence, respectively. This traic is characterized by the steady-state distribution and the transition probabilities. Probability o the PU absence is described by P 0 Prs = 0} = 1 u and the probability o PU presence is P 1 Prs = 1} = u. The transition probabilities describe the probability o switching rom the current state s to next state ŝ ater a constant slot time T. The probability o transition rom state x to state y is denoted as P xy with our possible probabilities given by tuple P P 00, P 01, P 10, P 11 } with u = 01 P 01+P 10, P 00 + P 01 = 1 and P 10 + P 11 = 1. Deine s s 1, s,, s M ) and ŝ ŝ 1, ŝ,, ŝ M ), where s i is the current PU traic state and ŝ i is the next PU traic state or the i-th channel. Note each PU ollows its own Markov process, i.e., } s i is deined by its own traic parameters u i), P i) 00, P i) 10.. A SENSNG ORDER ALGORTHM TO MAXMZE THROUGHPUT N MULT-CHANNEL DSA NETWORKS We begin by explaining in detail our proposed sensing order algorithm and its relationship to the throughput in our system model. The DSA throughput gains o this sensing order algorithm are dependent on the accuracy o the traic statistic estimates o the PUs on each channel. Thereore, we show the need o a sensing order conidence metric which will govern the level o impact o estimation errors on the throughput loss. This sensing order conidence metric is derived rom the CR
LU et al.: TRAFFC-AWARE CHANNEL SENSNG ORDER N DYNAMC SPECTRUM ACCESS NETWORKS 3 Fig. 1. An example o time and requency domain DSA transmission structure or M = 4 PU channels. Current state s = 1, 1, 0, 0), and next state ŝ = 0, 1, 0, 1). Sensing order A = 1,, 3, 4). bounds o the maximum likelihood traic estimator adopted in this work. Deployed SU networks will manually speciy or tune this metric according to network conditions number o channels, MAC structure, etc.) and desired throughput. Once the sensing order conidence is speciied, SU networks will know how accurately they will need to estimate the traic beore proceeding to sense channels and transmit. We show how a convex optimization problem can be ormulated to minimize the number o traic samples to take in a certain time period to estimate PU traic statistics to a desirable accuracy, which is determined by the sensing order conidence. Lastly, we will discuss how to use the combination o traic estimation and the proposed sensing order algorithm when the traic is non-stationary. A. SU Network Throughput and Sensing Order Algorithm Deine a sensing order A = a 1, a,, a M ) or SUs in M channels, which is a permutation o the set U, i.e., U is a set that contains all the permutations o the sequence 1,,, M), and U = M!. For example, Fig. 1 shows a sensing order a 1, a, a 3, a 4 ) = 1,, 3, 4) or 4 channels. The SUs sense the PU channels in this order using the required sensing time T i) or the i-th channel. Assume SUs have the current state inormation o PU channels, s i or the i-th channel. Ater sensing the next time slot, the SUs will obtain the inormation o the next state ŝ i. it is sensed as available, the rest o the slot time, T i T aj), can be used to transmit j=1 data. Otherwise, the SUs will ail to ind an opportunity to transmit in their respective queues. We deine the eective rate or data transmission or all channels in one slot given the next state inormation ater sensing as the system throughput RA) [7], [16], [17] M i T j=1 RA) = T aj) ) C ai ŝ ai ), 1) T whereas x) is the indicator unction deined as x) = 1 or x = 0, and x) = 0, otherwise. The expected throughput ERA)} is deined as the average throughput, where E } denotes the expectation. The problem is to maximize ERA)} by deciding on the proper sensing order A. We introduce the ollowing theorem. Theorem 1: The sensing order to maximize ERA)} can be achieved by sensing the channels in descending order o the sensing actor F i = Ci Prŝi=0} T i), 1 i M. Proo: Assuming we sense channels in a sequence A = a 1, a,, a l, a l+1,, a M ) in descending order o F i. Deine A = a 1, a,, a l+1, a l,, a M ) which changes any two consecutive elements a l and a l+1 in A, 1 l M 1. Subtracting ERA )} rom ERA)} we have ERA)} ERA )} = } } T a l+1) T a l) E C al ŝ al ) E T T C a l +1ŝ al+1 ) = T a l+1) C al Prŝ al = 0} T a l) T T C a l+1 Prŝ al+1 = 0}. ) Since we sense the channels in descending order o F i, we have F al F al+1, which implies that ERA)} ERA )} 0, and thereore ERA )} ERA)}. Hence, interchanging any two consecutive elements a k, a l ) with F ak F al will decrease the expected throughput. However, any sequence B U can be obtained by a inite interchanging algorithm starting rom sequence A see Appendix A), with each interchanging step o two consectuive elements a k, a l ), k < l. Thereore, since F ak F al, k < l, we have ERA)} ERB)}, which proves the theorem. For example, given M = 5, assume sequence A is sorted by the proposed sensing actor. we would like to interchange A to a sequence B = a, a 3, a 5, a 1, a 4 ), we can ollow the process A = a 1, a, a 3, a 4, a 5 ) a, a 1, a 3, a 4, a 5 ) a, a 3, a 1, a 4, a 5 ) a, a 3, a 1, a 5, a 4 ) a, a 3, a 5, a 1, a 4 ) = B. Each step interchanges only two consecutive elements a k, a l ), k < l, so it decreases the expected throughput, which implies ERA)} ERB)}. The sensing actor F i is composed o three parts: the capacity, sensing time, and idle probability o the PU channel, as the throughput is related to all o these values. Since the capacity and sensing time are known beore deciding
4 EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X Algorithm 1 Proposed Sensing Order Algorithm 1: procedure SENSNGA, C i, T i), P m i), P i) ) : i 1 3: while i M do 4: spectrum sensing at current time slot to obtain s i 5: i i + 1 6: end while 7: i 1 8: while i M do 9: i s i == 1 then 10: F i = C i 11: else 1: F i = C i P m i) P i) i) 00 +1 P m )P i) ) 10 T i) 1 P i) i) i) )P 00 +P P i) ) 10 T i) 13: end i 14: i i + 1 15: end while 16: Sort sequence F 1,..., F M } in descending order as the sequence A 17: return A 18: end procedure a sensing order, the accuracy o the estimation o PU idle probabilities aects the sensing order the most. The idle probability Prŝ i = 0} can be obtained by parameter 1 u i). With the knowledge o the present state s i, we can estimate the idle probability as Prŝ i = 0} = P i) i) s i0, where P s is i0 the transition probability rom the present state to state zero or the i-th channel. n addition, i we consider sensing errors given by alse alarm and mis-detection probabilities, denoted by P i) and P m i) or the i-th channel, respectively 1 ) in our model, the idle probability given the current estimated state s i can be derived as Prŝ i = 0} = P m i) P i) i) 00 + 1 P m )P i) i) )P 00 + P i) 1 P i) 10, i s i = 1, P i) 10, i s i = 0, see detailed derivation in Appendix D). The proposed sensing order algorithm is shown in Algorithm 1. To analyze i the errors o idle probability estimations negatively impact the correctness o our sensing order algorithm, we next deine and analyze the sensing order conidence metric. B. Sensing Order Conidence Metric and ts Derivation We start with the derivation o our proposed sensing order conidence metric without sensing errors, i.e., P i) = P m i) = 0 i M. First, we motivate the introduction o this metric. Then, we ormally introduce our adoption o an ML PU state transition probability estimator and derive its CR bound. Lastly, we show how the estimator and its bounds can be used to derive the sensing order conidence metric. 1 The sensing error is assumed to be independent or each traic sample. The estimated PU traic sample is denoted by s. t ollows that s = 1 i i) s = 1 and no mis-detection error occurred, or ii) s = 0 and a alse alarm error occurred. Similarly, s = 0 i i) s = 1 and a mis-detection error occurred, or ii) s = 0 and no alse alarm error occurred. For urther discussions and derivations o traic estimation in terms o sensing errors, please reer to [3]. 3) 1) Throughput Loss Derivation: As discussed earlier, when PU traic statistics are not perectly known, the sensing order algorithm will not always produce the best sequence, and, as a result, incur DSA throughput loss. We deine the normalized throughput loss L as the normalized dierence between the expected throughput with estimated transition probabilities and expected maximal throughput when transition probabilities are perectly known. This is expressed as L = 1 ERÂ)} ERA o )}, 4) where  is the sensing order sequence obtained by the proposed sensing actor with estimated transition probabilities, and A o is the sensing order sequence obtained by the proposed sensing actor with perect transition probabilities. The expected maximal throughput is derived as ERA o )} = Prs, ŝ}ra o ), 5) s where Prs, ŝ} is the joint probability or current state s and next state ŝ. The expected estimated throughput is derived in a similar way as M! } ERÂ)} = E s κ i seŝrŝa i )} = s M! ŝ κ i s Prs, ŝ}rŝa i ), 6) where κ i s = Pr = Ai s} is the conditional probability o the estimated order equals to the i-th order A i U, given current state s later we will denote κ i s as the sensing order conidence metric), and RŝA i ) is the throughput with sensing order A i given the next state ŝ, deined as 1) replacing A with A i. Beore deining and deriving κ i s or order A i, we irst discuss the estimators or estimating transition probabilities and their analysis in the next section. ) ML Transition Probability Estimators: Assume we acquire N PU traic samples 3 that populate a state vector z N = z 1, z,, z N ) under perect sensing rom a PU channel, where z i 0, 1}, 1 i N. We ormulate the likelihood unction Lz N ) using the Markov chain property to be N 1 Lz N ) = Prz N } = Prz 1 } Prz i+1 z i } N 1 = u z1 1 u) 1 z1 ŝ P ziz i+1 = u z1 1 u) 1 z1 P n0 00 P n1 01 P n 10 P n3 11 = u z1 1 u) 1 z1 P n0 00 1 P 00) n1 P n 10 1 P 10) n3, 7) where n 0, n 1, n, and n 3 represent the number o state transitions o 0 0), 0 1), 1 0), and 1 1) 3 Recall that in this paper a PU sample is the spectrum sensing result in a time slot. Thereore, the total number o samples is equivalent to the time duration with the unit o a time slot.
LU et al.: TRAFFC-AWARE CHANNEL SENSNG ORDER N DYNAMC SPECTRUM ACCESS NETWORKS 5 respectively. t is known in ML estimation that the unknown variables P 00 and P 10 are obtained by solving Then the estimators are derived as log Lz N ) = 0, P 00 8) log Lz N ) = 0. P 10 9) ˆP 00 = n 0, n 0 + n 1 10) ˆP 10 = n, n + n 3 11) where ˆP 01 = 1 ˆP 00 and ˆP 11 = 1 ˆP 10. 3) ML Estimator CR Bound: We adopt the CR bound, the analytical lower bound or the variance o any unbiased ML estimator, to quantiy the perormance o our adopted estimators. ML estimators achieve this bound as the number o traic samples approaches ininity when certain conditions are satisied [18, Ch. 1]. The CR bound is obtained through the ollowing Lemma. Lemma 1: The CR bounds or estimators ˆP 00 and ˆP 01 are V 0 and V 1, respectively, which are expressed as V 0 = P 001 P 00 ) P 0 N 1), 1) V 1 = P 101 P 10 ) P 1 N 1). 13) Proo: See Appendix B. Note that as N goes to ininity, the bound goes to zero. This demonstrates that the accuracy o the estimator increases as the number o samples used or estimation increases. 4) The Sensing Order Conidence Metric κ i s: n this section, we irst derive analytically the sensing order conidence metric κ i s using the CR bound o the estimators derived in Section -B3. n addition, we deine the average sensing order conidence metric κ o obtaining the best order A o. We start irst with the two channel case, i.e., M =. We consider sensing two channels in any order a i, a i+1 ) with perect traic estimation and â i, â i+1 ) with traic estimation errors. Given the current state s = s ai, s ai+1 ), the sensing order conidence or two channels only or the sensing order not being aected by traic estimation errors, is given as κ i s,. t is deined as κ i s, Prâ i, â i+1 ) = a i, a i+1 )} = Pr ˆFai ˆF } ai+1 = Pr = Pr = Pr = Pr C ai ˆP a i) T ai) ˆP ai) ˆP ai+1) ˆP a i) +P ai) C a a i+1 ˆP i+1) T ai+1) ɛ i = Pr a ˆP i) b ɛ i0 i ˆP a i+1) P ai) b ɛ i0 i } ɛ ip ai+1) b 0 i+10 e ai) ɛ i e ai+1) ɛ i P ai+1) ) P ai+1) ˆP ai+1) 0 } } P ai), 14) where b i = s ai, ɛ i = Ca i+1 T a i) C ai T a i+1 ), ˆFai = Ca ˆP a i) i b i 0 is the T a i ) is the estimated transition a estimated sensing actor, ˆP i) probability, and e ai) ai) = ˆP b P ai) i0 is the traic estimation error or the a i -th channel. We model the traic estimation errors e ai) and e ai+1) as Gaussian random variables, which is a general assumption or ML estimation i the number o samples used in the estimator is large } enough [18, Ch. 1]. Notice that E e ai)} a = E ˆP i) P ai) = 0 and E e ai+1)} } a = E ˆP i+1) P ai+1) = 0. We deine Var e ai)} } a = Var ˆP i) σa i and Var e ai+1)} = } a Var ˆP i+1) σa i+1. The distribution o the sum o two Gaussian independent variables is also normally distributed as J = e ai) ɛ i e ai+1) N 0, σa i + ɛ i σ a i+1 ). Hence, 14) is expressed as κ i s, = Q ɛ ip ai+1) b P ai) i+10 b i0, 15) σa i + ɛ i σ a i+1 where Q ) is the tail probability unction o the standard normal distribution. n addition, we derive the approximate closed-orm expression or κ i s, in 15) by using the CR bound 1) and 13) or σa i and σa i+1, respectively, which is expressed as κ i s, Q P a i ) b i 0 ɛ i P ai+1) P ai) 1 P a i ) b i 0 ) P a i ) b i N ai 1) + ɛ i P a i+1 ) b i+1 0 1 P a i+1 ) b i+1 0 P a i+1 ) b i+1 N ai+1 1) ), 16) where N ai is the number o samples or the a i -th channel. Next, we consider the sensing order conidence metric or M channels. Assume the i-th sensing order is A i = a 1, a,, a M ) with corresponding state vector s. The sensing order conidence metric κ i s o the estimated best sensing order  = â 1, â,, â M ) is achieved as long as the order or any two consecutive sequence is not aected by the estimation errors, which is represented as κ i s = Pr = Ai } = Pr[â 1, â ) = a 1, a )] [â, â 3 ) = a, a 3 )] [â M 1, â M ) = a M 1, a M )]}. 17) Note that the order or any two consecutive sequence is not aected by other consecutive sequences, thereore the events in 17) are independent, i.e., κ i s = = M 1 M 1 Prâ i, â i+1 ) = a i, a i+1 )} = M 1 κ i s, Q ɛ ip ai+1) b P ai) i+10 b i0, 18) σa i + ɛ i σ a i+1 by using κ i s, rom 16). Finally, the sensing order conidence metric o the best sensing order, denoted as κ [o] s, is derived by applying the best
6 EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X sensing order A o to 18). n addition, the average sensing order conidence metric or the best sensing order is expressed as κ = E s κ [o] s }. 19) C. Minimizing the Required Number o Samples or Traic Estimation: A Convex Optimization Problem From the estimation point o view, using more PU traic samples, i.e., a longer traic estimation time, will increase the accuracy o obtaining the transition probabilities. Although higher sensing order conidence can be achieved with more traic samples or longer estimation time, it is realistic to consider energy-eicient, delay-aware traic estimation. To do so, we strive to ind the minimum number o samples or each channel to satisy the desired sensing order conidence constraint, since each channel has its own traic statistics. The optimization problem becomes min M N i subject to κ [o] s η, 0) where N i i M, is the number o samples used or estimating the i-th channel, and η is the minimum tolerated sensing order conidence metric. Expression κ [o] s is a unction o N i since it contains the variances or the estimated transition probabilities, which are shown through the CR bounds in 1) and 13). Note that the problem o 0) is purely theoretical, as in a practical network implementation o the optimal sensing order A o is not known. The objective unction is a linear combination o M variables. Thereore, it is a convex unction. The constraint can be shown to be a convex constraint 4 by the ollowing Lemma. Lemma : κ [o] s is a concave unction. Proo: See Appendix C. Since both the objective unction and its constraint are convex, we show that the optimization problem is a standard convex optimization problem which can be solved by numerous methods e.g. interior point method [19]) using computer optimization tools e.g. CVX [0]) eiciently. D. Discussion: Traic Estimation and The Sensing Order Algorithm in Non-Stationary Traic We assume stationary PU traic in our system model, which means the traic o all channels is estimated at once beore the proposed sensing order algorithm starts. However, in the real wireless scenario, the traic statistics may change slowly according to the utilization o PUs. This implies we need an algorithm which can track the traic statistics together with the proposed sensing order algorithm simultaneously. A sliding window method is a good candidate to solve this problem. For example, in the initial phase, we adopt a long time period to estimate each channel and obtain their statistics with high accuracy. Then, we start the tracking phase by sensing the channels using the proposed sensing order. Ater sensing each channel, we collect the samples and update the 4 A standard convex constraint is ζ, where is a convex unction. t is equivalent to ζ, where is a concave unction. statistics within a window length o N w samples. The samples located outside the window are abandoned. The statistics are updated gradually as time goes assuming the statistics do not vary rapidly with time, which is a reasonable assumption. V. NUMERCAL RESULTS n this section, we veriy our proposed algorithm and analysis with computer simulations. The simulation model here ollows directly the one introduced in Section. First, we compare the perormance o our proposed sensing order algorithm with the state o the art sensing order algorithms in [5] [7]. Then, we veriy the CR bound o PU traic estimators and compare it with the simulated variance o our proposed ML estimators o the PU traic parameters. Furthermore, we show the analytical and simulated results o the average sensing order conidence metric with respect to the number o samples used in traic estimation and the DSA network throughput loss with respect to the average sensing order conidence metric. Lastly, we demonstrate two design examples o using traic estimation and the proposed sensing order algorithm. The examples will show the minimum required number o samples or PU traic estimation given a sensing order conidence constraint. A. DSA Throughput with Perect PU Traic Estimation Fig. shows the throughput comparisons or dierent sensing order algorithms assuming perect PU traic estimation. We choose M = 10 PU channels in our scenario. For the system parameters, we choose the constant traic parameters P 00 and P 10, which are vectors containing the elements P i) 00 and P i) 10, respectively, or the i-th channel i M. For the physical layer parameters, we choose constant capacity vector C and sensing time vector T, which contains the elements C i and T i), respectively, or the i-th channel i M. T i) is chosen to be less than or equal to 50 ms and C i is less than or equal to 10 bits/s, which are reasonable values ollowing [6]. Each simulation result is calculated by averaging over 10 4 realizations. Under perect sensing or the current PU states, Fig. a) demonstrates that as the PU slot time T increases, the DSA throughput increases as well. This is because as long as T increases, there is more time or SU transmissions, i.e., enlarging T T i). Furthermore, we can observe that the analytical result o our proposed sensing order algorithm, derived in 5), matches the simulation result. t is clear rom Fig. a) that our proposed sensing order algorithm outperorms other well-known sensing order algorithms, such as the combined algorithm which merges the two algorithms proposed in [5], [6], the capacity-based algorithm in [7], and the random order algorithm. The combined algorithm adopts the sorting actor Prŝ i=0} T i) to sense the channels. The capacity-based algorithm considers simply sensing the channels in descending order o C i. The random order algorithm senses the channels randomly. By considering the traic and physical layer sensing parameters together, our proposed algorithm perorms the best over the other sensing order algorithms.
LU et al.: TRAFFC-AWARE CHANNEL SENSNG ORDER N DYNAMC SPECTRUM ACCESS NETWORKS 7 Fig. b) takes imperect sensing or current PU states into account. First, we can observe the throughput degradation when we have sensing errors with P i) = P m i) = 0.1, i M. we do not include sensing errors into our sensing actor when sensing errors are actually present), i.e., we set P i) = 0 and P m i) = 0 in Algorithm 1 reer to lines 10 and 1), the perormance degradation rom the perect sensing case is even worse. Second, since the combined method also takes transition probabilities into consideration, its throughput will degrade as well. However, the capacity and random order method does not rely on transition probabilities. Thereore, they show the same perormance as the case without sensing errors. To validate the stochastic perormance o the sensing order algorithms, we plot the experimental cumulative distribution unction CDF) o the throughput R in Fig. 3 under perect sensing, i.e. P i) = P m i) = 0, i M. The CDF curve is obtained based on 100 randomly generated parameter sets S = C, T, P 00, P 10 } with 10 4 random realizations or parameters s, ŝ} in each set. To obtain values or each element in S: i) parameters P i) 00 in P 00 and P i) 10 in P 10 are chosen uniormly and randomly rom U0, 1), where Ua, b) denotes the uniorm distribution within a range o minimum and maximum value o a and b, respectively, or the i-th channel i M; ii) the capacity C i in C and sensing time T i) in T are also chosen to be uniormly distributed in the range U0, C avg ) and U0, T avg ), where C avg and T avg are the average capacity and average sensing time, respectively. Because the curve representing our proposed algorithm lays on the right-hand side o the other curves, we conclude that our algorithm also outperorms state o the art approaches in a stochastic manner. B. PU Traic Estimation Perormance Under Perect Sensing To measure the traic estimation perormance, we consider only one channel as multiple channels can ollow the same estimation process. Fig. 4 depicts the PU traic estimation perormance. The traic parameters are chosen as P 00 = P 10 = 0.7 in Fig. 4a), and N = 00, P 01 = 0.1 in Fig. 4b). The metric we use to evaluate the perormance is the root-mean-square RMS) error o the estimated transition probabilities, i.e., taking the square root o the mean-squarederror MSE) o the estimated parameters. Fig. 4a) demonstrates that as the number o samples increases, the RMS error decreases. mproved accuracy or the estimation is obtained by taking more traic samples into consideration. Furthermore, the estimator ˆP 00 is shown to have better perormance than ˆP 10. This can be explained by ocusing at u = 0.3 in this example. Since P 0 = 1 u = 0.7, we know that there is higher chance or zero to be present in the sequence. This increases the number o the estimation inormation n 0 and n 1 or ˆP 00, which results in more accurate estimation. n addition, the RMS error achieves the CR bound as long as the number o samples is large enough, which veriies the ML estimation property stated in [18]. Fig. 4b) plots the RMS error with respect to u. Note that CR bound remains a lower bound or the PU estimators. But or the estimator ˆP 00, the RMS error Throughput bits/s) 0 18 16 14 1 10 8 Proposed Sim.) 6 Proposed An.) 4 Combined Capacity Random order 0 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 0.5 Slot Time s) a) Throughput within the PU slot time T without sensing errors. The DSA throughput is plotted or the proposed algorithm, combined algorithm [5], [6], capacity algorithm [7], and random order algorithm. Throughput bits/s) 0 18 16 14 1 0.3 10 Proposed w/o SE 8 Proposed not considering SE 6 Proposed considering SE 4 Combined Capacity Random order 0 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 0.5 Slot Time s) b) Throughput within the PU slot time T with sensing errors SE) P i) = P m i) = 0.1 i M. The DSA throughput is plotted or the proposed algorithm without sensing errors, the proposed algorithm not considering sensing errors assuming, conversely, P i) = 0 and P m i) = 0 in F i in Algorithm 1, the proposed algorithm considering sensing errors set P i) = 0.1 and P m i) = 0.1 in F i in Algorithm 1, the combined algorithm [5], [6], the capacity algorithm [7], and the random order algorithm. Fig.. DSA throughput perormance evaluation considering both with and without sensing errors. The chosen parameters are M = 10, P 00 = 0.1, 0., 0.5, 0.3, 0.6, 0.3, 0., 0.7, 0.8, 0.9), P 10 = 0.7, 0., 0.3, 0.5, 0.4, 0.8, 0.9, 0.1, 0.6, 0.), T = 0, 10, 30, 40, 10, 0, 40, 40, 30, 50) ms, and C =, 8, 5, 1, 6, 3, 9, 7, 10, 1) bits/s.
8 EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X Cumulative Distribution Function F R r) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Proposed 0. Combined 0.1 Capacity Random access 0 0 5 10 15 0 5 30 35 40 r bits/s) Fig. 3. Experimental CDF comparisons o the DSA throughput obtained rom each realization under perect sensing. The CDF is plotted or our proposed algorithm, combined algorithm [5], [6], capacity algorithm [7], and random order algorithm. The used parameters are M = 10, T avg = 5 ms, T = 0.1 s, C avg = 5 bits/s, P i) 00 U0, 1), and P i) 10 U0, 1). deviates away rom the CR bound when u is high. This is because, as u grows larger, the channel has a higher probability o being occupied. Most o the traic samples are sensed busy, and thereore the measured transition numbers n 0 and n 1 are close to zero. The collection o such a small amount o transition numbers is not suicient or the accurate PU traic estimation, especially when the number o samples is not large enough e.g. N = 00 in our simulations). n addition, ˆP 10 becomes more accurate as u increases, since it has more inormation on n and n 3 given a constant number o samples. However, as we explained in Fig. 4a), the perormance o ˆP 00 and ˆP 10 depends on u. This phenomenon is clearly shown here, i.e., ˆP10 is better than ˆP 00 when u > 0.5, and ˆP 00 is better than ˆP 10 when u < 0.5. C. Sensing Order Conidence Metric and DSA Throughput Loss Under Perect Sensing Fig. 5 plots the average sensing order conidence, i.e., 19) versus the number o samples used or traic estimation. n this simulation, we apply the same number o samples used or PU traic estimation in each channel, which is more convenient or demonstrating the results. The simulation results are shown under two dierent traic parameter sets named Test 1 and Test. We deine the probability o PU presence vector as u = u 1), u ),, u M) }. From the simulation results, we can irst observe that the average sensing order conidence increases when the number o samples increases. This is because, once we estimate the traic parameters more accurately, the resulting sensing order will be with high probability closer to the best one. Secondly, we can observe that the simulation results match the analytical results. The results demonstrate that our approximation or the ML estimation errors being Gaussian distributed and application o the CR bound or the variance are accurate enough or deriving the average sensing order conidence. Fig. 6 plots the normalized throughput loss L with respect to the average sensing order conidence κ. The igure shows RMS Error 0.09 0.08 0.07 0.06 0.05 0.04 0.03 P 00 Sim.) CR Bound or P 00 P 10 Sim.) CR Bound or P 10 0.0 100 150 00 50 300 350 400 450 500 Number o Samples a) RMS error o estimated transition probabilities with the number o samples. The parameters used are P 00 = 0.7, P 10 = 0.7. RMS Error 0.1 0.1 0.08 0.06 0.04 0.0 P 00 Sim.) CR Bound or P 00 P 10 Sim.) CR Bound or P 10 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability o PU Presence b) RMS error o estimated transition probabilities with the probability o PU presence u. The parameters used are N = 00, P 01 = 0.1. Fig. 4. Traic estimation perormance evaluation: CR bound and simulation results Sim.). that, as the average sensing order conidence increases, the throughput loss decreases. This is because increasing the average sensing order conidence translates to a higher probability o obtaining the best sensing order i.e., we have a larger chance to achieve the maximal throughput). Note the analytical result or L matches the simulation result when the sensing order conidence metric κ i s is known. As κ i s is derived by 18), L deviates rom the analytical results to the simulation result when the average sensing order conidence is small. This can be explained by the Gaussian assumption we make or the ML estimators errors. When the number o samples is not large enough, the ML estimators errors will not approach a Gaussian distribution. Thereore κ i s is not accurate in this region. The igure provides a guideline or choosing the sensing order conidence metric in a system perspective, since we can get the inormation on the throughput loss in our system given a corresponding average sensing order conidence. As in the example shown in Fig. 6, i we would like to design a system with a normalized throughput loss o
LU et al.: TRAFFC-AWARE CHANNEL SENSNG ORDER N DYNAMC SPECTRUM ACCESS NETWORKS 9 Average Sensing Order Conidence 1 0.95 0.9 0.85 0.8 Test 1 Sim.) 0.75 Test 1 An.) Test Sim.) Test An.) 0.7 50 150 50 350 450 Number o Samples Fig. 5. Average sensing order conidence with the number o samples. Analytical An.) results and simulation Sim.) results are presented. The parameters used are M = 6, u = 0., 0.1, 0.6, 0.5, 0.4, 0.3), T = 0, 10, 30, 40, 10, 0) ms, C =, 8, 5, 1, 6, 3) bits/s. Test 1 adopts P 00 = 0.87, 0.90, 0.76, 0.7, 0.73, 0.83). Test adopts P 00 = 0.80, 0.90, 0.40, 0.50, 0.60, 0.70). Normalized Throughput Loss 0.4 0.35 0.3 0.5 0. 0.15 0.1 0.05 Sim. Approx. κ s i An.) i Perect κ s An.) 0 0 0.1 0. 0.3 0.4 0.5 0.6 Average Sensing Order Conidence Fig. 6. Normalized throughput loss with the average sensing order conidence. Simulation Sim.) results are compared to the analytical An.) results. Both approximated Approx.) and perect sensing order conidence are plugged into the analytical expression or normalized throughput loss. The parameters used are T = 0.15 s, M = 6, u = 0., 0.1, 0.6, 0.5, 0.4, 0.3), P 00 = 0.95, 0.94, 0.9, 0.89, 0.88, 0.93), T = 0, 10, 30, 40, 10, 0) ms, C =, 8, 5, 1, 6, 3) bits/s. less than 1% note we have done the simulations up until this point), we need to design a traic estimation algorithm which keeps the average sensing order conidence higher than 57%. D. Design Examples n this section, we show two design examples or solving the proposed convex optimization problem under perect sensing. For a given state vector s, we need to determine the minimum number o samples or PU traic estimation to achieve certain conidence. Note that conidence maps to equivalent throughput loss. We consider two designs: Design 1: min N subject to κ [o] s η; 1) Design See Section -C): min M N i subject to κ [o] s η. ) The irst design optimizes a single variable N used or all channels, while the second design optimizes M variables used or each channel. Obviously Design 1 solves a sub-optimal solution while Design solves the optimal solution since each channel should be optimized according to its own traic parameters, i.e., Design 1 is a special case o Design. t has been proved in Section -C that both designs are solving a convex optimization problem, so we can solve it numerically by using optimization tools, e.g., those available in MATLAB. Denote the optimal solutions N o and Ni o i M or Design 1 and Design, respectively. Since the number o samples can only be taken in integers, the optimal solution should be rounded to the nearest integer which is greater than itsel, i.e., N o and Ni o. Note that the rounding process preserves the convex constraint, since the sensing order conidence unction is an increasing unction with respect to number o samples. We compare these two solutions in terms o total number o samples needed or estimation, i.e., M N o and M, or Ni o Design 1 and Design respectively. Fig. 7 presents the total number o samples with optimization threshold η. First, as the sensing order conidence constraint threshold increases, the total number o samples increases. Since we have a tighter constraint, we need more PU traic samples to raise the estimation accuracy. Second, we see that the optimal solution requires much less traic samples than the sub-optimal solution. The optimal solution saves at most 48.36% samples rom the sub-optimal solution or η = 0.9. But the optimal solution takes longer time to be solved, so there is a tradeo between the duration o the traic estimation process and the computational complexity. V. CONCLUSONS n this work, we propose a sensing order algorithm to maximize the throughput o secondary users transmitting on multiple primary user channels. The algorithm is based on deriving the sensing actor which is dependent on the traic statistics o the primary users and sensing parameters o the secondary user, assuming a discrete-time Markov chain channel utilization model and a slotted transmission structure or primary users. Furthermore, we propose the use o a maximum likelihood estimator to estimate the traic statistics and quantiy the estimation perormance using the Cramér-Rao bound. Finally, we analyze the perormance o the proposed sensing order algorithm together with the traic estimation. We derive the sensing order conidence metric o obtaining the best sensing order analytically by approximating the estimator errors with the Gaussian distribution and applying the Cramér- Rao bound. The system throughput loss is derived according to the sensing order conidence, which links to the traic
10 EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X Total Number o Samples 1500 150 1000 750 500 50 Design 1 Design 0 0.5 0.6 0.7 0.8 0.9 Optimization Threshold Fig. 7. Total number o samples with optimization threshold. Comparison o optimizing single number o sample used or every channel with optimizing all number o samples used or each channel. The used parameters are M = 6, u = 0., 0.1, 0.6, 0.5, 0.4, 0.3), P 00 = 0.87, 0.90, 0.76, 0.7, 0.73, 0.83), T = 0, 10, 30, 40, 10, 0) ms, C =, 8, 5, 1, 6, 3) bits/s, current state vector s = 1, 1, 1, 1, 1, 1). Algorithm Sequence nterchanging Algorithm 1: procedure NTERCHANGEA, B) : i 1 3: while i M do 4: Search k or a k == b i 5: j k 6: while j i 1 do 7: X j a j nterchange a j and a j 1 in A 8: X j 1 a j 1 9: a j X j 1 10: a j 1 X j 11: j j 1 1: end while 13: i i + 1 14: end while 15: return A 16: end procedure estimation duration. We also orm a convex optimization problem in order to select the number o samples needed to sense each channel or a certain level o sensing order conidence. APPENDX A SEQUENCE NTERCHANGNG ALGORTHM Assume we have two sequences A = a 1, a,, a M ) and B = b 1, b,, b M ), where A and B are two dierent permutations o the set M. Denote a j as the j-th element in A and b j as the j-th element in B. n order to interchange A to B, we can ollow Algorithm. Note in each step o interchanging a l, a m ) we always ollow the rule o l < m. APPENDX B CRAMÉR-RAO BOUND DERVATON We derive the CR bound as given by 1) and 13) by deriving the Fisher inormation matrix. Let θ = [P 00, P 10 ] T. The bounds o the estimators are deined as the diagonal terms o the reciprocal o the Fisher inormation matrix, which is written as } N P 00, P 10 ) = E z θ log Lz N). 3) The subscript N implies the Fisher inormation is calculated rom N number o traic samples, and E z } represents the expectation calculated over all permutations o z N. The bounds are expressed as V 0 = NP 00, P 10 )[, ], N P 00, P 10 ) 4) V 1 = NP 00, P 10 )[1, 1], N P 00, P 10 ) 5) where represents determinant o a matrix. n order to derive the elements in the Fisher inormation matrix, we need to introduce Lemma 3 beorehand. Lemma 3: The expectation o transition numbers with n 1 PU traic samples given a discrete-time Markov chain z n are expressed as E z n 0 z n )} = P 0 P 00 n 1), 6) E z n 1 z n )} = P 0 P 01 n 1), 7) E z n z n )} = P 1 P 10 n 1), 8) E z n 3 z n )} = P 1 P 11 n 1). 9) Proo: Consider the transition number n 0 z n ). Note that n 0 z n ) is a unction o the state variable and its expectation can be expressed recursively as E z n 0 z n )} = E z n 0 z n 1 )} + P 0 P 00, n 3, 30) with the initial condition E z n 0 z )} = P 0 P 00. By solving 30), we derive E z n 0 z n )} = P 0 P 00 n 1). Similarly, we apply the same procedure or E z n 1 z n )}, E z n z n )}, E z n 3 z n )} using its own recursive equation, and we veriy the results in 7) 9), respectively. Note that Lemma 3 also shows that the ML estimators or the transition probabilities are unbiased, i.e., E ˆP 00 } = P 00 and E ˆP 10 } = P 10. Thereore, we can apply CR bound as our lower bound or the variance o the estimators. The elements in the Fisher inormation matrix are written as } N P 00, P 10 )[1, 1] = E z log Lz N ) P 00 = E z n0 z N ) P 00 N P 00, P 10 )[, ] = E z + n } 1z N ) 1 P 00 ) = P 0N 1) P 00 1 P 00 ), 31) P 10 = E z n z N ) P 10 } log Lz N ) + n } 3z N ) 1 P 10 ) = P 1N 1) P 10 1 P 10 ), 3)
LU et al.: TRAFFC-AWARE CHANNEL SENSNG ORDER N DYNAMC SPECTRUM ACCESS NETWORKS 11 } N P 00, P 10 )[1, ] = E z log Lz N ) = 0, P 00 P 10 33) } N P 00, P 10 )[, 1] = E z log Lz N ) = 0, P 10 P 00 34) by applying 6) 9) into 31) 3) with N traic samples, and the determinant is N P 00, P 10 ) = N P 00, P 10 )[1, 1] N P 00, P 10 )[, ] = P 0 P 1 N 1) P 00 1 P 00 )P 10 1 P 10 ). 35) Finally, applying 31), 3) and 35) into 4) 5), we obtain the CR bound as We want to prove that κ [o] s V 0 = P 001 P 00 ) P 0 N 1), 36) V 1 = P 101 P 10 ) P 1 N 1). 37) APPENDX C PROOF OF CONCAVTY OF κ [o] s ) = M 1 ɛ ip a i+1 ) b Q i+1 0 P a i ) b i 0 is σ ai +ɛ i σ ai+1 ) a concave unction. Deine g i 0, h i ɛ i P a i+1 ) b i+1 0 1 P a i+1 ) ) b i+1 0 P a i+1 ) b P a i ) i+1 b i 0 ɛip a i+1 ) b i+1 0 P a i ) b i 0 1 P a i ) b i 0 P a i ) b P a i ) i b i 0 ɛip a i+1 ) b i+1 0 ) 0, and P ai) b i ) the absence/presence probability o the a i -th channel such ) that Q ɛ ip a i+1 ) b i+1 0 P a i ) b i 0 σ ai +ɛ i σ ai+1 = Q 1 g i N i 1 + h i N i+1 1 N i, N i+1 ). To prove N i, N i+1 ) is a concave unction we need to introduce the ollowing Theorem. Theorem ) : Any real unction in the orm x, y) = Q 1 g x + h y is a concave unction, given g, h, x, y 0. Proo: The suicient and necessary condition or x, y) to be concave is x, y) 0, i.e., a negative semideinite matrix. Thereore, we need to calculate all the elements in x, y). Note that Qx) = x x 1 π e m as dm = 1 π e m dm. 1 π e m dm, thus x, y) = g x + h y ) 1 By the irst undamental theorem o calculus, the irst partial derivative o x, y) is expressed as x = g g π x x + h y y = h π y ) 3 e 1 g x + h y ) 1, 38) g x + h ) 3 e 1 g x + h y ) 1, 39) y and the second partial derivative o x, y) is expressed as x = g g π x 4 x + h ) 7 e 1 g x + h y ) 1 y g x x + h ) + 3 g y x + h ) ) g 1 y g, 40) y = h g π y 4 x + h y g y x + h y ) + 3 x y = gh g π x y x + h y 3 g x + h ) 1 y ) 7 e 1 g x + h y ) 1 g x + h ) ) h 1 y h, 41) ) 7 e 1 g x + h y ) 1 ). 4) Thus, we obtain x, y) = 1 g π x + h y ) 7 W, where W is a matrix with W ij representing its i-th row and j-th column element, i, j 1, }. The elements W ij are calculated to be g W 11 =gx x 4 x + h ) + 3 g y x + h ) ) g 1 y g, 43) g W =hy y 4 x + h ) + 3 g y x + h ) ) h 1 y h, 44) W 1 = W 1 = ghx y 3 g x + h ) 1 ). 45) y To prove x, y) 0 is equivalent to prove W 0. We introduce the ollowing Lemma which gives a suicient and necessary condition o a matrix being positive semideinite. Lemma 4: Given a matrix W, W is positive semideinite i and only i W 11 0 and W 0. Proo: See [1, Ch. 10]. Note that g W 11 = gx 4 x + h ) xh y y + g ) + g ) 0, 46) W = W 11 W W 1 W 1 g = ghx 4 y 4 x + h ) g xh + yg) y x + h ) y g + xh + yg) x + h ) ) + 1 0, 47) y rom Lemma 4 we have W 0, which implies W 0. Thereore x, y) 0 completes the proo. Since N i, N i+1 ) is a concave unction by Theorem, log N i, N i+1 ) is also a concave unction by the composition rule [19]. Hence, the sum o concave unctions M 1 log N i, N i+1 ) = log M 1 N i, N i+1 ) is a concave unction. Taking the exponential o log M 1 N i, N i+1 ) we can derive M 1 N i, N i+1 ), showing it is a concave unction again by the composition rule, which completes our proo.
1 EEE JOURNAL ON SELECTED AREAS N COMMUNCATONS, VOL. X, NO. X, X 01X APPENDX D DERVATON OF TRANSTON PROBABLTES CONSDERNG SPECTRUM SENSNG ERRORS First we note that the probability vectors or uture state ŝ and current state s can be written as [ ] [ ] [ ] Prŝ = 0} P00 P = 10 Prs = 0}. 48) Prŝ = 1} P 01 P 11 Prs = 1} Furthermore, the probability vector under sensing errors or estimated current state s can be shown as [ ] [ ] [ ] Pr s = 0} 1 P P = m Prs = 0}. 49) Pr s = 1} P 1 P m Prs = 1} Now, consider a reverse Markov chain describing transition rom s to s. The probability vectors or this reverse Markov chain are [ ] [ ] [ ] Prs = 0} 1 P P = m Pr s = 0}. 50) Prs = 1} P 1 P m Pr s = 1} Embedding 50) into 48) we can derive the probabilistic relation or ŝ and s as [ ] [ ][ ] [ ] Prŝ = 0} P00 P = 10 1 P P m Pr s = 0} Prŝ = 1} P 01 P 11 P 1 P m Pr s = 1} [ ] Pr s = 0} Q, 51) Pr s = 1} where Q [Q sŝ ] is the observed transition probability matrix given as [ ] 1 P )P Q= 00 + P P 10 P m P 00 + 1 P m )P 10. 5) 1 P )P 01 + P P 11 P m P 01 + 1 P m )P 11 Thereore, Prŝ = 0} = Q 10 = P m P 00 + 1 P m )P 10 i s = 1, and Prŝ = 0} = Q 00 = 1 P )P 00 + P P 10 i s = 0. REFERENCES [1] S. Haykin, Cognitive radio: Brain-empowered wireless communications, EEE J. Select. Areas Commun., vol. 3, no., pp. 01 0, Feb. 005. [] M.-V. Nguyen and H. S. Lee, Eective scheduling in inrastructurebased cognitive radio networks, EEE Trans. Mobile Comput., vol. 10, no. 6, pp. 853 867, Jun. 011. [3] W. Gabran, C.-H. Liu, P. Pawełczak, and D. 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Cambridge University Press, 004. [0] M. Grant and S. Boyd, CVX: Matlab sotware or disciplined convex programming, version 1.1. CVX Research, nc., 011. [1] A. J. Laub, Matrix Analysis or Scientists & Engineers. Society or ndustrial and Applied Mathematics, 005. Chun-Hao Liu received the BSc and MSc degrees in electronics engineering rom National Chiao Tung University and National Taiwan University, in 007 and 009, respectively. He is currently working toward the PhD degree in the Department o Electrical Engineering, University o Caliornia, Los Angeles. His research interests are signal processing and algorithms design or wireless communications. He is working on development o algorithms and analysis or cognitive radio networks. Jason A. Tran S 08) received his BSc in Electrical Engineering at the University o Caliornia, rvine, in 010 and his MSc in Electrical Engineering rom the University o Caliornia, Los Angeles, in 013. He is currently pursuing a PhD degree in Electrical Engineering at the University o Southern Caliornia. He has worked at the Aerospace Corporation and held internships at Boeing and Qualcomm. His current research interests lie in the design, analysis, and optimization o algorithms and protocols or wireless networks.
LU et al.: TRAFFC-AWARE CHANNEL SENSNG ORDER N DYNAMC SPECTRUM ACCESS NETWORKS 13 Przemysław Pawełczak S 03-M 10) received the MSc degree rom Wrocław University o Technology, Wrocław, Poland, in 004 and the PhD degree rom Delt University o Technology, Delt, The Netherlands in 009. Between 009 and 011 he was a postdoctoral researcher at the Cognitive Reconigurable Embedded Systems Lab, University o Caliornia, Los Angeles. Between 01 and 013 he was a research sta member at Fraunhoer nstitute or Telecommunications, Heinrich Hertz nstitute, Berlin, Germany. Since 013 he is a tenure track) assistant proessor at the Embedded Sotware Group o Delt University o Technology. His research interests include crosslayer analysis o opportunistic spectrum access networks. Dr. Pawelczak was a vice-chair o EEE SCC41 Standardization Committee between 009 and 011. Since 010 he has been a co-chair o the demonstration track o EEE DySPAN. He was the recipient o the annual Telecom Prize or Best PhD Student in Telecommunications in The Netherlands in 008 awarded by the Dutch Royal nstitute o Engineers and a recipient o NWO Veni grant in 01. Danijela Cabric S 96 M 07) received the Dipl. ng. degree rom the University o Belgrade, Serbia, in 1998, and the M.Sc. degree in electrical engineering rom the University o Caliornia, Los Angeles, in 001. She received her Ph.D. degree in electrical engineering rom the University o Caliornia, Berkeley, in 007, where she was a member o the Berkeley Wireless Research Center. n 008, she joined the aculty o the Electrical Engineering Department at the University o Caliornia, Los Angeles as an Assistant Proessor. Dr. Cabric received the Samueli Fellowship in 008, the Okawa Foundation Research Grant in 009, Hellman Fellowship in 01 and the National Science Foundation Faculty Early Career Development CAREER) Award in 01. She serves as an Associate Editor in EEE Journal on Selected Areas in Communications Cognitive Radio series) and EEE Communications Letters, and TPC Co- Chair o 8th nternational Conerence on Cognitive Radio Oriented Wireless Networks CROWNCOM) 013. Her research interests include cognitive radio systems and spectrum sensing, VLS architectures o signal processing and digital communication algorithms, and their perormance analysis and experiments on embedded system platorms.