Optimal Random Access and Random Spectrum Sensing for an Energy Harvesting Cognitive Radio with and without Primary Feedback Leveraging

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1 1 Otimal Random Access and Random Sectrum Sensing for an Energy Harvesting Cognitive Radio with and without Primary Feedback Leveraging Ahmed El Shafie, Member, IEEE, arxiv: v3 [cs.it] 27 Ar 2014 Abstract We consider a secondary user SU with energy harvesting caability. We design access schemes for the SU which incororate random sectrum sensing and random access, and which make use of the rimary automatic reeat request ARQ feedback. We study two roblem-formulations. In the first roblem-formulation, we characterize the stability region of the roosed schemes. The sensing and access robabilities are obtained such that the secondary throughut is maximized under the constraints that both the rimary and secondary queues are stable. Whereas in the second roblem-formulation, the sensing and access robabilities are obtained such that the secondary throughut is maximized under the stability of the rimary queue and that the rimary queueing delay is ket lower than a secified value needed to guarantee a certain quality of service QoS for the rimary user PU. We consider sectrum sensing errors and assume multiacket recetion MPR caabilities. Numerical results show the enhanced erformance of our roosed systems. Index Terms Cognitive radio, energy harvesting, queueing delay. I. INTRODUCTION Cognitive radio technology rovides an efficient means of utilizing the radio sectrum [2]. The basic idea is to allow secondary users to access the sectrum while roviding certain guaranteed quality of service QoS erformance measures for the rimary users PUs. The secondary user SU is a battery-owered device in many ractical situations and its oeration, which involves sectrum sensing and access, is accomanied by energy consumtion. Consequently an energy-constrained SU must otimize its sensing and access decisions to efficiently utilize the energy at its disosal. An emerging technology for energy-constrained terminals is energy harvesting which allows the terminal to gather energy from its environment. An overview of the different energy harvesting technologies is rovided in [3] and the references therein. Data transmission by an energy harvester with a rechargeable battery has got a lot of attention recently [4] [13]. The otimal online olicy for controlling admissions into the data buffer is derived in [4] using a dynamic rogramming framework. In [5], energy management olicies which stabilize the data queue are roosed for single-user communication Part of this work has been resented in the 8th International Conference on Wireless and Mobile Comuting, Networking and Communications WiMob, 2012 [1] A. El Shafie is with Wireless Intelligent Networks Center WINC, Nile University, Giza, Egyt. ahmed.salahelshafie@gmail.com and some delay-otimaroerties are derived. Throughut otimal energy allocation is investigated in [6] for energy harvesting systems in a time-constrained slotted setting. In [7], [8], minimization of the transmission comletion time is considered in an energy harvesting system and the otimal solution is obtained using a geometric framework. In [9], energy harvesting transmitters with batteries of finite energy storage caacity are considered and the roblem of throughut maximization by a deadline is solved for a static channel. The authors of [10] consider the scenario in which a set of nodes shares a common channel. The PU has a rechargeable battery and the SU is lugged to a reliable ower suly. They obtain the maximum stable throughut region which describes the maximum arrival rates that maintain the stability of the network queues. In [11], the authors investigate the effects of network layer cooeration in a wireless three-node network with energy harvesting nodes and bursty traffic. In [12], Sultan investigated the otimal cognitive sensing and access olicies for an SU with an energy queue. The analysis is based on Markov-decision rocess MDP. In [13], the authors investigated the maximum stable throughut of a backlogged secondary terminal with energy harvesting caability. The SU randomly accesses the channel at the beginning of the time slot without emloying any channel sensing. The secondary terminal can leverage the availability of rimary feedback and exloit the multiacket recetion MPR caability of the receivers to enhance its throughut. In this aer, we develo sectrum sensing and transmission methods for an energy harvesting SU. We leverage the rimary automatic reeat request ARQ feedback for secondary access. Due to the broadcast nature of the wireless channel, this feedback can be overheard and utilized by the secondary node assuming that it is unencryted. The roosed rotocols can alleviate the negative imact of channel sensing because the secondary access is on basis of the sensed rimary state as well as the overheard rimary feedback. The roblem with deending on sectrum sensing only is that sensing does not inform the secondary terminal about its imact on the rimary receiver. This issue has induced interest in utilizing the feedback from the rimary receiver to the rimary transmitter to otimize the secondary transmission strategies. For instance, in [14], the SU observes the ARQ feedback from the rimary receiver as it reflects the PU s achieved acket rate. The SU s objective is to maximize its throughut while guaranteeing a certain acket rate for the PU. In [15], the authors use a artially observable Markov decision rocess POMDP

2 2 to otimize the secondary action on the basis of the sectrum sensing outcome and rimary ARQ feedback. Secondary ower control based on rimary feedback is investigated in [16]. In [17] and [18], the otimal transmission olicy for the SU when the PU adots a retransmission based error control scheme is investigated. The olicy of the SU determines how often it transmits according to the retransmission state of the acket being served by the PU. The contributions of this aer can be summarized as follows. We investigate the case of an SU equied with an energy harvesting mechanism and a rechargeable battery. We roose a novel access and sensing schemes where the SU ossibly senses the channel for a certain fraction of the time slot duration and accesses the channel with some access robability that deends on the sensing outcome. The SU may access the channerobabilistically without sensing in order to utilize the whole slot duration for transmission. Furthermore, it leverages the rimary feedback signals. Instead of the collision channel model, we assume a generalized channel model in which the receiving nodes have MPR caability. We roose two roblem-formulations. In the first roblem-formulation, we characterize the stability region of the roosed schemes. The sensing and access robabilities are obtained such that the secondary throughut is maximized under the constraints that both the rimary and secondary queues are stable. Whereas in the second roblem-formulation, we include a constraint on the rimary queueing delay to the otimization roblem for delay-aware PUs. The sensing and access robabilities are obtained such that the secondary throughut is maximized under the stability of the rimary queue and that the rimary queueing delay is ket lower than a secified value needed to guarantee a certain QoS for the PU. We comare our systems with the conventional access system in which the SU senses the channel and accesses unconditionally if the PU is sensed to be inactive. The numerical results show the gains of our roosed systems in terms of the secondary throughut. The rest of the aer is organized as follows. In the next section, we discuss the system model adoted in this aer. The secondary access without incororating the rimary feedback is investigated in Section III. In Section IV, we discuss the feedback-based scheme. The case of delay-aware PUs is investigated in Section V. We rovide numerical results and conclusions in Section VI. II. SYSTEM MODEL We consider the system model shown in Fig. 1. The model consists of one PU and one SU. The channel is slotted in time and a slot duration equals the acket transmission time. The PU and the SU have infinite buffer queues, Q and Q s, resectively, to store fixed-length data ackets. If a terminal transmits during a time slot, it sends exactly one acket to its receiver. The arrivals at Q and Q s are indeendent and l s Q Q s l e Q e Fig. 1. Primary and secondary queues and links. The PU has data queue Q, whereas the secondary terminal has data queue Q s and energy queue Q e. There is a feedback channel between the rimary receiver PR and the PU to acknowledge the recetion of data ackets. This feedback channel is overheard by the secondary transmitter. Both the PR and the secondary receiver SR may suffer interference from the other link. identically distributed i.i.d. Bernoulli random variables from slot to slot with means λ and λ s, resectively. The SU has an additional energy queue, Q e, to store harvested energy from the environment. The arrival at the energy queue is also Bernoulli with mean λ e and is indeendent from arrivals at the other queues. The Bernoulli model is simle, but it catures the random availability of ambient energy sources. More imortantly, in the analysis of discrete-time queues, Bernoulli arrivals see time averages BASTA. This is the BASTA roerty equivalent to the Poisson arrivals see time averages PASTA roerty in continuous-time systems [19]. It is assumed that the transmission of one data acket consumes one acket of energy. We adot a late arrival model as in [10], [11], [13], [20] where an arrived acket at a certain time slot cannot be served at the arriving slot even if the queue is emty. Denote by V t the number of arrivals to queue Q at time slot t, and Z t the number of deartures from queue Q at time slot t. The queue length evolves according to the following form: Q t+1 = Q t Z t + +V t 1 where z + denotes maxz,0. Adequate system oeration requires that all the queues are stable. We emloy the standard definition of stability for queue as in [20], [21], that is, a queue is stable if and only if its robability of being emty does not vanish as time rogresses. Precisely, lim t Pr{Q t = 0} > 0. If the arrival and service rocesses are strictly stationary, then we can aly Loynes theorem to check for stability conditions [22]. This theorem states that if the arrivarocess and the service rocess of a queue are strictly stationary rocesses, and the average service rate is greater than the average arrival rate of the queue, then the queue is stable. If the average service rate is lower than the average arrival rate, the queue is unstable [20]. PR SR

3 3 Instead of the collision channel model where simultaneous transmission by different terminals leads to sure acket loss, we assume that the receivers have MPR caability as in [23] [25]. This means that transmitted data ackets can survive the interference caused by concurrent transmissions if the received signal to interference and noise ratio SINR exceeds the threshold required for successful decoding at the receiver. With MPR caability, the SU may use the channel simultaneously with the PU. τ T s P MD P FA t Sensing duration Slot duration Probability of sensing the channel Misdetection robability False alarm robability Probability of direct channel access if the channel is not sensed III. SECONDARY ACCESS WITHOUT EMPLOYING PRIMARY FEEDBACK The first roosed system is denoted by Φ NF. Under this rotocol, the PU accesses the channel whenever it has a acket to send. The secondary transmitter, given that it has energy, senses the channel or ossibly transmits the acket at the head of its queue immediately at the beginning of the time slot without sensing the channel. We exlain below why direct transmission can be beneficial for system erformance. The SU oeration can be summarized as follows. If the secondary terminal s energy and data queues are not emty, it senses the channel with robability s from the beginning of the time slot for a duration of τ seconds to detect the ossible activity of the PU. If the slot duration is T, τ < T. If the channel is sensed to be free, the secondary transmitter accesses the channel with robability f. If the PU is detected to be active, it accesses the channel with robability b. If at the beginning of the time slot the SU decides not to sense the sectrum which haens with robability 1 s, it immediately decides whether to transmit with robability t or to remain idle for the rest of the time slot with robability t = 1 t. 1 This means that the transmission duration is T seconds if the SU accesses the channel without sectrum sensing and T τ seconds if transmission is receded by a sensing hase. We assume that the energy consumed in sectrum sensing is negligible, whereas data transmission dissiates exactly one unit of energy from Q e. We study now secondary access in detail to obtain the mean service rates of queues Q s, Q e and Q. The meaning of the various relevant symbols are rovided in Table I. For the secondary terminal to be served, its energy queue must be nonemty. If the SU does not sense the channel, which haens with robability 1 s, it transmits with robability t. If the PU s queue is emty and, hence, the PU is inactive, secondary transmission is successful with robability P, whose exression as a function of the secondary link arameters, transmission time T, and the data acket size is rovided in Aendix A. If Q 0, secondary transmission is successful with robability see Aendix A. If the SU decides to sense the channel, there are four ossibilities deending on the sensing outcome and the state of the rimary queue. If the PU is sensed to be free, secondary transmission takes lace with robability f. This takes lace with 1 Throughout the aer X =1 X. f b P P P 1s 1s Probability of channel access if the channel is sensed to be free Probability of channel access if the channel is sensed to be busy Probability of successfurimary transmission to the rimary receiver if the secondary terminal is silent Probability of successfurimary transmission to the rimary receiver with concurrent secondary transmission Probability of successful secondary transmission if the PU is silent and transmission occurs over T seconds Probability of successful secondary transmission if the PU is silent and transmission occurs over T τ seconds Probability of successful secondary transmission if the PU is active and transmission occurs over T seconds Probability of successful secondary transmission if the PU is active and transmission occurs over T τ seconds TABLE I LIST OF SYMBOLS INVOLVED IN THE QUEUES MEAN SERVICE RATES. robability 1 P FA if the PU is actually silent. In this case, the robability of successful secondary transmission is P 1s, which is lower than P for roof, see [1]. On the other hand, if the PU is on, the robability of detecting the channel to be free is P MD and the robability of successful secondary transmission is 1s. If the channel is sensed to be busy, the secondary terminal transmits with robability b. Sensing the PU to be active occurs with robability P FA if the PU is actually inactive, or with robability 1 P MD if the PU is actively transmitting. The robability of successful secondary transmission is P 1s when the PU is silent and 1s when the PU is active. Given these ossibilities, we can write the

4 4 following exression for the mean secondary service rate. µ s = 1 s t Pr{Q = 0,Q e 0}P +1 s t Pr{Q 0,Q e 0} + s f Pr{Q = 0,Q e 0}1 P FA P 1s + s f Pr{Q 0,Q e 0}P MD 1s + s b Pr{Q = 0,Q e 0}P FA P 1s + s b Pr{Q 0,Q e 0}1 P MD 1s Based on the above analysis, it can be shown that the mean service rate of the energy queue is µ e = 1 s t Pr{Q s 0} + s f P MD Pr{Q s 0,Q 0} +1 P FA Pr{Q s 0,Q = 0} + s b P FA Pr{Q s 0,Q = 0} +1 P MD Pr{Q s 0,Q 0} A acket from the rimary queue can be served in either one of the following events. If the SU is silent because either of its data queue or energy queue is emty, the rimary transmission is successful with robabilityp. If both secondary queues are nonemty, secondary oeration roceeds as exlained above. In all cases, if the SU does not access the channel, the robability of successfurimary transmission is P, else it is. 2 Therefore, µ = 1 Pr{Q s 0,Q e 0} P +Pr{Q s 0,Q e 0} [ 1 s t P c +1 t P + s P MD f +1 f P ] + s 1 P MD b P c +1 b P The maximum rimary throughut is P, i.e., µ P, which occurs when the PU oerates alone, i.e., when the SU is always inactive. Following are some imortant remarks on the roosed access and sensing scheme. Firstly, the roosed access and sensing scheme can mitigate the negative imact of sensing errors. Secifically, the SU under the roosed rotocol randomly accesses the channel if the PU is either sensed to be active or inactive. Hence, the false alarm robability and the misdetection robability are controllable using the sectrum access robabilities. These access robabilities can take any 2 We assume that the access delay of the SU does not affect the rimary outage robability. This is valid as far as 1 τ e e, which is true here T as τ T. For details, see Aendix A value between zero and one. Hence, the SU can mitigate the imact of the sensing errors via adjusting the values of the access robabilities. Accordingly, this would enhance the secondary throughut and revent the violation of the PU s QoS. Secondly, when the MPR caabilities of the receivers are strong which means P and is P is for i = 0,1, the SU does not need to sense the channel at all, i.e., s = 0. This is due to the fact that the SU does not need to emloy channel sensing as it can transmit each time slot simultaneously with the PU without violating the rimary QoS because the receivers can decode ackets under interference with a robability almost equal to the decoding robability when nodes transmit alone. As in [1], [10], [13], we assume that the energy queue is modeled as M/D/1 queue with mean service and arrival rates µ e = 1 and λ e, resectively. Hence, the robability that the secondary energy queue being nonemty is λ e /µ e = λ e [26]. Based on this assumtion, the energy queue in the aroximated system emties faster than in the actual system, thereby lowering the robability that the queue is nonemty. This reduces the secondary throughut by increasing the robability that the secondary node does not have energy. Therefore, our aroximation result in a lower bound on the secondary service rate or throughut [1], [13]. We denote the system under the aroximation of one acket consumtion from the energy queue each time slot as S. Since the queues in the aroximated system, S, are interacting with each other, we resort to the concet of the dominant system to obtain the stability region of system S. The dominant system aroach is first introduced in [27]. The basic idea is that we construct an aroriate dominant system, which is a modification of system S with the queues decouled, hence we can comute the dearture rocesses of all queues. The modified system ensures that the queue sizes in the dominant system are, at all times, at least as large as those of system S rovided that the queues in both systems have the same initial sizes. Thus, the stability region of the new system is an inner bound of system S. At the boundary oints of the stability region, both the dominant system and system S coincide. This is the essence of the indistinguishability argument resented in many aers such as [10], [20], [25], [27]. Next, we construct two dominant systems and the stability region of aroximated system S is the union of the stability region of the dominant systems. We would like to emhasize here that system S is an inner bound on the original system, Φ NF. A. First Dominant System In the first dominant system queue, denoted bys 1,Q s transmits dummy ackets when it is emty and the PU behaves as it would in the original system. Under this dominant system, we have Pr{Q s = 0} = 0. Substituting by Pr{Q s = 0} = 0 into 4, the average rimary service rate after some simlifications can be given by µ = P λ e s t + s P MD f + s P MD b 5

5 u lm m u2 m 3 u0 u1 l m m Fig. 2. Markov chain of the PU under dominant system S 1. State selftransitions are omitted for visual clarity. Probabilities µ = 1 µ and λ = 1 λ. where =P P c 0. The Markov chain modeling the rimary queue under this dominant system is rovided in Fig. 2. Solving the state balance equations, it is straightforward to show that the robability that the rimary queue has k ackets is [ ] k 1 λ µ ν k = ν 0 6 µ λ µ where λ = 1 λ and µ = 1 µ. Using the condition k=0 ν k = 1, ν 0 = 1 λ µ 7 For the sum k=0 ν k to exist, we should have λ < µ. This is equivalent to Loynes theorem. SincePr{Q = 0} = 1 λ µ, [ 1 λ µ s =λ e s t P + s b P FA P 1s + s f P FA P 1s µ ] Let + λ µ s t + s f P MD 1s + s b P MD 1s 8 µ = P +D P 9 where denotes vector transosition, P=[ t, b, f ] and ]. D= λ e [ s, s P MD, s P MD Substituting from 9 into 8, λ µ s = 1 A λ P + P +D P P +D P G P 10 [ ] where G = s t, sp MD 1s, sp MD 1s and A = [ ]. s P, s P FA P 1s, s P FA P 1s After some mathematical maniulations, we get µ s = P λ A P+P DA P+λ G P P +D P 11 The ortion of the stability region based on the first dominant system is characterized by the closure of the rate airs λ,λ s. One method to obtain this closure is to solve a constrained otimization roblem such that λ s is maximized for each λ under the stability of the rimary and the secondary queues. The otimization roblem is given by max. s,p=[ t, b, f ] µ s, s.t. 0 t, s, f, b 1, λ µ 12 For a fixed s, the otimization roblem 12 can be shown to be a quasiconcave rogram over P. We need to show that the objective function is quasiconcave over convex set and under convex constraints. From 9, µ is affine and hence convex over P for a fixed s. The Hessian of the numerator of µ s is given by H = AD +DA. Let y be an arbitrary 3 1 vector. The matrix H is negative semidefinite if y Hy 0. Since the matrices AD and DA are generated using a linear combination of a single vector, the rank of each is 1 and therefore each of them has at least two zero eigenvalues. The trace of each is negative and equal to Λ= λ e 2 sp + 2 sp FA P 1s P MD + 2 sp MD P FA P 1s 0. Hence, AD and DA are negative semidefinite with eigenvalues 0,0,Λ. Accordingly, y AD y 0, y DA y 0 and their sum is also negative. Based on these observations for a fixed s, the numerator of 11 is nonnegative 3 and concave over P and the denominator is ositive and affine over P; hence, µ s is quasiconcave, as is derived in Aendix B. Since the objective function of the otimization roblem is quasiconcave and the constraints are convex for a fixed s, the roblem is a quasiconcave rogram for each s. We solve a family of quasiconcave rograms arameterized by s. The otima s is chosen as the one which yields the highest objective function in 12. The roblem of maximizing a quasiconcave function over a convex set under convex constraints can be efficiently and reliably solved by using the bisection method [28]. Based on the construction of the dominant system S 1 of system S, it can be noted that the queues of the dominant system are never less than those of system S, rovided that they are both initialized identically. This is because the SU transmits dummy ackets even if it does not have any ackets of its own, and therefore it always interferes with PU even if it is emty. The mean service rate of rimary queue is thus reduced in the dominant system and Q is emtied less frequently, thereby reducing also the mean service rate of the secondary queue. Given this, if the queues are stable in the dominant system, then they are stable in system S. That is, the stability conditions of the dominant system are sufficient for the stability of system S. Now if Q s saturates in the dominant system, the SU will not transmit dummy ackets as it always has its own ackets to send. For λ < µ, this makes the behavior of the dominant system identical to that of system S and both systems are indistinguishable at the boundary oints. The stability conditions of the dominant system are thus both sufficient and necessary for the stability of system S given that λ < µ. To get some insights for this system under the first dominant system, we consider the roblem when λ /P is close to 3 The non-negativity of the numerator and the denominator of µ s follow from the definition of the service rate.

6 6 unity and with significant MPR caabilities, 4 which means that the rimary queue is nonemty most of the time and therefore the otimal sensing decision is s = 0. Note that, in general, this case rovides a lower bound erformance on what can be obtained in S 1. When s = 0, the maximum secondary stable throughut is given by solving the following otimization roblem: [ max. λ e t 1 λ ] P + λ, s.t. λ µ 13 0 t 1 µ µ The roblem is convex and can be solved using the Lagrangian formulation. The access robability t is uerbounded by F { F = min 1, P } λ 14 λ e The second term in F must be nonnegative for the roblem to be feasible. The otimal access robability is thus given by { c t {F,max = min P P λ 1 P /P }},0 λ e 15 with 0 λ µ. From the otimal solution, we notice the following remarks. As λ increases, the secondary access robability, t, decreases as well. This is because the ossibility of collisions increases with increasing the access robability or increasing the secondary access to the channel and since the PU is busy most of the time, the ossibility of collisions and acket loss increase as well. In addition, by observing the otimal solution in 15, we notice that as the secondary energy arrival, λ e, increases, the access robability decreases. This is because accessing the channel most of the time with the availability of energy may cause high average acket loss for the PU. We note that as the caability of MPR of the rimary receiver, i.e.,p c, increases, the access robability of the secondary queue increases. This occurs because the ossibility of decoding the rimary acket under interference is almost equal to the robability of decoding without interference when the MPR caability of the rimary receiver is high. Therefore, the secondary throughut increases. In addition, as the ability of the secondary receiver of decoding the secondary ackets under interference, which is reresented by /P, increases, the access robability of the secondary terminal increases as far as the rimary queue stability condition is satisfied. where Pr{Q e 0}=λ e. Under this dominant system, the SU otimal sensing decision is s = 0. This is because the PU is always nonemty. Hence, Q s mean service rate in 16 is rewritten as µ s = t λ e 17 The robability of Q s being nonemty is λ s /µ s. Hence, the rimary queue mean service rate is given by µ = 1 λ s λ e P + λ [ ] s λ e t P c +1 t P 18 µ s µ s After some simlifications, the rimary mean service rate is given by µ = P λ s 19 Note that µ is indeendent of t. The ortion of the stability region of S based on S 2 is obtained by solving a constrained otimization roblem in which µ is maximized under the stability of the rimary and the secondary queues. Since the rimary mean service rate is indeendent of t, the stability region of the second dominant system is given by solving the following otimization feasibility roblem max. P λ s 0 t 1 s.t. λ s t λ e, Hence, the otimal access robability is t λ s λ e with λ s λ e. Based on 21, the solution of the roblem is a set of values which satisfies the secondary queue stability constraint. We note that as the secondary mean arrival rate, λ s, increases, the lower limit of t increases as well. This is because the SU must increase its service rate, which increases with the increasing of the access robability, to maintain its queue stability. We also note that one of the feasible oints is λ s = µ s, which means a saturated SU since the arrival rate is equal to the service rate, Pr{Q s 0}=λ s /µ s =1. This system is equivalent to a system with random access without emloying any channel sensing with backlogged saturated rimary and secondary transmitters. B. Second Dominant System In the second dominant system, denoted by S 2, queue Q transmits dummy ackets when it is emty and the SU behaves as it would in systems. By substituting withpr{q = 0} = 0 into 2, the average secondary service rate is given by [ ] µ s = s t + s f P MD 1s + s b P MD 1s Pr{Q e 0} 16 4 As roosed in [13], the rimary arameters λ, P and λ /P can be efficiently estimated by overhearing the rimary feedback channel. Since the stability region of system S is the union of both dominant systems, the stability region of the roosed rotocol always contains that of the random access without emloying any sectrum sensing. Based on this observation, we can say that at high rimary arrival rate, or at high robability of nonemty rimary queue, the random access without emloying any sensing scheme is otimal, i.e., the SU should not emloy channel sensing in such case. This is because the PU is always active and therefore there is no need to sense the channel and waste τ seconds of the data transmission time.

7 7 IV. FEEDBACK-BASED ACCESS In this section, we analyze the use of the rimary feedback messages by the cognitive terminal. This system is denoted by Φ F. In the feedback-based access scheme, the SU utilizes the available rimary feedback information for accessing the channel in addition to sectrum sensing. Leveraging the rimary feedback is valid when it is available and unencryted. In the roosed scheme, the SU monitors the PU feedback channel. It may overhear an acknowledgment ACK if the rimary receiver correctly decodes the rimary transmission, a negative acknowledgment NACK if decoding fails, or nothing if there is no rimary transmission. We introduce the following modification to the rotocol introduced earlier in the aer. If a NACK is overheard by the SU, it assumes that the PU will retransmit the lost acket during the next time slot [29]. Being sure that the PU will be active, the secondary terminal does not need to sense the channel to ascertain the state of rimary activity. Therefore, it just accesses the channel with some robability r. If an ACK is observed on the feedback channel or no rimary feedback is overheard, the SU roceeds to oerate as exlained earlier in Section III. We assume the feedback ackets are very short comared to T and are always received correctly by both the rimary and secondary terminals due to the use of strong channel codes. It is imortant to emhasize here the benefit of emloying rimary feedback. By avoiding sectrum sensing, the secondary terminal does not have to waste τ seconds for channel sensing. It can use the whole slot duration for data transmission. As roven in [1], [30], this reduces the outage robability of the secondary link. Therefore, by differentiating between the rimary states of transmission, i.e., whether they are following the recetion of an ACK or not, the SU can otentially enhance its throughut by eliminating the need for sectrum sensing when the PU is about to retransmit a reviously lost acket. Note that we denote the system oerating exactly as system S with rimary feedback leveraging as S f. A. First dominant system As in the revious section, under the first dominant system, denoted by S f 1, Q s transmits dummy ackets when it is emty and the PU behaves as it would in system S f. The PU s queue evolution Markov chain under the first dominant system of this rotocol is shown in Fig. 3. The robability of the queue having k ackets and transmitting for the first time is π k, where F in Fig. 3 denotes first transmission. The robability of the queue having k ackets and retransmitting is ǫ k, where R in Fig. 3 denotes retransmission. Define α as the robability of successful transmission of the PU s acket in case of first transmission and Γ is the robability of successful transmission of the PU s acket in case of retransmission. It can be shown that both robabilities are given by: α = P λ e s t + s P MD f + s P MD b 22 Γ = P λ e r 23 ϵ ϵ ϵ ϵ G a G G G a G 0 1 R 2 R R R l G a a G G 0 1F 2 F F 3F la a 0 a a Fig. 3. Markov chain of the PU for the feedback-based access scheme under dominant system S f 1. Probabilities Γ = 1 Γ and α = 1 α. State self-transitions are not deicted for visual clarity. η λ α +1 λ Γ π η λ Γ ǫ 0 π 1 ǫ 1 π k,k 2 π λ 1 α λ π λ +1 λ Γ 1 λ η π λ η 1 α ǫ k,k 2 π 1 λ 1 α k=1 π k k=1 ǫ k [ ] k λ 1 η 1 η 2 1 λ η [ ] k λ 1 η 1 η 2 1 λ η π λ Γ η λ = λ π λ η λ 1 α = λ Γ 1 α TABLE II STATE PROBABILITIES FOR THE FEEDBACK-BASED ACCESS SCHEME. Solving the state balance equations, we can obtain the state robabilities which are rovided in Table II. The robability π 0 is obtained using the normalization condition k=0 π k + ǫ k = 1. It should be noticed thatλ < η, whereη is defined in Table II, is a condition for the sum k=0 π k + ǫ k to exist. This condition ensures the existence of a stationary distribution for the Markov chain and guarantees the stability of the rimary queue. The service rate of the SU is given by:

8 [ µ s = λ e π 0 s t P + s b P FA P 1s + s f P FA P 1s + π k s t + s f P MD 1s k=1 ] + s b P MD 1s + ǫ k r k=1 24 Let I = [0,0,0,1], H = λ e I, J = [D,0], Û = λ J +λ H, η = P + ˆP Û, ˆP = [t, f, b, r ], Γ = P + H ˆP, α = P ˆP J, and r = I ˆP. After some algebra, and substituting by the state robabilities in Table II, the secondary data queue mean service rate in 24 can be rewritten as µ s = [P λ K +õp I +λ P C ]ˆP+ ˆP ΨˆP P +H ˆP 25 [, where K= s P, s P FA P 1s, s P FA P 1s,0] Ψ= ÛK + λ CH õji, C = [ s, sp MD 1s, sp MD 1s,0], and õ = λ. It is straightforward to show that the Hessian matrix of the numerator of 25 is 2ˆP ˆP ΨˆP=Ψ+Ψ which is a negative semidefinite matrix and therefore the numerator is concave. 5 The denominator is affine over ˆP. Since for a given s the denominator is affine and the numerator is concave over ˆP, 25 is quasiconcave over ˆP for each s. For a fixed λ, the maximum mean service rate for the SU is given by solving the following otimization roblem using exression 24 for µ s µ s max. s, f, t, b, r s.t. 0 s, f, t, b, r 1 λ η 26 The otimization roblem is a quasiconcave otimization roblem given s which can be solved efficiently using the bisection method [28]. For roof of quasiconcavity of the objective function, the reader is referred to Aendix B. The constraint λ η is affine over ˆP for a fixed s. Since the objective function is quasiconcave given s and the constraint is convex given s, 26 is quasiconcave rogram for a fixed s. Based on the construction of the dominant system S f 1, the queues of the dominant system are never less than those of system S f, rovided that they are both initialized identically. This is because the SU transmits dummy ackets even if it does not have any ackets of its own, and therefore it always interferes with PU even if it is emty. The mean service rate of rimary queue is thus reduced in the dominant system and Q is emtied less frequently, thereby reducing also the mean service rate of the secondary queue. Given this, if the queues are stable in the dominant system, then 5 Ψ is a negative semidefinite because it comoses of three matrices ÛK, λ CH and õji each of which is a negative semidefinite matrix. These matrices are negative semidefinite because each of them is a nonositive matrix all elements are nonositive with rank 1. they are stable in system S f. That is, the stability conditions of the dominant system are sufficient for the stability of system S f. Now if Q s saturates in the dominant system, the SU will not transmit dummy ackets as it always has its own ackets to send. For λ < η, this makes the behavior of the dominant system identical to that of system S f and both systems are indistinguishable at the boundary oints. The stability conditions of the dominant system are thus both sufficient and necessary for the stability of system S f given that λ < η. B. Second dominant system The second dominant system ofs f is denoted bys f 2. Under S f 2, the PU sends dummy ackets when it is emty. This system reduces to a random access scheme without emloying any sectrum sensing and without leveraging the rimary feedback. This is because the PU is always nonemty and the otimal sensing decision is not to sense the channel at all. Moreover, the access robability of the SU is fixed over all rimary states. Hence, s = 0 and r = t. Accordingly, the second dominant system of S f is exactly the second dominant system of S. The stability region of system S f is the union of both dominant systems. Following are some imortant notes. First, the stability region of the first dominant system of S or S f always contains that of the second dominant system. This can be easily shown by comaring the mean service rate of nodes in each dominant system. Second, the stability region of systems S and S f are inner bounds for the original systems Φ NF and Φ F, resectively, where the energy queue is oerating normally without the assumtion of one acket consumtion er time slot. Third, when the SU is lugged to a reliable ower source, the average arrival rate is λ e = 1 ackets er time slot. Under this case, the stability region of systems S and S f coincide with their corresonding original systems Φ NF and Φ F, resectively. This is because the energy queue in this case is always backlogged and never being emty regardless of the value of µ e. Hence, in general, the case of λ e = 1 energy ackets er time slot is an outer bound for the roosed systems, Φ NF and Φ F, as the SU can always send data whenever its data queue is nonemty. Next, we analyze the case of sectrum access without emloying any sensing scheme to give some insights for system S f. Note that the results obtained for this case are tight when λ /P is close to unity and the MPR caabilities are strong. This is because, under this condition, the robability of the rimary queue being emty at a given time slot is almost zero and therefore the otimal sensing decision which avoids wasting τ seconds of the transmission time is s = 0. C. The case of s = 0 Under this case, the mean service rate of the SU is given by ] µ s =λ e [π 0 t P + π k t + ǫ k r k=1 k=1 8 27

9 9 Substituting with robability of summations in Table II, the secondary mean service rate is given by [ ] µ s =λ e π 0 t P +λ t +Γ α r 28 For a fixed λ, the maximum service rate for the SU is given by solving the following otimization roblem: max. t, r µ s s.t. 0 t, r 1, λ η 29 The otimization roblem is quasiconcave quasiconcave objective with a linear constraint and can be solved using bisection method [28]. Fixing r makes the otimization roblem a convex rogram arameterized by r. The otimal r is taken as that which yields the highest value of the objective function. Let l = λ e. We obtain the following otimization roblem for a given r : max. 0 t 1 P λ l r λ +λ s.t. λ η Γ Γ P P t λ l Γ 2 t 30 The objective function of 30 is concave over convex set under linear constraints and therefore a concave rogram. It can be solved using the Lagrangian formulation. Setting the first derivative of the objective function to zero, the root of the first derivative is given by t = P P λ l r λ +λ c P P 2λ l 31 Since λ η = λ α +λ Γ and using 22, the access robability is uerbounded as t P λ l r λ λ l The otimal solution is then given by t =min { P λ l r λ λ l, P λ l r λ +λ 2λ l V. DELAY-AWARE PRIMARY USERS P P 32 } 33 In this section, we investigate the rimary queueing delay and lug a constraint on the rimary queueing delay to the otimization roblems. That is, we maximize µ s under the constraints that the rimary queue is stable and that the rimary acket delay is smaller than or equal a secified value D 1. 6 The value of D 1 is alication-deendent and is related to the required QoS for the PU. Delay analysis for interacting queues is a notoriously hard roblem [20]. To byass this difficulty, we consider the secial case where the secondary data queue is always backlogged or saturated 7 while the rimary queue behaves exactly as it would in the 6 Note that based on the adoted arrival model, the minimum rimary queueing delay is 1 time slot, i.e., D = 1 time slot. 7 This case equivalent to the first dominant systems of S and S f. original systems Φ NF and Φ F. This reresents a lower bound or worst-case scenario on erformance for the PU comared with the original systems in which the secondary data queue is not backlogged all the time. Next, we comute the rimary queueing delay under each system. A. Primary Queueing Delay for System S with Saturated SU Let D be the average delay of the rimary queue. Using Little s law and 6, D = 1 λ k=1 kν k = 1 λ µ λ 34 For the otimal random access and sensing, we solve the following constrained otimization roblem. We maximize the mean secondary service rate under the constraints that the rimary queue is stable and that the rimary acket delay is smaller than or equal a secified value D. The otimization roblem with µ given in 5 and µ s in 8 can be written as max s, f, b, t µ s s.t. 0 s, f, b, t 1 λ µ D D 35 The delay constraint in case of system S with backlogged SU can be converted to a constraint on the rimary mean service rate. That is, D = 1 λ µ λ D can be rewritten as µ λ + 1 λ D. The intersection of the stability constraint and the delay constraint is the delay constraint. That is, the set of λ which satisfies the delay constraint is {λ : λ µ 1 λ D } = {λ µ 1/D 1 1/D µ }, whereas the set of λ which satisfies the stability constraint is {λ : λ µ }. The intersection of both sets is given by {λ µ 1/D 1 1/D } {λ : λ µ } = {λ : λ µ 1/D 1 1/D}, both sets are equal when the delay constraint aroaches, i.e., D. Hence, the delay constraint subsumes the stability constraint. The otimization roblem is quasiconcave given s because µ s is quasiconcave the roof is in Aendix B and the delay constraint is linear on the otimization arameters. At high λ /P and strong MPR caabilities, the robability of the rimary queue being emty at a given time slot is almost zero and therefore the otimal sensing decision is s = 0. In this case, we can get the otimal solution of the otimization roblem. The otimization roblem can be stated as [ max. λ e t 1 λ ] P + λ, s.t. λ µ, D D 0 t 1 µ µ where 36 µ = P λ e t 37 The otimization roblem 36 is convex and can be solved using the Lagrangian formulation. The delay constraint subsumes the stability constraint, λ < µ, and t is uerbounded by

10 10 U { U = min 1, P 1 λ D +λ λ e } 38 The second term in U must be nonnegative for the roblem to be feasible. The otimal access robability is thus given by { c t {U,max = min P P λ 1 P /P }},0 λ e 39 From the otimal solution 39, we can establish here a similar argument about the imact of each arameter on the secondary access robability as the one beneath 33. However, the difference here is that we have the imact of the delay constraint which has the following affect on the secondary access robability. As the delay constraint, D, increases, the access robability of the SU decreases to avoid increasing collisions with the PU which causes rimary throughut loss. If the amount of collisions is high, the delay constraint may be violated if the SU accesses with an access robability higher than t. B. Primary Queueing Delay for System S f with Saturated SU Alying Little s law, the rimary queueing delay is given by D = 1 kπ k +ǫ k 40 λ k=1 Using the state robabilities rovided in Table II, D = α ηη λ 2 +1 λ 2 1 α η η λ 1 λ 1 ηγ 41 For a fixed λ, the maximum mean service rate for the SU is given by solving the following otimization roblem using exression 24 for µ s max s, f, t, b, r µ s s.t. 0 s, f, t, b, r 1 λ η D D 42 Note that µ s is given in Eqn. 25. The otimization roblem can be shown to be a concave rogram for a given s and r. More Secifically, for a fixed r, the denominator in 25 becomes a constant. Since the numerator is concave for a given s as shown beneath 25, the objective function of 42 is concave. The delay constraint can be rewritten as E= η Γ η λ 2 +λ W η+ Dη λ η 1Γ λ 0 η η 43 where W=λ +λ Γ and D 1. The second derivative of E for a given r with resect to η is given by 2 ηe= 2Γ D 1λ 2 +η3 η ηe is always nonnegative. Hence, E is convex over η for a fixed r. Since η is affine over ˆP for a fixed s, E is then convex over ˆP. This comletes the roof of concavity of the otimization roblem 42 for a given s and r. Note that we solve a family of concave roblems arameterized by s and r. The otimaair r, s is taken as the air which yields the highest objective function in 42. VI. NUMERICAL RESULTS AND CONCLUSIONS In this section, we rovide some numerical results for the otimization roblems resented in this aer. A random access without emloying sectrum sensing is simly obtained from system S by setting s to zero. Let S R and S f R denote the random access system without emloying any sectrum sensing without and with feedback leveraging, resectively. We also introduce the conventional scheme of sectrum access, denoted by S c. In this system, the SU senses the channel each time slot for τ seconds. If the PU is sensed to be inactive, the SU accesses with robability1. If the PU is sensed to be active, the SU remains silent. The mean service rates for this case are obtained from Section III with t = 0, s = 1, f = 1 and b = 0. We define here two variables δ = Pc P and δ 1s = Pc 1s, both of them are less than 1 as shown in Aendix P 1s A. Fig. 4 shows the stability region of the roosed rotocols. Systems S R and S f R are also lotted. The arameters used to generate the figure are: λ e = 1 energy ackets/slot, P = 0.7, = 0.1, P = 0.8, = 0.1, P 1s = 0.6, 1s = 0.3, P FA = 0.01, and P MD = We can note that rimary feedback leveraging exands the stability region. It is also noted that randomly accessing the channel without channel sensing and with rimary feedback leveraging can outerform system S for some λ. This is because in system S f R the SU does not sense the channel at the following time slot to rimary acket decoding failure at the rimary destination. Therefore, the SU does not waste τ seconds in channel sensing and it is sure of the activity of the PU. Fig. 5 rovides a comarison between the maximum secondary stable throughut for the roosed systems and the conventional system. The arameters used to generate the figure are: λ e = 0.4 energy ackets/slot, P = 0.7, P c = 0.1, P = 0.8, = 0.1, P 1s = 0.6, 1s = 0.075, P FA = 0.05, and P MD = For the investigated arameters, over λ <0.475 ackets/slot, the roosed rotocols outerform the conventional system. Whereas over λ ackets/slot, all systems rovide the same erformance. This is because at high rimary arrival rate, the robability of the rimary queue being emty is very low and the PU will be active most of the time slots. Hence, the SU senses the channel each time slot and avoids accessing the channel when the PU is sensed to be active and at retransmission states. That is, t = 0, s = 1, r = 0, b = 0 and f = 1. We note that feedback leveraging always enhances the secondary throughut.

11 11 Figs. 6 and 7 show the imact of the MPR caability at the receiving nodes on the stable throughut region. Without MPR caability, collisions are assumed to lead to sure acket loss. Therefore, a collision model without MPR corresonds to the case of the robabilities of correct recetion being zero when there are simultaneous transmissions. As shown in Fig. 6, the secondary service rate is reduced when there is no MPR caability. As the strength of MPR caability increases, the stability regions exand significantly. It can be noted that the erformance of S and S f are equal when the MPR caability is high. This is due to the fact that the SU does not need to emloy channel sensing or feedback leveraging as it can transmit each time slot simultaneously with the PU because the secondary receiver can decode ackets under interference with a robability almost equal to the robability when it transmits alone. The figure is lotted for different MPR strength of the secondary receiver, namely, for P 1 =δ =δ 1s =0, P 2 =δ = δ 1s =1/8, P 3 = δ =δ 1s =1/4 and P 4 =δ =δ 1s =1/2. The arameters used to generate the figure are: λ e = 0.4 energy ackets/slot, P = 0.7, P c = 0.1, P = 0.8, P 1s = 0.6, P FA = 0.05, andp MD = Fig. 7 demonstrates the imact of the MRR caability of the rimary receiver on the stability region of system S. As can be seen, the increases of increases the secondary stable throughut for each λ. The arameters used to generate the figure are: λ e = 0.8 energy ackets/slot, P = 0.7, P = 0.8, = 0.1, P 1s = 0.6, 1s = 0.075, P FA = 0.05, and P MD = 0.01 and for different values of. Fig. 8 shows the imact of the energy arrival rate on the secondary stable throughut for the considered systems. The arameters used to generate the figure are: λ = 0.4 ackets/slot, P = 0.7, = 0.1, P = 0.8, = 0.1, P 1s = 0.6, 1s = 0.075, P FA = 0.05, and P MD = As exected, the secondary service rate increasing with increasing λ e. We note that there are some constant arts in systems S R and S f R at high λ e. This is due to the fact that increasing the energy arrivals at the energy queue may not boost the secondary throughut because the SU even if it has a lot of energy ackets it cannot violate the rimary QoS. The violation of the rimary QoS may occur due to the resence of sensing errors. We also note that at low energy arrival rate, all systems have the same erformance. This is because the secondary access robabilities and the rate in each system are limited by the mean arrival rate of the secondary energy arrival rate. Fig. 9 demonstrates the imact of varying the rimary queueing delay constraint, D, on the secondary service rate. The arameters used to generate the figure are: λ e = 0.4 energy ackets/slot, P = 0.7, = 0.1, P = 0.8, = 0.1, P 1s = 0.6, 1s = 0.075, P FA = 0.05, and P MD = 0.01 and two different values of the rimary queueing delay constraint. As is clear from the figure, the secondary service rate is reduced when the rimary queueing delay constraint is more strict. λ s [ackets/slot] Fig. 4. λ s [ackets/slot] S f S S f R S R λ [ackets/slot] Stability region of the roosed systems. X: Y: S f S S c Equaerformance λ [ackets/slot] Fig. 5. Stability region of the roosed systems. The conventional system, S c, is also lotted for comarison uroses. λ s [ackets/slot] Fig P 1 P2 P 3 S S f λ [ackets/slot] P 4 Stability region of the roosed systems.