Optimal Collaborative Spectrum Sharing for a Cognitive Multi-Antenna Relay With Full-Duplex Capability and Stability Constraint

Collaborative spectrum sharing (CSS) between primary and secondary users (PU and SU) is an effective way of utilizing limited radio spectrum. In CSS, cognitive SU acts as a relay for PU, which facilitates the PU to send its packet with a lower error probability by the help of the SU, and consequently, the SU has more chances to find a vacant spectrum. When SU is equipped with multiple antennas, it can further efficiently utilize the radio spectrum by adopting a growing self-interference cancelation technique for full-duplex (FD) transmission. In this paper, both an aggressive and a passive secondary usage of the spectrum are proposed, the operational principles of which are defined using spectrum-sharing probability <inline-formula> <tex-math notation="LaTeX">$\phi $ </tex-math></inline-formula> in FD CSS environments. For the spectrum-sharing probability <inline-formula> <tex-math notation="LaTeX">$\phi $ </tex-math></inline-formula>, SU for each of the methods sends its own signal and the PU’s (relayed) signal together by using superposition transmission. For the remaining probability <inline-formula> <tex-math notation="LaTeX">$1-\phi $ </tex-math></inline-formula>, the two methods work differently: SU sends its own signal only in the aggressive mode while it sends only the relaying signal for PU in the passive mode. We formulate an FD CSS problem with various physical-layer parameters and the proposed operating modes as a function of the spectrum-sharing probability. Our goal is to maximize the secondary stable throughput while keeping a primary traffic constraint. Closed-form optimal solutions on <inline-formula> <tex-math notation="LaTeX">$\phi $ </tex-math></inline-formula> are provided in the paper, the value of which heavily depends on the primary traffic volume, the operating modes, the relative locations of the collaborative nodes and the transmit power budget at SU. The analytical results in the paper are verified with numerical investigation, and the performance enhancement by the proposed methods is evaluated in comparison with benchmark systems. The results show that the aggressive mode is promising if SU has a relatively small transmit power and the primary traffic load is low, while the passive mode is suitable when the primary traffic load is high.


I. INTRODUCTION
As an effective way of cognitive utilization of the limited radio spectrum, collaborative spectrum sharing (CSS) between primary and secondary users (PU and SU) has attracted much research interest [1]- [7]. Unlike ordinary cognitive radio systems, cognitive SU in CSS acts as a relay for PU, which facilitates the PU to send its packet with a lower error probability by the help of the relay (i.e. SU) and consequently can provide more chance for SU to find The associate editor coordinating the review of this manuscript and approving it for publication was Rongbo Zhu . a vacant spectrum. In CSS, a time slot is usually divided into two consecutive mini-slots, which consist of PU and SU transmission phases. If PU transmits its signal at the first mini-slot, SU may sense and relay it to the PU's destination at the second mini-slot in which the SU's own signal can also be transmitted simultaneously with the PU's signal by being superimposed. If PU is idle at the first mini-slot, SU can send only its own data at the second mini-slot. SU in CSS can always access the spectrum either as a collaborative relay [5] or as a pure sender. However, the SU can also always interfere with the PU by either sending two signals simultaneously or missing detection of PU's existence and sending its VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ own signal. Thus, resource allocation at the SU is essential in CSS in order to prevent PU's signal quality from dropping below a predefined target. CSS therefore lies somewhere in between underlay and interweave cognitive radio systems [8].
With proper resource allocation, SU in CSS is shown to achieve significant performance improvement compared to SU in ordinary cognitive radio systems [1]. Though the time slot is halved in CSS, PU's transmission performance is still well protected to a desired level thanks to the power gain obtained by SU acting as a relay. CSS can be further improved if letting PU transmit at both the first and the second mini-slots and/or letting the relay of PU's signal be made at the instant SU receives it without waiting for the next mini slot. In [9], the former method is shown to significantly improve the achievable throughput and can reduce the outage probability by optimally splitting PU's power between the first and the second transmission as well as by adopting cooperative beamforming between PU and SU. On the other hand, the second one can be fulfilled by introducing a full-duplex (FD) capability at SU, which of course needs only a single slot in relaying. In-band FD of simultaneous radio transmission and reception on the same frequency band is getting implementable with the help of advanced interference cancelation (IC) techniques and is shown to achieve an approximately twofold capacity increase for isolated radio links [10].
Feasibility of FD relay systems has been addressed in many works. Especially, in MIMO FD relay systems, self-interference (SI) can be eliminated through multiple-antenna signal processing and close-to-twofold capacity is achieved compared to the half-duplex counterparts [11]- [13]. In [14], zero-forcing (ZF) loopback interference suppression at the relay node is proposed. In [15] and the references therein, to suppress SI and interference due to multiple users, ZF and block diagonalization precodings are used. However, the feasibility results are not always applicable due to the varying interference cancelation levels in different realistic environments. In this respect, [16] addresses an FD MIMO transceiver design that is aware of aging and inaccuracy of FD hardware components. On the other hand, assuming that residual loopback interference exists due to imperfect cancelation, [17] provides a theoretical FD precoder design problem that can be solved in closed-form expressions for MIMO source and relay with an energy harvesting destination. In [18], system-level FD feasibility is also investigated in a stochastic geometry-based network of multiple-antenna base stations and FD relay nodes.
FD-based cognitive radio systems also have a great potential for improving the spectrum efficiency. For FD-capable SUs at which the result of sensing whether PU exists is affected by residual self-interference, a special ''listen-totalk'' sensing protocol is proposed in [19], a waveform-based sensing approach is considered in [20], and an FD wireless sensing is provided in [21] when PU and SU are non-timeslotted and not synchronized. In [22], the sum mean-squarederror of multiple cognitive FD SU pairs is investigated and minimized under the SU power constraints and the interference constraint on PU, which is an example for FD underlay. In [23], an underlaying cognitive radio system with multiple FD SU relays is investigated and closed-form expressions for the channel capacity and outage probability are provided. In [24], an overlay version of FD CSS is presented, where SU relays the PU signals superimposed with its own signal, and the achievable rate region given the primary and cognitive power constraints is investigated. Reference [25] provides an FD-based one-way OFDM relaying system where the SU receives/transmits the PU signals over a subset of subcarriers in two phases while transmitting its own signals over the remaining subcarriers. The results in [24] and [25] are extended to cover an FD-based two-way OFDM relaying system in [26].
In this paper, we propose a hybrid version of FD CSS that falls in the middle of overlay (or interweave) and underlay cognitive radios and it also becomes an overlay system in a trivial case. In overlay FD CSS in [24] and its variants, the SU relays PU's signal as well as sends its own signal by using zero-forcing (ZF) beamforming transmission for two distinct destinations. The simultaneous transmission of SU's and PU's signals may improve the spectrum utilization but sometimes degrade the performance by splitting the transmit power of SU between the two signals. The performance degradation occurs especially when PU and SU are far away from each other or the radio channel between them is in a deep fading but SU tries to relay PU's signal. This paper is motivated by the need to let SU be able to switch the collaboration mode between compositely relaying PU's signal and either only transmitting its own signal (in an aggressive mode) or purely relaying PU's signal (in a passive mode). The switching in this paper is controlled by spectrum-sharing control probability φ (0 ≤ φ ≤ 1). When PU is sensed as active, SU works as an overlay FD CSS relay described above with probability φ. On the other hand, with probability 1 − φ, the SU uses either of the following two strategies: the aggressive and the passive usage of the spectrum. With the aggressive one, the SU transmits purely its own signal but interference to PU's receiver is avoided by ZF. With the other, the SU purely relays PU's signal. The first strategy has its name because the SU aggressively occupies the spectrum with probability 1 − φ even though PU exists, while the other one has its name becayse the SU passively utilizes the spectrum for only part of the time (i.e., for the time probability φ) when acting as a CSS relay. If we let φ = 1, the proposed FD CSS is equivalent to the system presented in [24]. When PU is sensed to be idle, SU turns its receiver off and works as a pure transmitter of its own signal (now not operating in FD mode).
As a measure of performance, we investigate the stable throughput of the proposed FD CSS, which keeps the PU also stable respect to the input arrival rates of data. We assume that PU and SU are equipped with a buffer for the input data packets, respectively. For a given arrival data rate at PU, an optimal spectrum-sharing probability φ * is obtained for the above explained strategies, respectively. And the stability region that is defined as a set of arrival data rates for PU and SU, with which the queue lengths of both PU and SU are finite, is presented. Since the arrivals of the data at PU and SU are bursty and the interference between PU and SU is thus probabilistic, stable throughput is a meaningful measure of network-level performance. In order to treat the interacting queues in the system, the stochastic dominance concept in [27] is used for investigating the throughput. Such stability studies for half-duplex (cognitive) relay systems have been widely found in the literature [1], [3], [28]. For FD systems, however, stability studies are rather limited so far. In [29], the stability throughput of an FD Aloha network is characterized. The stability of secure communication is investigated for two-user FD broadcast channel in [17]. The stability of an FD relay queue that assists two-user multiple access with multi-packet reception capability at the receiver is discussed in [30]. For a relay with two queues, each of which is for the respective pair of users (i.e., the respective source-destination pair), the per-user stable throughput is presented in [31].
The main contributions of this paper are summarized as follows.
• We formulate the stability throughput in FD CSS environments with various physical-layer parameters and operating modes as a function of the spectrum-sharing probability. The spectrum-sharing probability determines the resource allocation between PU and SU as well as the throughput in CSS environments.
• We propose two operating modes of FD CSS that give further operational flexibility to CCS nodes and those modes are shown to improve the performance in combination with the optimal spectrum-sharing probability provided in the paper.
• We define our optimization problems by finding an optimal spectrum-sharing probability that maximizes SU's throughput under the constraint of keeping PU's throughput at a given level. And we provide closed-form solutions on optimal spectrum-sharing probabilities, which are numerically verified in the paper, and also present the corresponding stability regions. Numerical investigation are presented for the proposed analysis and show that the aggressive mode with the optimal-spectrum sharing provided in the paper improves the performance (compared to the fixed overlay system, i.e., letting φ = 1) if the traffic load of PU is low. On the other hand, if PU's traffic load is high, the passive mode is shown to enhance the performance. To the best of the authors' knowledge, such design and analysis of combining collaboration modes and performance optimization have not been conducted for FD CSS systems.
The remainder of the paper is organized as follows. Section II provides a comprehensive description on the overall system model, the physical-layer model in detail, the definition of stability and the optimization problem focused on herein. The probabilities of successful decoding (PrSDs) are also defined in Section II, playing a critical role in the throughput analysis. According to the relationship between PrSDs, we choose one representative condition as an assumption to provide a context for the results in Sections III and IV. The other conditions, which are rarely satisfied in usual FD CSS environments, are also considered in Appendix A. Section III presents an analysis and a closed-form solution of spectrum-sharing for the aggressive case. The passive case is investigated in Section IV. Section V introduces benchmark CSS systems for a performance comparison and provides numerical verification of the optimal spectrum-sharing provided in the paper. The usefulness of the proposed FD CSS to enhance the stability throughput compared to the benchmark systems is also illustrated in Section V. Section VI concludes the paper. Appendix A provides closed-form optimal spectrum-sharing and the corresponding stability regions for miscellaneous conditions for PrSDs that are not treated in the main body of the paper. Finally, Appendix B provides the stability throughput of a benchmark system for comparison.

A. OVERALL MODEL
We consider a communications network (shown in Figure 1) that consists of a primary transmitter-receiver pair (PT-PR) operating on a licensed radio frequency (RF) channel and a secondary pair (ST-SR) being capable of performing cognitive radio (CR) functions. With the CR capability, ST in CSS works as a relay for the PT-PR pair as well as sends its own signal if it does not degrade PT's transmission quality. We assume that PT, PR and SR are equipped with single antennas, while ST is equipped with M receive and N transmit antennas. Both PT and ST have a buffer of infinite capacity to store their own incoming packets, respectively. Time is slotted and the transmission of each packet is assumed to take one time slot. The packet arrival processes are independent and jointly strictly stationary with mean λ p (packets per slot) and λ s for PT and ST, respectively. Q p and Q s denote their respective queue lengths.
At the start of each slot, given the operating mode (either aggressive or passive), ST senses whether PT is active or not (equivalently, busy or idle). We assume that the sensing   result could be wrong and let p d and p f denote the detection and false alarm probabilities, respectively. We ignore the loss in bandwidth due to sensing duration in this paper, but it could be accounted for in a straightforward way if the sensing duration is given and fixed. When PT is sensed to be busy, ST, with probability φ, acts as an FD collaborative relay for the PT-PR pair, in which ST simultaneously sends its own data and relays PT's data by using zero-forcing (ZF) and cophasing techniques. With probability 1 − φ, ST is assumed to operate in either of two methods: aggressive or passive methods. In the aggressive mode, ST sends only its own data without relaying PT's data, for which ST uses ZF weights to nullify possible interference to PR. In the passive mode, on the other hand, ST purely relays PT's signal instead of sending its own signal, for which ST employs a beamforming technique to reinforce the active PT-PR channel. When PT is sensed to be idle, ST sets its receiver as inactive and sends its own data. In this case, ST does not operate in the FD mode. It is noted that the CR operation assumed in this paper may be a hybrid overlay (or interweave) and underlay method in the sense that ST transmits its signal not only when PT is sensed to be idle, but also if PT is busy by using a ZF technique if the transmission is not harmful to the primary pair.
Working as an FD relay, ST is assumed to use an amplifying-and-forwarding (AF) relaying technique and hence does not need a buffer to store the relaying packets. We also assume that ST has perfect knowledge on the channels between pairs: ST-SR, ST-PR and PT-PR and on the self-interfering channel from the ST's transmitter to the receiver. With the channel information, ST manages to compute ZF weights that nullify unnecessary interference to the receivers or to construct so-called a co-phasing weight that is used to align ST-PR channel into PT-PR channel. With the co-phasing weight, PR with a single antenna can receive both of the signals from PT and ST coherently. The self-interference is assumed to be perfectly canceled by ZF techniques described below. We investigate two options for the self-interference cancelation (SIC): receive-side and transmit-side cancelation, respectively. We assume that the SIC is perfect in this paper.

B. PHYSICAL LAYER MODEL
As shown in Figure 1, let h p and g p denote the channel for PT-PR and the interfering channel for PT-SR, respectively, and let h r , h s and g s denote M × 1, 1 × N and 1 × N channel vectors for the pairs PT-ST, ST-SR and ST-PR, respectively. Let G r denote the M × N self-interference channel at ST. We assume that radio propagation over any transmitter-receiver pair is an independent stationary Rayleigh flat-fading channel.
| 2 ] = 1, respectively. And let P p and P s denote transmitting power of PT and ST, respectively. When ST transmits in an FD relaying mode, it does not purely transmit its own signal x (s) but sends composite signal s (s) constructed by x (s) and x (p) that is possibly a part of received signal y ST described as follows. Let θ (p) be a binary variable that indicates whether PT is active (θ (p) = 1) or inactive (θ (p) = 0). Then the received signal at ST after applying a weight vector w R for multiple (M ) receiving antennas is where θ (d) denotes optional SIC (either θ (d) = 1 when receiver-side SIC is applied or θ (d) = 0 when transmitter-side SIC is applied) and n ST is an additive white Gaussian noise (AWGN) vector at ST. Hereafter we assume that the AWGN at each receiver antenna is independently and identically distributed according to CN (0, N 0 ).
With y ST , the composite signal s (s) to be transmitted by ST is constructed (please see details depicted in Table 1, where proposed aggressive-and passive-mode CSS denoted by agg and pss in the table, respectively) according to the buffer state, the sensing result and the physical layer options. Let H 0 andĤ 1 denote the sensing results on the transmitting state of PT: idle and busy, respectively. When Q s = 0 and the sensing result isĤ 0 , ST transmits only x (s) by constructing T is a ZF weight described shortly in the following. If the sensing result isĤ 1 when Q s = 0, y ST and x (s) together form s (s) . In this case, PT's signal is relayed with probability φ and hence where w T is a ZF weight that is used to limit the interference to SR and is described below shortly, α is a power allocation factor between relaying signal y ST and ST's signal x (s) , and h p is a normalized co-phasing AF weight given bŷ Hereĥ p is used to align the channel of relayed y ST into the channel of signal traveling from PT to PR and to regulate the relaying power of y ST equal to P s α. On the other hand, though 68652 VOLUME 8, 2020 = w (p) T √ P sĥp y ST in the passive spectrum sharing. It is noted that if we take α = 0 in (2) and choose the passive mode, s (s) becomes either w

2) WEIGHT VECTORS USED AT MULTIPLE ANTENNAS OF ST
We now describe how to select w R , w T and w (s) T . The weight vectors are obtained differently according to SIC options whether SIC is applied at the transmitter side or the receiver side at ST. If SIC is performed at the transmitter side of ST, (1) and w R = h H r is an optimal combining weight (that is, it provides maximal-ratio combining [32]). If w R is given, w respectively. The solution weights are summarized in Table 2(a). On the other hand, when SIC is performed at the receiver side of ST, w (p) T and w (s) T are obtained by ZF weights without the SIC constraint |w R G r w| 2 = 0 in (4) and (5), respectively, and given in the first two rows in Table 2(b). Then w R , which cancels the self-interference, can be obtained by the solution weight of which is given in the third row in Table 2(b).

3) RECEIVED SIGNAL FORMS AT PR AND SR
When ZF weights w R , w T and w (s) T are used as described above, the received signal at PR is where n PR is AWGN at PR. And the received signal at SR is where n SR is AWGN at SR.

4) PROBABILITY OF SUCCESSFUL DECODING
We assume that a receiving node can decode a packet successfully if the received instantaneous signal-to-interferenceplus-noise ratio (SINR or sometimes SNR if the interference does not exist) is greater than or equal to a certain threshold i∈{p,s} (the subscript p representing for PR and s for SR). Let H 0 and H 1 denote the actual state of PT: idle and busy, respectively. The probability of successful decoding (PrSD) at each receiver is denoted by q (i) j,k , and summarized and described in Tables 3 and 4, respectively. In the notation of PrSD, superscript (i) is in {(p), (c), (s) }. Superscript (p) means PrSD at PR when the signal is only from PT, (c) means PrSD at PR when the signal is from both PT and ST and combined at PR, and (s) means PrSD at SR. And subscripts j, k are used in the form of q (i) j,k where j ∈ {z, b} denotes a class of weight applied to transmit antennas and k ∈ {f , l} VOLUME 8, 2020 denotes a power allocation state. Subscript z means that the transmitting signal is processed with ZF weights in ST to relay PT's signal as well as to transmit its own signal, while b means that the signal is processed with beamforming weights in ST just to relay PT's signal since it operates in the passive mode or Q s = 0. Furthermore, subscript f means that ST transmits with its full power P s either to relay PT's signal or to send its own signal, but l means that the power is split and allocated by α and 1−α between the relaying signal and ST's signal, respectively.
It is also noted that the tildes inq (s) z,· at the fifth row in Table 3 (a) indicate that the signal desired at SR is directly interfered with the signal from PT, while the other PrSDs without tildes could be obtained from interference-free environments by co-phasing or ZF techniques even though the two transmitters PT and ST are simultaneously active. It is finally noted that PrSDs q b,f can be obtained straightforwardly as in [33] but with integral parts, for which we have used a numerical method to evaluate q (c) z,l and q (c) b,f for the performance investigation. For notational brevity, we introduce some inequalities regarding the PrSDs: In the following sections III and IV, we provide the main results by assuming q The results for the other cases are also summarized in Appendix A, where the assumed inequality cp ≥ 0 or c ≥ 0 is violated and a trivial form (either 0 or 1) of spectrum-sharing probability is obtained.
z,f and other ones will also be provided in Section V.

C. STABILITY CONSTRAINT AND PROBLEM DESCRIPTION
Let µ p and µ s denote the average service rate of the queue in PT and ST, respectively. For queues where the arrival and service processes are jointly strictly stationary and ergodic, Loynes' theorem [34] states that the queue at each transmitting node is stable if and only if the average arrival rate is strictly less than the average service rate. Based on Loynes' theorem, let us define that a queue is stable if λ i < µ i , i ∈ {p, s}. Moreover, a system is said to be stable if and only if all the queues in the system are stable.
Our goal is to find an optimal spectrum-sharing ratio φ (denoted by φ * ) that maximizes λ s (letting λ * s denote optimal λ s corresponding to φ * ), while keeping the stability constraints in both PT and ST (that is, µ p > λ p and µ s > λ s , respectively), given λ p . p d and p f are also assumed to be given and fixed. φ * obviously depends on various physical layer parameters and operational options. As a result of the optimization, we can obtain a stability region for the proposed system.

D. IMPLEMENTATION ISSUES AND ASSUMPTIONS
The complexity of practical implementation of the proposed method mainly lies in keeping the coexistence synchronization between PU and SU and gathering the channel information for obtaining φ. Like other cooperative MIMO systems, PT and ST should be synchronized especially with respect to the PR's view. Distributed synchronization in wireless networks has a pretty long history and is well summarized in [35]. The 802.22 MAC introduces a new superframe structure to facilitate incumbent protection, synchronization and self-coexistence [36]. If Coexistence Beaconing Packet (CBP) [37] is used for providing synchronization in FD CSS, ST receives CBP from its neighbor PT and it adjusts the start time of the superframe according to specific rules, where the superframe may consist of tens or hundreds of time slots.
In this paper, ST is assumed to determine optimal φ with the relevant channel knowledge. We assume that PT and ST send reference signals. PR is assumed to extract channel information from PT-PR and ST-PR channels and report them to PT. PT then relays this information to ST. On the other hand, SR is assumed to measure and send (to ST) the channel information for ST-SR channel. If the receiver at ST can measure the self-feedback channel from the reference signal, ST can have the whole channel knowledge required to obtain optimal φ. Moreover, we also assume that PT informs ST of its buffer state, traffic arrival rate, a desired service target, and its transmission power level. Then ST can estimate the related probabilities of successful decoding with the channel knowledge and determine the region (i.e., among the four categories depicted in the following Figure 3) that it falls into. With the region information, the proper mode can be selected to maximize the achievable secondary service rate. Along with the mode selection procedure, φ should be determined together. In practical systems, these parameters are changing dynamically and exchanging such information on a shorter-time scale will also add a practical overhead to the system.
In [38] and some practices, FD is implemented on a single antenna. It assumes special circuits that can measure and cancel the feedback signals. Unlike this method, we apply precoding and decoding weights for cancelling the feedback signals, which need an additional antenna at either transmitting or receiving side. Our assumption has been widely accepted in MIMO and full-duplex literature. If we assume the cancellation circuit block like in [38], we do not need an additional antenna for removing the feedback signals and the number of required antennas decrease by 1.

III. OPTIMAL SPECTRUM SHARING WITH THE STABILITY CONSTRAINTS: THE AGGRESSIVE CASE
The primary service rate µ p at PT's queue depends on the state of Q s and the action taken by ST. µ p is then given by On the other hand, the secondary service rate µ s at ST's queue is depending on the state of Q p and the sensing result. µ s is given by When maximizing λ s subject to given λ p and the stability constraints, the states of Q p and Q s are interacting since PrSD depends on the queue states due to interference, which makes the optimization not straightforward. To decouple the effect of queue states, we use in this paper a popular dominant system approach originally provided in [27] and [39]. In this approach, one of the transmitters sends dummy packets when its queue is empty, while the other transmits according to its traffic. The dominant system generally provides a lower bound of the stable throughput. It is however shown to exhibit an exact stable throughput, especially when the number of interacting queues is two, by taking the union of two dominant systems: the so-called first and second dominant system [27]. Hereafter, superscripts (1) and (2) denote the first and the second dominant system, respectively. And let φ * 1 and φ * 2 denote the respective optimal spectrum sharing ratios.

A. FIRST DOMINANT SYSTEM: ST TRANSMITS DUMMY PACKETS
The first dominant system is identical to the original system except that ST transmits a dummy packet whenever Q s empties. All other assumptions remain unchanged in the dominant system. Thus, in this first dominant system, ST's queue never empties and the service rate for PT's queue is given as a function of φ: Since cp ≥ 0, Loynes' stability condition λ p < µ (1) p (φ) imposes the following constraints on λ p and φ, respectively.
where λ (1) p,U denotes an upper-bound of λ p that can be achieved by taking φ = 1, and φ L indicates a lower-bound of φ, which can be found by re-arranging λ p < µ On the other hand, the average service rate of ST in this dominant system can be obtained as When λ p (< λ (1) p,U ) is given, a maximum stable throughput of ST then can be found by solving where φ L = φ L + for sufficiently small positive real and φ L is introduced to ensure the optimization on a closed constraint-set. We assume 0 ≤ φ L ≤ 1 without loss of generality. By differentiating µ where A is a constant given by Since s ≥ 0, the derivative in (16) is non-positive for all feasible φ in this case. Thus µ (1) s (φ) is a non-increasing function and φ * 1 = φ L .
2) WHEN A > 0 dµ (1) s dφ = 0 has two possible roots, one of which is always negative and the other is given bỹ ≤ 0 when φ =φ and henceφ is an optimal φ that maximizes µ (1) s (φ). Thus, optimal φ * 1 can be summarized in (19), as shown at the bottom of this page.
The stable region for the secondary dominant system S (1) thus can be expressed by and ST's optimal average service rate µ (1) s (φ * 1 ) can be summarized in (21), as shown at the bottom of this page.

B. SECOND DOMINANT SYSTEM: PT TRANSMITS DUMMY PACKETS
In the second (or primary) dominant system, PT transmits a dummy packet whenever Q p empties, while ST behaves in the same way as in the original system. In the second dominant system, the average service rate of ST is On the other hand, the average service rate of PT is given by A , From the stability condition λ p < µ (2) p (φ), we have a constraint on λ s such that By taking λ s = 0, we can have an upper bound on λ p : When λ p < λ (2) p,U is given, λ s should be satisfied with both of the stability constraints: λ s < µ (2) s (φ) and λ s < λ s,U (φ)}. Then, a maximum stable throughput of ST can be found by solving φ * 2 can be obtained by an optimal solution of problem in (26). If the detection probability is positive (i.e., p d > 0), µ (2) s (φ) is a strictly decreasing linear function of φ since˜ s ≥ 0. On the other hand, by differentiating λ where C is a constant given by 1) WHEN C ≤ 0 In this case, both µ 2) WHEN C > 0 λ (2) s,U (φ) monotonically increases and a maxmin solution is found by equating µ (2) s (φ) and λ (2) s,U (φ), from which we have and the corresponding v * is given as in (31), as shown at the bottom of this page. Thus, optimal φ * 2 can be summarized as and the corresponding v * is given as in (33), as shown at the bottom of this page. The stable region for the second dominant system S (2) thus can be expressed by

C. STABLE REGION: UNION OF S (1) AND S (2)
The joint stability region for PT and ST is given by Sinceλ (1) p,U ≤λ (2) p,U , an optimal spectrum sharing φ * characterized by φ * 1 and φ * 2 can be determined by

IV. OPTIMAL SPECTRUM SHARING WITH THE STABILITY CONSTRAINTS: THE PASSIVE CASE
In the aggressive mode, ST sends only its own signal with probability 1 − φ even though PT is sensed to be busy. However, in the passive mode, ST just relays PT's signal without sending its own signal by probability 1 − φ if PT is sensed to be busy. The passive strategy intends to always relay PT's signal if it exists, which avoids congestion in PT's queue and hence makes PT idle more frequently. With the passive strategy, the primary service rate µ p at PT's queue is given by On the other hand, the secondary service rate µ s at ST's queue is given, depending on the state of Q p , by

A. FIRST DOMINANT SYSTEM WITH THE PASSIVE STRATEGY
For the first dominant system with the passive strategy, the service rate for PT's queue is given bȳ Hereafterā is used to indicate a variable with the passive strategy, the counterpart of which is a with the aggressive strategy. For given λ p , Loynes stability condition λ p <μ (1) p (φ) imposes the following constraints on λ p and φ, respectively.
On the other hand, the average service rate of ST in the first dominant system can be obtained as When λ p (<λ (1) p,U ) is given, a maximum stable throughput of ST can be found by solving whereφ L =φ L + for sufficiently small positive real . We assume 0 ≤φ L ≤ 1 without loss of generality by differentiatingμ (1) s (φ) with respect to φ, we can have dμ (1) where E is a constant given by It is noted that when c = 0,μ (39) is not affected by φ and thus we assume c > 0 in finding φ * 1 in this subsection. (44) is monotonic for 0 ≤ φ ≤ 1 and hence non-positive,μ (1) s (φ) is non-increasing and consequentlyφ * 1 = 0.
Since F(φ) is monotonically decreasing for 0 ≤ φ ≤ 1 and henceμ (1) s (φ) is a concave function, a maximum of which is attained at F(φ) = 0. F(φ) = 0 has two possible roots, one of which is always negative and the other is given bỹ Thus, optimalφ * 1 can be summarized as (47), as shown at the bottom of the next page. The stable region for the first dominant systemS (1) then can be expressed bȳ and ST's optimal service rate ofμ (1) s (φ * 1 ) can be summarized in (49), as shown at the bottom of the next page.

B. SECOND DOMINANT SYSTEM WITH THE PASSIVE STRATEGY
The service rate for ST's queue for the second dominant system is given bȳ On the other hand, the average service rate of PT is given bȳ When c = 0,μ (2) p (φ) is not affected by φ and we also assume c > 0 in this subsection. By taking λ s = 0, we can have an upper bound on λ p : From (51) with the stability condition λ p <μ (2) p (φ), we also can have When λ p <λ (2) p,U , λ s should be satisfied with both of the stability conditions: λ s <μ (2) s (φ) and λ s <λ (2) s,U . Letv(φ) = min{μ (2) s (φ),λ (2) s,U }. Then, a maximum stable throughput of ST can be found by solving If the detection probability is positive (i.e. p d > 0),μ (2) s (φ) is a strictly increasing linear function of φ sinceq (s) z,l ≥ 0. On the other hand, by differentiatingλ (2) s,U with respect to φ, we have dλ (2) The derivative in (55) is non-positive for all feasible φ. A maxmin solution ofv(φ) is found by equatingμ (2) s (φ) and λ (2) s,U . Thus we can havē and the correspondingv * is given in (57), as shown at the bottom of this page. The stable region for the second dominant systemS (2) thus can be expressed bȳ C. STABLE REGION WITH THE PASSIVE STRATEGY: The joint stability region for PT and ST with the passive strategy and optimal spectrum sharingφ * are respectively given byS and for 0 ≤ λ p ≤λ

V. NUMERICAL INVESTIGATION A. SIMULATION SETUP
In this section, we numerically verify the optimality of the proposed spectrum sharing probability and then the stable throughput of proposed aggressive-and passive-mode CSS (denoted by agg and pss in the figures, respectively) is evaluated with simulation. In addition, in order to gain insight into the performance achieved by the proposed methods, the performances of three reference models as follows are also presented and compared:  • A CSS method provided in reference [24], which is equivalent to the proposed aggressive mode if we let φ = 1 without optimizing (denoted by ref24 in the figures); • A no-relaying system, in which ST never relays PT's signal even though ST itself is idle and always transmits its own signal unless its buffer is empty (denoted by no-relay in the figures). However, ST should use a zero-forcing technique in order to not interfere with PT when sending its signal. The stability region of the no-relaying system is also presented in Appendix B; • CSS without sensing, which is obtained if we let p d = p f = 1 in the proposed methods (denoted by aggb and pssb for the aggressive and the passive modes, respectively, where ''b'' implies ''blind''). This variant is considered to evaluate the effect of the sensing capability adopted in CSS on the performance improvement. It is noted that in the blind aggressive mode, ST is assumed to share the buffer information of PT and it relays the primary signal though φ = 0 if its own buffer is empty. For numerical investigation, we mainly consider two different network configurations (as shown in Figure 2), where the primary locations are geometrically fixed at (0,0) and (4,0) for PT and PR, respectively, but the secondary locations are essentially different. For case 1, ST and SR are located in between PT and PR as specified in Figure 2 and for case 2, ST and SR are far away from PT and PR to the left. For both the cases, the distances between pairs of transmitter and receiver are the same. Case 1 may represent a typical CSS scenario where four nodes are closely located. On the other hand, case 2 in which ST is far from PT and PR is an extreme scenario. In case 2, it is relatively inefficient for ST to relay PT's signal but ST is assumed to urgently need to use the spectrum.
b,f , respectively. Hence, case 2 with P s /N 0 = 15 dB falls into region (D), where the collaboration between PT and ST looks not effective since ST is far away from PT. Though not presented in this paper, the total area considered in Figure 3 falls into a region of type (A) if P s /N 0 = 5 dB is used with the same simulation setup. This is why we mainly focus on analyzing the condition of q b,f }, are rarely realized in practice since they are possible when σ PT,ST should be 10 dB less than σ PT,PR and σ ST,SR , and P s is greater than P p by at least 16 dB and most power of P s should be used for relaying PU's signal. However, it is noted again that the stable throughputs for all the other conditions are also provided in Appendix.VII. Figures 4 and 5 illustrate the optimality of the optimal spectrum-sharing control probabilities (φ * ) provided in the paper. Figure 4 for case 1 with P s /N 0 = 5 dB can be used to check φ * in (36) and (60) for the aggressive and the passive mode, respectively. On the other hand, Figure 5 for case 2 with P s /N 0 = 15 dB can be used to check φ * in (86) and (87). Each plot in the figures presents the achievable secondary service rate max{µ (1) s (φ), v(φ)} and max{μ (1) s (φ),v(φ)} as a function of φ for different λ p 's for the aggressive and the passive mode, respectively. Both of the figures numerically show us optimal φ that attains the maximum secondary service rate. In Figure 4, the maximum secondary service rate for the aggressive mode seems to be  obtained at φ = 0.6519, φ = 1 and φ = 1 when λ p = 0.2, λ p = 0.5 and λ p = 0.8, respectively. For the passive mode, it is obtained at φ = 1, φ = 1, and φ = 0.9354, respectively. The optimal φ's for case 1 are the same as the φ * provided in (36) and (77), respectively. In Figure 5, the secondary service rate seems to be maximized when φ = 0 for the aggressive mode and when φ = 1 for the passive mode, which can verify the optimal sharing provided in (86) and (87), respectively. Figures 6 and 7 provide optimal spectrum-sharing (φ * ) as a function of λ p for case 1 and 2, respectively. In the figures, φ * 's are presented with different secondary transmission power P s /N 0 = 5, 10, 15 dB, respectively. For case 1 in Figure 6, it is seen that φ * = 1 in the mid-range of λ p regardless of the collaboration modes. It means that ST always transmits the primary and its own signal together if PT is sensed as active and its buffer is not empty. In the range of low primary traffic (low λ p ), the lower P s /N 0 results in the smaller φ * for the aggressive mode, which implies FIGURE 6. φ * as a function of λ p for case 1 with P s /N 0 = 5, 10, 15 dB; p d = 0.9. FIGURE 7. φ * as a function of λ p for case 2 with P s /N 0 = 5, 10, 15 dB; p d = 0.9. VOLUME 8, 2020 that the smaller transmission power in ST encourages the aggressive transmission (that is, the larger 1 − φ * ) of ST to maximize its throughput. On the other hand, in the range of high primary traffic (high λ p ), the higher P s /N 0 results in the smaller φ * for the passive mode. In the aggressive mode, the larger 1 − φ * causes the more interference from ST to PR while resulting in the less interference from PT to SR in the passive mode. In Figure 6, a high P s /N 0 reduces the interference by adjusting optimal spectrum-sharing, in which the interference from ST to PR is treated significantly when λ p is low, while the interference from PT to SR is important when λ p is high. For case 2 in Figure 7, P s /N 0 = 10 and 15 dB result in q

B. VERIFICATION
b,f (the condition indicated by region (D) in Figure 3). However, P s /N 0 = 5 dB preserves the main relationship q z,f (the condition indicated by region (A) in Figure 3), even in case 2. With P s /N 0 = 10 and 15 dB, it is seen that φ * = 0 for the aggressive mode and φ * = 1 for the passive mode for the allowable λ p .
The stable throughput of the proposed CSS certainly relies on the sensing accuracy (both p d and p f ). We adopt the energy detection scheme presented in [37] and then if target p d is given, p f is obtained by using the equation given in [37, eq. (13)] as follows: where Q(·) is the complementary distribution function of the standard Gaussian, Q −1 (·) is its inverse, γ is the SNR of sensed signal, and we have set the number of samples for detection to T = 2. Figure 8 shows the effect of the sensing accuracy on the stable throughput. The x-axis represents the detection probability target. Different primary traffic loads λ p = 0.1, 0.3, 0.5, 0.7, 0.9 are considered, for which the accuracy target (p d ) that maximizes the secondary throughput seems to be 0.73, 0.85, 1, 1, 1, 1, respectively. Detection should be more accurate if λ p becomes larger due to avoiding interference. Since p f is an increasing function of p d in (61), a small p d consequently contributes to giving more transmission chances to ST, which helps ST increase its throughput especially in the condition of low primary traffic. With a congested condition of high primary traffic, decreasing false alarm may increase the interference to PT, which also results in shrinking the secondary throughput. Thus, an optimal target for p d may exist according to the input traffic. The search for such an optimal p d is however beyond the scope of this paper. We hereafter use the target p d = 0.9 in the following simulation unless otherwise noted. It is seen that if p d is sufficiently small, the passive mode is slightly better than the aggressive mode when λ p = 0.5, 0.7, 0.9. Figures 9 and 10 provide numerical comparisons of the stable throughput of the proposed CSS with ref24 and no-relay systems for case 1. In Figure 9, P s /N 0 = 5dB is assumed.   In the figure, the aggressive and passive modes provide better performance than ref24 for λ p < 0.270 and λ p > 0.793, respectively. For 0.270 ≥ λ p ≥ 0.793, agg, pss, ref24 show exactly the same performance. Thus if we compare the aggressive and passive modes, the aggressive mode achieves better performance if λ p < 0.270 and the passive mode is better if λ p > 0.793. This is because in the range of low primary traffic the aggressive mode works well, while the passive mode is well adaptive in high primary traffic environments, as also indicated in Figure 6. If the primary arrival rate is higher than 0.677 and 0.683 (for agg and pss, respectively), the sensing capability (unless it is perfect) is harmful to the performance, whereas ignoring the sensing results (i.e., blind collaboration) rather improves the throughput. If λ p < 0.158, no-relay is better than the passive mode as well as ref24, but the aggressive mode outperforms the no-relaying system unless λ p < 0.091, in which the two methods achieve the same stable throughput. In Figure 10 where P s /N 0 = 15dB is assumed, it is seen that the proposed methods work better than ref24 and no-relay systems. Comparing Figures 9 and 10, we can see that optimal spectrum sharing becomes less effective when the secondary power P s /N 0 is sufficiently high since the performances of agg, pss and ref24 become the same for the wider range of λ p . In Figure 10, the aggressive and the passive mode provide better performance than ref24 for λ p < 0.077 and λ p > 0.913, respectively. Table 5 summarizes the effect of the secondary power on performance enhancement by the proposed optimal spectrum-sharing methods. In the table, the secondary service rates achievable by agg, pss and ref24 are reported as a function of P s /N 0 with assuming P p /N 0 = 15 dB and 4 different levels of λ p (0.1, 0.4, 0.7, 0.9). If P s /N 0 ≥ 14 dB, all three systems show exactly the same performance regardless of the input traffic level. When λ p = 0.1, the aggressive method improves the secondary performance if P s /N 0 ≤ 12 dB and maximally achieves 19% enhancement over the passive mode and the reference method when P s /N 0 = 0 dB. When λ p = 0.9 and 0.7, the passive method accomplishes better performance than the other ones if P s /N 0 ≤ 12 dB and P s /N 0 ≤ 2 dB, respectively. With λ p = 0.4, all three methods show the same performance. Thus, it is seen that the proposed optimal sharing is effective when the secondary user has low transmission power especially when the level of primary traffic is not modest but either low or high. Figure 11 provides numerical comparison of the stable throughput for case 2 with P s /N 0 = 15 dB. As noted previously, case 2 with P s /N 0 = 15 dB gives cp < 0 and c < 0 since q

C. PERFORMANCE COMPARISON
b,f . And thus the analysis given in sections III and IV are not valid. Optimal sharing and throughputs for this case are additionally given in Appendix A-A. In this case, φ * = 0 for the aggressive mode andφ * = 1 for the passive mode as provided in (86) and (87) and verified in Figure 5. φ * = 0 in the aggressive mode implies that ST always (with sharing probability 1 − φ * = 1) sends its own data without relaying the primary signal if its buffer is not empty. Otherwise, referring to (9), ST relays the primary signal, in which the noise is mainly amplified since ST is far away from PT and hence the co-phasing performance is severely degraded (as we know from the small q (c) b,f ). However, as discussed in Appendix B, the aggressive method is not worse than its no-relay counterpart, which is also seen in the figure. It reveals that the proposed method adaptively works like the no-relaying system when the relay of PU's signal makes the performance worse. On the other VOLUME 8, 2020 hand,φ * = 1 in the passive mode represents that ST always relays the primary signal by using ZF, which results in power loss in sending its own signal and hence causes performance degradation compared to the aggressive mode as well as no-relay. Figure 12 illustrates the relative performance of the proposed method for case q z,l = 0.3745. With this configuration, the spectrum sharing in the aggressive mode is trivially given by φ * 1 = φ * 2 = 0. Thus, the aggressive modes with and without sensing, as well as no-relay, are seen to provide similarly better performance than both the passive mode and ref24 for all the tested λ p 's. It is noted that the performances simulated for case q  Figure 3) would show similar results, which can be seen in Figure 12 since φ * 1 = φ * 2 = 0 also for this case. One exception is that if λ p is sufficiently large (especially, λ p > q (p) z,f ) then the performance of the proposed passive mode becomes better than that of the no-relaying system.

VI. CONCLUSION
In this paper, aggressive and passive usages of a secondary spectrum are proposed and the respective stability performance is investigated in FD CSS environments. Each of the utilization modes has its own favorable condition: the aggressive mode is better especially if the primary traffic is low; otherwise, the passive mode becomes better. The two modes work similarly for spectrum-sharing probability φ. For the remaining probability 1−φ, the two modes act differently: SU sends its own signal only in the aggressive mode while it sends only the relaying signal for PU in the passive mode.
Closed-form optimal solutions on φ are also provided in the paper, the value of which heavily depends on the primary traffic volume, the operating modes, the relative positions of the collaborative nodes and the transmit power budget at SU. For a future wireless CSS network where distributed nodes (for example, sensor nodes) have small transmit power and the primary traffic load is low, the aggressive mode is a promising operational strategy. For other CSS networks where participating nodes have sufficient power and the the primary traffic load is heavy, the passive mode is a suitable candidate.

APPENDIXES APPENDIX A OPTIMAL SPECTRUM SHARING PROBABILITIES ACCORDING TO PrSD CONDITIONS
adopting zero-forcing transmission. Thus, the departure rate of PT and ST is respectively given by The stability region S thus can be expressed by It is noted that letting φ = 0 in the aggressive mode makes the stable throughput of the first dominant system represented by (11) and (14) exactly equal to (85) and (87), respectively. Since the stability region is the union of the stable throughputs from the first and the second dominant system, the stable throughput of the aggressive mode is always greater than or equal to that of the no-relaying system.