Energy-efficiency maximization bidirectional direct and relay transmission

Energy-efficient transmission (EET) has become a very important problem in wireless communication. Optimal power allocation (OPA) is one of the general methods to achieve EET. But when OPA is only considered in EET, it maybe cannot analyze EET problems accurately. This paper aims at analyzing energy-efficient bidirectional direct and relay transmission (RT) through joint optimization of transmit power (TP) and transmit time (TT) allocation. In RT, direct links (DLs) are existed. The EET problems are given with three optimization cases: (i) maximizing sum throughput (ST) to maximize energy efficiency (EE), in such case, optimal TT (OTT), optimal relay position (ORP), and outage probability analysis are given; (ii) minimizing total energy consumption (TEC) to maximize EE, in such case, optimal TP (OTP), OTT with gradient-descent algorithm, and influences of asymmetry and asymmetry transmission tasks are given; and (iii) maximizing ST and minimizing TEC simultaneously to maximize EE, in such case, Dinkelbach’s algorithm and one-by-one optimal algorithm are given. Simulation results are presented to validate theoretical analysis. Results reveal that relay technique and DLs in RT can improve system’s EE.


Introduction
With rapid development of wireless communication, traditional communication system cannot satisfy the requirement of green communication and spectrum resources shortage [1,2]. A large amount of interests have been focused on improving energy efficiency (EE) and spectral efficiency (SE). At the same time, lots of works have been done with different transmission techniques to improve them separately or simultaneously, such as cognitive radio [3,4], orthogonal frequency division multiplexing (OFDM) [5], multi antenna technique [6], millimeter wave technique [7], simultaneous wireless information and power transfer (SWIPT) technique [8,9], and non-orthogonal multiple access [10,11]. Among all of these techniques, relay technique has attracted a lot of attention for achieving spatial diversity [12]. The major existing relay protocols are amplify-and-forward (AF) and decode-and-forward (DF). In AF relay protocol, relay node simply amplifies and forwards the received signal without any decoding operation [13]. Also, AF relay protocol is easy DLs in RT, and joint optimization of TT and TP allocation are considered simultaneously, rather than only considering parts of them in existing work. At the same time, ORP problem is investigated from the perspective of maximizing EE for the system EE can be improved with ORP [29]. This problem also has not been considered in the most of existing work.
• Secondly, to achieve a more comprehensive EE analysis in this paper, EET problems are given with three optimization cases, i.e., maximizing sum throughput (ST) to maximize EE, minimizing TEC to maximize EE, and maximizing ST and minimizing TEC simultaneously to maximize EE, rather than only considering minimizing TEC to maximize EE in [19] and [22,23], or only considering maximizing ST to maximizing EE in [18] and [29,30].
• Thirdly, with the three optimization cases: (1) OTT, ORP, and outage probability analysis are given with maximizing ST to maximize EE; (2) OTP, OTT with gradient-descent algorithm, and influences of asymmetry and asymmetry transmission tasks are given with minimizing TEC to maximize EE; (3) Dinkelbach's algorithm and one-by-one optimal algorithm are given with maximizing ST and minimizing TEC simultaneously to maximize EE.
The rest of this paper is organized as follows. Section 2 describes the system model. Section 3 provides the throughput analysis, energy consumption model, and problem formulation. Sections 4, 5, and 6 presents the three optimization cases to maximize EE. Section 7 gives simulation results and followed by conclusions in Section 8.

System model
In this section, the models of bidirectional DT and RT are explained. In DT, there are two source nodes S 1 and S 2 . In RT, there is also a relay node R except for two source Table 1 Existing literature comparison with this paper Optimal transmit power (OTP) [15][16], [19][20][21][22][23], [25][26][27][28], this paper Optimal transmit time (OTT) [19][20], [22][23], [27][28], this paper Circuit power (CP) consumption [15][16], [20][21][22][23], [26][27][28], this paper Power amplifier (PA) efficiency [15], [20][21][22][23], [26][27][28], this paper Direct links (DLs) [18], [24][25][26][27][28], this paper Optimal relay position (ORP) [17], [27], [29][30], this paper nodes S 1 and S 2 . At the same time, the relay node R lies between two source nodes S 1 and S 2 . To achieve a bidirectional transmission, it needs two time slots in DT, it needs four time slots in OWRT, and it needs three time slots in TWRT. The relay protocol is AF, and nodes' work mode is half-duplex. The links experience independent block Rayleigh fading and remain unchanged during one block. The following optimization and analysis are based on one block duration T t . The transmission signal for the node S 1 is x 1 with variance E{x 1 2 } = 1, and the transmission signal for the node S 2 is x 2 with variance E{x 2 2 } = 1. The TP for the node S 1 is P 1 , and the TP for the node S 2 is P 2 . The channel gain between the nodes S 1 and S 2 is h 3 , the channel gain between the nodes S 1 and R is h 1 , and the channel gain between the nodes S 2 and R is h 2 . At the same time, the channel gains between the same two nodes are all reciprocal, and the nodes have the full CSI. The noise at the node S 1 is n 1 , the noise at the node S 2 is n 2 , and the noise at the node R is n r . Meanwhile, the DLs between two source nodes S 1 and S 2 in RT are existed and they can be exploited to convey information [25]. The noises are zero-mean symmetric complex Gaussian vector with variance 1. The system bandwidth is W. The receive signals are combined at two nodes S 1 and S 2 by maximum ratio combining (MRC) technique.
The DT model is shown in Fig. 1a. In the first time slot, source node S 1 transmits signal x 1 to node S 2 , and the receive signal at node S 2 is y d In the second time slot, source node S 2 transmits signal x 2 to node S 1 , and the receive signal at node S 1 is Fig. 1b. In the first time slot, source node S 1 transmits signal x 1 to node R and S 2 , and the receive signals at node R and S 2 are respectively y o 1r = √ P 1 h 1 x 1 + n r and y o 2 = √ P 1 h 3 x 1 + n 2 . In the second time slot, relay node R amplifies and forwards the receive signals y o 1r to node S 2 , and the receive signal at node S 2 is y o r2 = √ P r2 h 2 x 1r + n 2 , where P r2 is relay node's TP in the second time slot of OWRT, x 1r is node R amplifies and forwards signal in the second time slot of OWRT   [31]. In the third time slot, source node S 2 transmits signal x 2 to node R and S 1 , and the receive signals at relay node R and S 1 are respectively y o 2r = √ P 2 h 2 x 2 + n r and y o 1 = √ P 2 h 3 x 2 + n 1 . In the fourth time slot, relay node R amplifies and forwards the receive signals y o 2r to node S 1 , and the receive signal at node S 1 is y o r1 = √ P r1 h 1 x 2r + n 1 , where P r1 is relay node's TP in the fourth time slot of OWRT, x 2r is node R amplifies and forwards signal in the fourth time slot of OWRT with x 2r = A 2 y o 2r , A 2 is amplify coefficients, and [31]. The TWRT model is shown in Fig. 1c, and it transmits signal with the three-phase ANC protocol. The three-phase ANC protocol is also called time division broadcast channel (TDBC) protocol as [31] suggested. In the first time slot, source node S 1 transmits signal x 1 to node R and S 2 , and the receive signals at node R and S 2 are respectively y t 1r = y o 1r and y t 2 = y o 2 . In the second time slot, source node S 2 transmits signal x 2 to node R and S 1 , and the receive signals at relay node R and S 1 are respectively y t 2r = y o 2r and y t 1 = y o 1 . In the third time slot, relay node R broadcasts x r to nodes S 1 and S 1 , where x r = ζ 1 y t 1r + ζ 2 y t 2r is node R amplifies and forwards signal of TWRT. ζ 1 and ζ 2 are forward coefficients, 1 and o 2 can be seemed as the signal combining factors, and they determine how relay node R combines the two signals y t 1r and y t 2r , where 0 ≤ {o 1 , o 2 } ≤ 1 and o 1 + o 2 = 1 [31]. The original receive signals of TWRT y t r1 and y t r2 can be expressed as y t r1 = √ P r h 1 x r + n 1 and y t r2 = √ P r h 2 x r + n 2 , where P r is relay node's TP in TWRT. Since each of source node receives a copy of its own transmitted signal as interference, the signal transmitted from the other source node can be decoded after self-interference cancellation (SIC). Finally, the receive signals at the nodes S 1 and S 2 can be respectively expressed as y t r1 = √ P r h 1 ζ 2 √ P 2 h 2 x 2 + ζ 1 n r + ζ 2 n r + n 1 and y t r2 = √ P r h 2 ζ 1 √ P 1 h 1 x 1 + ζ 1 n r + ζ 2 n r + n 2 .

Throughput analysis, energy consumption model, and problem formulation
In this section, throughput analysis is given first, followed by the energy consumption model and problem formulation.

Throughput analysis
In this subsection, throughput analysis is given. As we have suggested in the system model, one block duration is T t and within which a round of bidirectional transmission is accomplished [22]. In DT and OWRT, T 1 and T 2 can be used to represent TT from node S 1 to S 2 and from node S 2 to S 1 , respectively. With throughput definition in [22], y d 1 , and y d 2 , throughput of DT can be given by 1 is throughput at node S 2 of DT, and C d 2 = T 2 W log 2 1 + γ d 2 is throughput at node S 1 of DT. γ d 1 and γ d 2 are signal-to-noise ratios (SNRs) at nodes S 2 and S 1 of DT, respectively. With noise variance is 1, they can be given by γ d 1 = P 1 |h 3 | 2 and γ d 2 = P 2 |h 3 | 2 .
With y o 2 , y o r2 , y o 1 , and y o r1 , throughput of OWRT can be given by 1 2 W log 2 1 + γ o 1 is throughput at node S 2 of OWRT, and C o 2 = T 2 2 W log 2 1 + γ o 2 is throughput at node S 1 of OWRT. The 1/2 is due to two time slots in completing a transmission process in each direction. γ o 1 and γ o 2 are respectively SNRs at nodes S 2 and S 1 of OWRT, and they can be given by . In TWRT, T twr can be used to represent TT for the whole information exchange. With y t 2 , y t r2 , y t 1 , and y t r1 , throughput of TWRT can be given by C t = C t 1 + C t 2 , where C t is ST of TWRT, C t 1 = T twr 3 W log 2 1 + γ t 1 is throughput at node S 2 of TWRT, and C t 2 = T twr 3 W log 2 1 + γ t 2 is throughput at node S 1 of TWRT. The 1/3 is due to equal TT is consumed in three time slots. γ t 1 and γ t 2 are respectively SNRs at nodes S 2 and S 1 of TWRT, and they can be given by

Energy consumption model
In this subsection, the TEC model is given. It should be noted that the TEC contains TPs and CPs. To reduce the TEC, the system may not use entire block duration T t for transmission. In each block, each node has three states: transmission, reception, and idle [32], which corresponding to CP consumptions of three states are respectively P ct , P cr , and P ci . For node S 1 , S 2 , and R all have three kinds of CP consumptions, we use subscript ı ∈ {1, 2, r} to represent nodes S 1 , S 2 , and R, respectively. In such case, the CPs of three nodes with three states can be expressed as P ct ı , P cr ı , and P ci ı , respectively. Also, the SIC in TWRT needs to consume CPs P sic . In such case, the CP consumptions for SIC of nodes S 1 and S 2 in TWRT can be expressed as P sic 1 and P sic 2 , respectively. The non-ideal PA efficiency is μ and μ ≥ 1 [33].
From the above illustrations, the TEC is related to TPs, CPs, and PA efficiency, and TT can be known. Then, the TEC in DT is E d = T 1 (μP 1 + P d c1 − P Dci ) + T 2 (μP 2 + P d c2 − P Dci ) + T t P Dci , where P d c1 = P ct 1 + P cr 2 and P d c2 = P ct 2 + P cr 1 represent total CPs in the first and second time slots of DT, respectively. T t = T 1 + T 2 + T in DT and OWRT, where T is the time in idle state. P Dci = P ci 1 + P ci 2 represents total idle CP in DT.
where P o c1 = 1 2 P ct 1 + P cr 2 + P cr r + P ct r + P cr 2 + P ci 1 and P o c2 = 1 2 P ct 2 + P cr 1 + P cr r + P ct r + P cr 1 + P ci 2 represent total CPs in the first and second two time slots of OWRT, respectively. P Rci = P ci 1 + P ci 2 + P ci r represents total idle CP in RT.
The TEC in TWRT is E t = T twr μ(P 1 +P 2 +P r ) 3 where P t c1 = 1 3 P ct 1 + P cr 2 + P cr r , P t c2 = 1 3 P ct 2 + P cr 1 + P cr r , and P t c3 = 1 3 P ct r + P cr 1 + P cr 2 + P sic 1 + P sic 2 represent total CPs in the first, second, and third time slot, and T t = T twr + T in TWRT.

Problem formulation
In this subsection, EE definition and problem formulation are given. The EE in this paper is defined the same as [34], and it can be given as η = C T E T , where C T is ST and E T is TEC. From the throughput analysis and energy consumption model, it can be known that both E T and C T have relations with TPs and TT, and we need to maximize it by maximizing C T and minimizing E T with OTP and OTT. Thus, energy-efficient optimization problem can be summarized as For a more comprehensive analysis, we divide the optimization problem into three optimization cases. Firstly, maximizing C T to maximize η when TPs are equally allocated as [29,30]. Secondly, minimizing E T to maximize η when C T is constant as [22,23]. Thirdly, maximizing C T and minimizing E T simultaneously to maximize η. The EE analysis of these three cases will be given in the following three sections.

Maximize sum throughput
Firstly, when TPs are equally allocated, which means P 1 = P 2 = P r1 = P r2 = P r = P, we try to maximize C T with OTT to maximize η. TPs equally allocation is usually existed in practical system for simplicity, i.e., IS-95-CDMA system [35]. At the same time, the throughput is an important index that the communication systems want to pursue. Thus, the discussion about maximizing C T to maximize η with TPs equally allocation maybe can effectively improve the EE of the IS-95-CDMA system under a certain situation.

Optimal transmit time
In this subsection, the OTT analysis is given. It should be noted that all the CPs are a fixed power cost from 0 to hundreds of mW [36]. However, although the CPs are a fixed power cost, they are concerned with TT. Then, we need to maximize C T through OTT to maximize η. In such case, Propositions 1 and 2 about OTT to maximize η can be obtained.

Proposition 1
When P 1 = P 2 = P r1 = P r2 = P r = P and different nodes' CPs for transmit, receive, and idle are the same, i.e., P ct 1 = P ct 2 , P cr 1 = P cr 2 , and P ci 1 = P ci 2 , the OTT in DT and OWRT to maximize η is T = 0.
Proof: With P 1 = P 2 = P r1 = P r2 = P r = P and different nodes' CPs are the same constants, can be further obtained. In such case, EE of OWRT η o can be given as The (a) step is for dividing T 1 + T 2 , and (b) step is for T 1 + T 2 = T t and T = 0 to maximize η o . The same conclusion of the OTT in DT is T = 0 can also be obtained with P 1 = P 2 = P r1 = P r2 = P r = P and P d c1 The proof is completed.
Proposition 2 When P 1 = P 2 = P r1 = P r2 = P r = P, the OTT in TWRT to maximize η is also T = 0. Proof: With P 1 = P 2 = P r1 = P r2 = P r = P, EE of TWRT η t can be given as The (a) step is for dividing T twr 3 , and (b) step is for T twr = T t and T = 0 to maximize η t . The proof is completed.

Remark 1:
It should be noted that comparing with DT and OWRT, the OTT in TWRT to maximize η t has no constraints for CPs. At the same time, the OTT in it is always T twr = T t when TPs are not functions of TT even with unequally power allocation.

Optimal relay position
In this subsection, ORP analysis is given. Considering the Rayleigh fading channel, when three nodes are in a line and the distances of two source nodes between relay node are normalized by the distance between two source nodes S 1 and S 2 , the channel gains can be expressed as |h 3 [31], where d is the distance between node S 1 and R, α is the path loss attenuation factor, and α ∈[ 2, 5] [37].
From (2), we can see that only γ o 1 is needed to be maximized with ORP to maximize η o when T = 0. Then, Proposition 3 about ORP in OWRT can be obtained.
From (3), we can also know that only the sum SNR of TWRT γ t = γ t 1 + γ t 2 + γ t 1 γ t 2 is needed to be maximized with ORP to maximize η t when T = 0. However, it also should be noted that comparing with OWRT, the o 1 and o 2 will influence γ t 1 and γ t 2 . Then, Proposition 4 about optimal o 1 , o 2 , and ORP in TWRT can be obtained.  The proof is completed.

Outage probability analysis
In this subsection, outage probability analysis is given for it is an important criterion to measure the performance of a link. With C th to avoid disruption for a given throughput, then C (·) < C th denotes the outage event, and Pr(C (·) < C th ) denotes the outage probability.
Let γ = P 1 = P 2 = P r1 = P r2 = P r = P, C d can be reformulated as It should be noted that bidirectional transmission system is in outage if any end-to-end transmission is in outage. The outage event of DT is given by , where T min = min{T 1 , T 2 } in DT and OWRT. Statistically, the variances of channel gains h l are δ 2 l . For Rayleigh fading, i.e., |h l | 2 are exponentially distribution with parameter δ −2 l [38], where subscript l ∈ {1, 2, 3} represent different channel gains. Then, λ 1 = δ −2 1 , λ 2 = δ −2 2 , and λ 3 = δ −2 3 can be obtained. Also, when x tends to 0, an equivalent infinitesimal approximation 1 − e −x ≈ x can be obtained. With them, outage probability of DT p out d can be calculated as With γ , C o can be reformulated as where f (x, y) = xy x+y . The outage event of OWRT is given by min{C o 1 , C o 2 } < C th , and it is equivalent to the event Assuming according to [38], the following equation can be obtained as With it, we can further obtain the following equation as Finally, the outage probability of OWRT p out o can be calculated as With γ , C t can be reformulated as where , the outage event of TWRT is given by min{C t 1 , C t 2 } < C th , and it is equivalent to the event where f 1 (x, y) = xy 3x+y = 1 3 . 3xy 3x+y = 1 3 f (3x, y). The above equivalent event give us an example when C t 1 ≤ C t 2 to get the outage probability of TWRT and we can use the same method to get the outage probability of TWRT when C t 1 > C t 2 . Assuming 3|h 1 T twr W − 1 m, and according to [38], the following equation can also be obtained as With it, we can further obtain the following equation as Finally, the outage probability of TWRT p out t can be calculated as With the same method to get (16), we can know that when C t 1 > C t 2 , the outage probabil-

Minimize total energy consumption
Secondly, when C T is constant with the assumption of C 1 = βC T and C 2 = (1−β)C T , we try to minimize E T with OTT and OTP to maximize η. The β ∈ (0, 1) is the transmission task distribution factor. C 1 and C 2 are the minimum transmission tasks at two directions  [22]. Minimizing TEC is one of the widely used methods to achieve green communication [1]. At the same time, the OPA has been widely studied in many practical wireless systems, i.e., with OPA to achieve EET of cellular system [19] and with OPA to maximize the average throughput and EE of energy harvesting system [21]. Thus, the discussion about the minimizing E T to maximize η with joint optimization of TT and TP allocation maybe can effectively improve the EEs of the cellular and energy harvesting systems.

Energy-efficient direct transmission
In this subsection, the problem with minimizing E d to maximize η d is given. With the requirements of C 1 , C 2 , maximum TP P max t , and total TT T t , EET in DT can be given as follows: Through throughputs, the TPs can be expressed as functions of TT, then the minimum TECs can be obtained with only optimizing TT. For throughputs are derived from Shannon capacity formula, which means the maximum achievable throughput can be obtained under given TPs, thus, the TPs derived through throughputs are also the minimum which can support the required transmission tasks.
can be obtained. Then through γ d 1 = P 1 |h 3 | 2 and γ d 2 = P 2 |h 3 | 2 , the minimum TPs in DT can be obtained as It can be seen that the optimal P opt 1 and P opt 2 are respectively increasing functions of γ d 1 and γ d 2 ; thus, they are respectively decreasing functions of T 1 and T 2 . As T 1 and T 2 decrease, the optimal P opt 1 and P opt 2 increase, and they may achieve P max t . To simplify the analysis, we only consider all the TPs that will not achieve P max t situation, and the TPs that achieve P max t situation can be seen in our recent work [27]. With (18), the (17) can be reformulated into only optimizing TT problem and it can be given by In (19), the term T t P Dci is independent of TT, and the terms are only concerned with T 1 and T 2 , respectively. Then, the second order derivative (SEC) E d 1 can be obtained.
With E d 1 (T 1 ) ≥ 0 and E d 2 (T 2 ) ≥ 0, E d 1 and E d 2 that are respectively convex functions of T 1 and T 2 can be known. Therefore, the objective function is convex with respect to T 1 and T 2 for the addition of two convex functions is still a convex function [39]. The Algorithm 1 Gradient-descent Algorithm 1: Given a starting point t i ∈[ 0, T t ], and tolerance ε. 2: Repeat 3: Obtain t i :=− f (t i ) according to (19a), (23a), (23b), and (26a). 4: Choose step size n i . 5: Update t i := t i + n i t i . 6: convex functions can be solved using standard solvers, such as inter-point methods, Newton method, and gradient-descent method. For the fast convergence of gradient-descent algorithm [39], we use it in this paper. For each t i , the gradient-descent algorithm can be summarized as Algorithm 1.
In Algorithm 1, we give an initial point t i ∈[ 0, T t ]. Then, it iteratively moves toward lower values of the functions by taking steps in direction of negative gradient − f (t i ) with chosen step size n i , and− f (t i ) can be obtained according to (19a), (23a), (23b), and (26a). Eventually, Algorithm 1 converges to the minimum for the functions are convex.

Energy-efficient one-way relay transmission
In this subsection, with minimizing E o to maximize η o problem is given. The EET in OWRT can also be given under the requirements of C 1 , C 2 , P max t , and T t . Observing E o , it can be found that the terms + P o c2 − P Rci are respectively only concerned with T 1 and T 2 , and the term T t P Rci is independent of TT. To make a further observation, it can be found that γ o 1 is not a function of P 2 and P r1 , while γ o 2 is not a function of P 1 and P r2 . Therefore, the optimization problem in OWRT can be equivalently reformulated into two subproblems as min P 1 ,P r2 ,T 1 With it, the TP in (20a) can be represented as P 1 + P r2 = P 1 + f (P 1 ). By setting the derivative of P 1 + f (P 1 ) to zero and with the help of its SEC, the optimal P opt 1 and P opt r2 with minimizing P 1 +P r2 can be obtained as where c 1 = |h 1 | 4 , c 2 = (|h 2 | 2 |h 3 | 2 + |h 1 | 2 h 32 ), h 32 = |h 2 | 2 − |h 3 | 2 , h 31 = |h 1 | 2 − |h 3 | 2 , and h 13 = |h 1 | 2 + |h 3 | 2 .
Following the same solving procedure as (20a), the optimal P opt 2 and P opt r1 with minimizing P 2 +P r1 can be obtained as , and h 23 = |h 2 | 2 + |h 3 | 2 . With (21) and (22), the (20) can be reformulated as where P . Then, the SEC can be obtained. With

Energy-efficient two-way relay transmission
In this subsection, with minimizing E t to maximize η t problem is given. The EET in TWRT can also be given under the requirements of C 1 , C 2 , P max t , and T t . At the same time, P 1 = P 2 = P in TWRT can be assumed as [31]. In such situation, the EET in TWRT can be given as follows: Following the same solving procedure as (20a), the optimal P opt and P opt r with minimizing 2P + P r can be obtained as  1 2 ), and B = c 7 (c 5 + c 6 ) + |h 2 | 2 (c 6 c 7 (c 5 + c 6 )) 1

.
With (25), the (24) can be reformulated as where C = 2P opt +P opt . Then, the can be obtained.

Remark 2:
There is a pity that the analytic expression of OTT cannot be obtained for the existence of exponential terms of TT, and that is why we use Algorithm 1 to get the OTT in this section.

Asymmetry and asymmetry transmission tasks
In this subsection, the influences of asymmetry and asymmetry transmission tasks at two directions are given. With β to show the transmission task distribution factor at two directions, Proposition 5 can be obtained.

Maximize energy efficiency
Thirdly, we try to maximize C T and minimize E T simultaneously to maximize η with the proposed optimal algorithms. Maximizing EE is a more scientific method to achieve green communication, and many communication techniques have been studied to improve the EEs of different wireless communication systems, i.e., the MIMO system [1,32]. Thus, the discussion about maximizing C T and minimizing E T simultaneously to maximize η maybe also can effectively improve the EE of the MIMO system. Actually, the proposed optimal algorithms in this section can also be used in the communication systems when the multiple optimal variables cannot be obtained simultaneously [40]. The usage of the proposed optimal algorithms in the communication systems with multiple optimal variables is only to try to show its application in the existing works, and the further detailed discussions about whether the usage can improve the performance of the related systems can be studied in the further work. This is the same as Sections 4 and 5 with the application of their solutions in the related systems.

Energy-efficiency maximization optimal algorithms
In this subsection, EE maximization optimal algorithms are given. First, maximizing η t is given for we have proved in Remark 1: that the OTT in it is always T twr = T t when the TPs are not functions of TT. Then maximizing η t problem is only concerned with TPs. With (3) and the optimization constraints of (24), maximizing η t problem can be given as where P c t = 3(P t c1 + P t c2 + P t c3 ). At the same time, with P 1 = P 2 = P, o 1 = o 2 = 0.5 in TWRT [31], ζ 1 , and ζ 2 , In (27), η t is non-convex in (P, P r ) for the numerator of η t is concave and the denominator of η t is linear with respect to P and P r , respectively. At the same time, η t is differentiable; thus, it is pseudo-concave in terms of P and P r , respectively. Since for any optimization problems, we can first optimize over some of the variables and then over the remaining ones [40]. Then, we can divide it into two sub-optimization problems which optimizing P and P r in alternative. The pseudo-concave properties can be proved with Hessians of them, i.e., 2 η t (P) ≤ 0 and 2 η t (P r ) ≤ 0.
For the pseudo-concave properties of η t , it can be seemed as , where f (x) is concave and g(x) is linear. Define the function F(ψ) as F(ψ) = max x∈S {f (x) − ψg(x)} with continuous and positive f, g, and compact S, then F(ψ) is convex with respect to ψ; also, it is strictly decreasing and it has a unique root ψ * . With it, the problem of finding F(ψ) can be solved with convex optimization approaches and it is shown that the problem of maximizing is equivalent to finding ψ * [36]. At the same time, since the original problem has been divided into two sub-optimization problems, it can be solved one-by-one. For each x, Dinkelbach's algorithm is employed for its properties in solving non-convex fractional programming problems. Dinkelbach's algorithm can be summarized as Algorithm 2, where the superscript (n) denotes the iteration number.
Algorithm 3 leads to global optimal values of each pseudo-concave function. In this regard, for TWRT, firstly, with P (n) , Algorithm 2 is adopted in order to find P (n+1) r . Secondly, with known P (n+1) r , P (n+1) is computed. Consequently, the one-by-one optimal algorithm is required in order to optimize P and P r , simultaneously. Algorithm 3 presents the one-by-one optimal alternating procedure which updates the optimization parameters until convergence.
Second, maximizing η d and η o problems are given. With C d (P 1 , P 2 , T 1 , T 2 ), E d (P 1 , P 2 , T 1 , T 2 ), and the optimization constraints of (17), maximizing η d problem can opt ). 5: . 6: n ← n + 1. Given P Observing (28), it can be found that the OTT and OTP cannot be obtained simultaneously for the TT and TPs have multiplicative terms. But we also can use Algorithms 2 and 3 to get them one-by-one. However, in practice, there is an optimization order problem with one-by-one optimization method especially for TT and TPs are all existed in it. To simplify the theoretical analysis, we let T 1 = T 2 = T = 1 2 T t in DT and OWRT for it has been suggested in [23] that with high transmission tasks the system will use entail where γ d 1 = P 1 |h 3 | 2 , γ d 2 = P 2 |h 3 | 2 , and P c d = P d c1 + P d c2 . Following the same analysis method as TWRT, and with 2 η d (P 1 ) ≤ 0, 2 η d (P 2 ) ≤ 0, (29a) which are respectively pseudo-concave in terms of P 1 and P 1 can be proved. Then, (29) can be solved with Algorithms 2 and 3 as TWRT.

Complexity analysis
In this subsection, the complexity analysis of the optimal algorithms is given. To analyze the computational complexity of Algorithm 3, we should note that Algorithm 3 employs Algorithm 2. But it also should be noted that the convergence rate of Algorithm 2 is independent of the complexity of finding x (n) opt for its super linear convergence. As the problems of finding x

Simulation results
In this section, simulations are conducted to confirm the validity of theoretical analysis. The simulation parameters are given as Table 2. Considering the Rayleigh fading channel, |h 3 | 2 = 1, |h 1 | 2 = d −α , and |h 2 | 2 = (1 − d) −α , where d ∈ (0, 1) are also used in the simulations [31]. Some of the parameters in the simulations are set to a constant, i.e.,

Maximize sum throughput
In this subsection, EEs with maximizing C T are given. At the same time, TPs are equally allocated. In Fig. 2, EEs with and without idle state are given which correspond to T = 1ms and T = 0, respectively. The with and without idle state corresponds to "w" and "w/o" in Fig. 2, respectively. From it, we can find that EEs when T = 1 ms are lower than those when T = 0, which corresponds to Propositions 1 and 2 that the OTT both in DT and RT to maximize EE is T = 0. In Fig. 3  system in [23] are also given. From it, we can know that EEs with DLs in RT are higher than those of RT-NDL, which shows the effectiveness of DLs in RT for its SE performance gain. In Fig. 4, EEs of RT with different d are given. In Fig. 4a, it can be found that EE of OWRT is the maximum when relay node lies in the middle of two source nodes, which has been proved in Proposition 3 for in such case γ o 1 is the maximum. In Fig. 4b, it can be found that the smaller the d the higher the EE of TWRT, which also has been proved in Proposition 4 for in such case γ t is bigger.
In Fig. 5, outage probabilities are given. The simulation results of outage probabilities are obtained with (6), (11) and (16). From Fig. 5, it can be found that outage performance of TWRT is the best and DT is the worst. This is because with the given C th to avoid disruption, the ST of TWRT is the best and ST of DT is the worst.

Minimize total energy consumption
In this subsection, EEs with minimizing E T are given. At the same time, TPs are optimally allocated. In Fig. 6, EEs when T = 1 ms and T = 0 are given. The T = 1 ms corresponds to the DT, OWRT, and TWRT with idle state, and it means the system can transfer into idle state when the transmission tasks have been completed. The T = 0 ms corresponds to the DT, OWRT, and TWRT without idle state, and it means the system always uses the entail block duration for transmission. The with and without idle states correspond to "w" and "w/o" in the Fig. 6, respectively. From Fig. 6, the following results can be found: (i) EEs when T = 0 are lower than those when T = 1 ms when transmission rate is low; (ii) EEs when T = 0 are equal to those when T = 1 ms when transmission rate is high. This is because when the transmission task is low and it has been completed, the without idle mode still uses entail block duration T t for transmission. In such case, TPs and other CPs, such as transmit and receive CPs, keep consuming. However, the with idle mode changes to the idle mode. In such case, TPs and other CPs can be saved even with idle CPs increase. However, idle CPs are smaller than TPs and other CPs. Finally, the EEs with idle mode are higher. This shows the necessity of TT optimization. While when the transmission rate is high, both with and without idle modes will use entail T t for transmission, and the EEs of them are the same. In Fig. 7, EEs with different β are given. From Fig. 7, firstly, it can be found that β does not influence the EEs of DT and OWRT, but it influences the EE of TWRT. Secondly, it can be found that the bigger the β, the lower the EE of TWRT. All of these have been proved in Proposition 5. In Fig. 7, EE of RT-NDL is also given. From it, we can know that EE of RT-NDL is the worst in RT even it compares with EE of TWRT when β = 0.6, which also shows the effectiveness of DLs in RT.
Comparing EEs in minimizing E T with maximizing C T , two results can be found: (i) EEs are no more a decreasing function. This is because the OTT in maximizing C T is always T = 0. While in minimizing E T , when the transmission tasks have been completed, the transmission models can change into idle state. Then the influences of CPs become more obvious, when SE is low, the CPs are bigger than TPs and EEs are increasing functions. While when SE is high, TPs are much bigger than CPs, the influences of CPs can be ignored, and EEs are decreasing functions; (ii) EEs of OPA are higher than those of equal power allocation, which shows the significance of OPA.
In Fig. 8, EEs with maximizing C T and minimizing E T simultaneously are also given. From Fig. 8, it can be found that EEs of maximizing C T and minimizing E T simultaneously are higher than that of only maximizing C T or only minimizing E T . This is because the OTPs in it need to consider C T and E T , simultaneously. In Fig. 8, EE of RT-NDL which is the worst in RT can also be obtained. From Figs. 2, 3, 4, 5, 6, 7, and 8, it can be found that among different transmission models, EE of TWRT is almost always the maximum and EE of DT is almost always the minimum. The TWRT has the maximum EE both for relay node's assistance and its time delay advantage. Also, it can be found that EE of RT-NDL is lower than the other RT with DLs. These two phenomenons suggest that relay technique and DLs in RT can improve system's EE.

Conclusion
In this paper, throughput, energy consumption, and EE with consideration of non-ideal PA efficiency, non-negligible CP consumption, and DLs in RT, through joint optimization of TPs and TT allocation, have been analyzed. The EET with three optimization cases have been given, and several characteristics could be found: (i) with maximizing C T to maximize EE, EEs without idle state are higher, EE of OWRT when relay node is closer to the middle of two source nodes is higher, and EE of TWRT when relay node is closer to node S 1 and o 1 = 0.5 is higher. At the same time, outage probability of TWRT is the best; (ii) with minimizing E T to maximize EE, EEs without idle state are lower when SE is low and transmission tasks ratio only influences the EEs of TWRT; (iii) with Dinkelbach's algorithm and one-by-one optimal algorithm to maximize EE, EEs are higher than that of only maximizing C T or only minimizing E T ; (iv) relay technique and DLs in RT can improve system's EE. All of these have been verified by the theoretical analysis and the simulations. For the energy harvesting property of SWIPT, the EEs of these three cases with SWIPT can be studied in the further work.