Adaptive Time Slot Resource Allocation in SWIPT IoT Networks

1  Introduction

With the continuous development in communication technology, Internet of Things (IoT) has been gradually integrated into many aspects of public life (e.g., public safety, smart city, enemy reconnaissance, and many other fields) [1–6]. According to statistics, the number of Internet of Things devices (IoDs) may reach 30 billion by 2025. The widespread application of IoT enables the communication between people and things and among things at any time and place [7,8]. In addition to increasing convenience, it also promotes intelligence development in public life. Thus, the proliferation of IoT has greatly improved the quality of life, and it is of great practical significance to study IoT.

Although IoT has become an irreplaceable part of public life, the IoT system faces challenges, such as limited energy storage of the IoDs and the scarcity of spectrum resources [9]. The emergence of simultaneous wireless information and power transfer (SWIPT) technology has provided a new method of solving the problem of the limited energy of the IoDs [10]. The application of the SWIPT technology allows IoDs to harvest energy from radio frequency (RF) waves and convert it into electrical energy, which can be stored in the battery of the device, thus maximizing the usable time of the device [11,12] and reducing the additional cost of battery replacement or manual charging of the device [13]. However, there is a limit to the amount of energy that can be stored in the battery, and excessive energy harvesting puts a greater burden on the operation of the device. Less than the appropriate amount of energy disrupts the normal operation of the device, and adequate energy harvesting using SWIPT IoT enables the entire network system to continue operating. Meanwhile, the normal communication of IoDs requires the support of large amounts of the radio spectrum, and the recent shortage of spectrum resources has restricted the development of IoT [14]. Therefore, reasonable allocation of the limited spectrum resources has become the focus of research in regard to SWIPT IoT.

A number of recent studies have considered the issue of resource allocation in regard to energy harvesting in SWIPT IoT. Zhu et al. [15] determined the user maximized minimum energy harvesting subject to the secrecy rate and total transmit power constraints in the presence of channel estimation errors, ensuring the fairness of user energy harvesting. Masood et al. [16] proposed an algorithm to achieve a compromise between energy efficiency and spectral efficiency in the SWIPT system, which is based on using the Lagrange multiplier method to find the optimal solution without iterative computation. Simulation results proved that the algorithm achieved the maximum energy transmission by using the optimal power split ratio. Han et al. [17] studied fair resource allocation in terms of the maximum–minimum energy requirement for wireless power transmission in edge computing systems. They solved the nonconvex mixed-integer nonlinearity that arises when using a logarithmic nonlinear energy harvesting model by using a continuous relaxation method. Yazdani et al. [18] proposed a parametric power control method that allows each sub-user to adjust its power according to the stored energy such that the lower limit of the sum of the uplink rates is maximized. Xu et al. [19] studied the resource allocation problem with perfect channel state information under the constraints of minimum energy harvesting and quality of service. Qi et al. [20] considered channel uncertainty, receiver nonlinearity, and decoding message cancellation for imperfect continuous messages in SWIPT-based non-orthogonal multiple access (NOMA) large scale IoT in the process of designing a resource allocation method.

Researchers have also considered joint optimization and allocation of multiple resources in SWIPT IoT. Prathima et al. [21] considered a collaborative cognitive radio (CR) network with two primary and secondary users, wherein power splitting and time-switching energy harvesting techniques were used and the system throughput and energy efficiency were maximized by using a particle swarm optimization algorithm. Acosta et al. [22] jointly considered the energy harvesting of the system, data rate, and the maximum power level allowed to interfere with the users to minimize the transmit power of the auxiliary base station. They solved the external and internal optimizations using a particle swarm optimization algorithm and a semidefinite relaxation algorithm, respectively. Tuan et al. [23] investigated resource allocation with respect to maximizing the total energy harvesting and energy harvesting efficiency under linear and nonlinear energy harvesting models in a SWIPT system consisting of a multi-input single-output interference channel to jointly optimize the beam formation vector of the transmitter and the power splitting ratio of the receiver. Hu et al. [24] maximized the secrecy rate in a multi-input multi-output bidirectional relay-assisted CR NOMA network by jointly optimizing the power allocation of the users and power splitting factor under the constraints of quality of service, energy harvesting, and transmission power. Tang et al. [25] jointly optimized the transmission rate and harvested energy in a SWIPT-enabled NOMA system using power splitting to simultaneously fulfill the minimum rate and harvested energy requirements for each user. Yang et al. [26] used resource management methods based on deep reinforcement learning to jointly optimize radio allocation and control strategies of transmission power in cognitive IoT. Ramzan et al. [27] simultaneously optimized device selection, unmanned aerial vehicle (UAV) relay allocation, and power splitting ratio in an IoT system based on UAV relay communication with energy harvesting, where the optimization results showed significant advantages in the above aspects. Xiao et al. [28] maximized the energy efficiency in RF energy-driven CR networks by jointly optimizing transmission time and power control and also proposed a resource allocation scheme called the co-channel interference approximation convex strategy. These studies investigated resource allocation in regard to SWIPT IoT and obtained adequate results. However, most researchers only focused on the uplink or downlink information transmission in SWIPT IoT systems without considering that most IoDs required alternating uplink and downlink information transmissions in real-time. This problem can be addressed by jointly allocating resources for uplink and downlink information transmissions and energy harvesting in SWIPT IoT such that the results obtained can be more in line with actual situations.

The main contributions of this study are summarized as follows:

•   An adaptive time slot (ATS) structure is designed that consists of a sensing time period at the beginning, time periods for several alternating uplink and downlink information transmissions, and a common time period for downlink energy harvesting and information transmission. The ATS structure allows the IoD to adaptively allocate time in a time slot to obtain the maximum uplink and downlink throughputs while satisfying the minimum energy harvesting requirements.

•   A SWIPT IoT ATS resource allocation (SIATS) algorithm is proposed. The algorithm first solves the coupling problem in the optimization function by using the method of pre-setting sensing time, time allocation factor, and time slot allocation parameters and transforms the optimization function into a convex function. Second, the optimal transmit power and channel assignment of the system is obtained by using the Lagrangian dual and gradient descent methods. Finally, by comparing the throughput results with different sensing times, time allocation factors, and time slot allocation parameters, the optimal time slot allocation for each IoD is determined such that the sum of the throughputs of all IoDs in the system is maximized.

•   The simulation compares the system throughput and energy harvesting when using the SIATS algorithm proposed in this paper with the fixed time slot allocation (FTS) algorithm, and analyzes in detail the effects of parameters such as sensing time, time allocation factor, and noise on the total system throughput and energy harvesting.

2  System Model

The SWIPT IoT network system is shown in Fig. 1, and it is composed of B IoT base stations, denoted as BS1,BS2,…,BSB. Each base station possesses a certain communication range and can communicate with and provide energy to the IoDs within its communication range in real-time. Each IoD is equipped with an antenna, an energy harvesting module, and a battery that can be charged with the power provided by the energy harvesting module.

Figure 1: SWIPT IoT network system

The spectrum resources of the SWIPT IoT network system are divided into C available channels, denoted by the set C={CH1,CH2,…,CHj,…,CHC}. The IoT base stations in this network can sense the available spectrum resources to ensure that different frequencies are used for the IoDs in their respective coverage areas. Let us suppose that the communication range of BS1 contains D IoDs, denoted by the set D={IoD11,IoD12,…,IoD1i,…,IoD1D}, where IoD1i denotes the ith IoD in the communication range of BS1. The communication range of BS2 contains E IoDs, denoted by the set E={IoD21,IoD22,…,IoD2i,…,IoD2E}, where IoD2i denotes the ith IoD in the communication range of BS2.

The communication mode between the IoT base station and IoD is the time division duplex, for which an ATS structure is designed, as shown in Fig. 2.

Figure 2: Adaptive time slot structure

The ATS consists of periods for sensing, uplink information transmission, downlink information transmission, and energy harvesting. The total duration of the time slot is T, where the first time period is used for sensing, and the duration is ts. The subsequent T−ts durations are divided into L time periods of equal length, which can be denoted as t1,t2,…,tL−1,tL. The time periods t1 to tL−1 are primarily used for information transmission, where m time periods are used for the uplink information transmission and n time periods are used for the downlink information transmission, and the values of m and n in a time slot are determined adaptively. Thus, the duration of the uplink information transmission in time periods t1 to tL−1 can be calculated by the following equation:

Tu=mT−tsm+n+1(1)

Furthermore, the duration of the downlink information transmission in the time period t1 to tL−1 can be calculated by the following equation:

To,d=nT−tsm+n+1(2)

The time period tL in the time slot is the common time for energy harvesting and downlink information transmission. The time period tL is split by a time allocation factor α∈[0,1]. This parameter varies with the channel conditions and IoD requirements in different time slots. Thus, the duration of time to perform energy harvesting in time period tL can be calculated as:

Tc,e=αT−tsm+n+1(3)

The duration of time for the downlink information transmission is calculated as:

Tc,d=(1−α)T−tsm+n+1(4)

Finally, the duration of the complete time slot for the downlink information transmission is calculated as:

Td=To,d+Tc,d=(n+1−α)T−tsm+n+1(5)

The following is an illustration of the IoT SWIPT network communication considering the IoT base station BS1 as an example; let us assume BS1 and BS2 are adjacent base stations, as shown in Fig. 1. Each IoD within the communication range of BS1 senses the state of all channels at the beginning of each time slot, and the results of the sensing are categorized into two cases.

(1) The first case: The IoD within the communication range of BS1 senses that the channel is idle. The channel is not used by the IoD within the communication range of BS1, and it is also not used by the IoD within the adjacent base station BS2. The probability of this case is assumed to be pjtrue.

(2) The second case: The IoD within the communication range of BS1 senses the channel is idle. The channel is not used by the IoD within the communication range of BS1, but it is used by the IoD within the neighboring base station BS2. The probability of this case is assumed to be pjfalse.

The two cases are discussed separately in the following section.

2.1 The First Case

Based on the previous assumptions, the probability pjtrue is computed as:

pjtrue=p(Hj(0))(1−perror,j(ts)),(6)

where p(Hj(0)) denotes the probability that CHj is idle and perror,j denotes the probability of a false alarm for CHj, which can be computed as follows:

perror,j(ts)=F(2SNRj+1⋅F−1(pp)+tsfsSNRj),(7)

where SNRj denotes the signal-to-noise ratio of the signal received by the IoD using CHj communication, fs denotes the sampling frequency of the signal, and F(⋅) denotes the standard Gaussian complementary cumulative distribution function, which can be expressed as follows:

F(x)=∫x∞12πe(−t22)dt(8)

Notably, F−1(⋅) is the inverse function of F(⋅).

The IoD within the communication range of BS1 selects one or more available channels to communicate with BS1 after the sensing, and this system ensures that one channel is available for each IoD. During the communication process between BS1 and the IoD, the uplink is primarily used to transmit the information from the IoD to BS1.

The rate of transmission corresponding to IoD1i for the uplink information transmission using CHj in this case can be determined using Shannon's formula, as follows:

ri,jtrue(Tu)=Wlog2⁡(1+Pi,j,u1hi,jN),(9)

where the superscript true in ri,jtrue denotes the first case; the subscript i represents the device IoD1i, and the subscript j represents the channel CHj used by the device. Pi,j,u1 denotes the transmit power of IoD1i when using CHj for the uplink information transmission, where the superscript 1 denotes the communication range of BS1 the subscript i denotes IoD1i, the subscript j denotes CHj used by the device, and the subscript u indicates uplink information transmission. In the above equation, W denotes the channel bandwidth, hi,j denotes the channel gain for communication between BS1 and IoD1i using CHj, and N denotes the variance of the additive Gaussian white noise.

The downlink in the communication process is primarily used to send data from BS1 to the IoD within its communication range and to provide energy to the IoD when needed. Thus, the rate of transmission for the downlink information transmission between IoD1i and BS1 using CHj is computed as:

ri,jtrue(Td)=Wlog2⁡(1+Pi,j,d1hi,jN),(10)

where Pi,j,d1 denotes the transmit power for the downlink information transmission between BS1 and IoD1i using CHj, wherein the subscript d indicates downlink information transmission.

2.2 The Second Case

Based on the previous assumptions, the probability pjfalse is computed as:

pjfalse=p(Hj(1))(1−pp),(11)

where p(Hj(1)) is the probability that CHj is busy and p(Hj(1))=1−p(Hj(0)). pp is the probability of target detection that adequately protects the IoD.

In this case, the signal emitted by BS2 during the uplink information transmission with IoD2i interferes with the uplink information transmission using the co-channel within the communication range of BS1. Therefore, the rate of transmission corresponding to IoD1i during the uplink information transmission using CHj is computed as:

ri,jfalse(Tu)=Wlog2⁡(1+Pi,j,u1hi,jPi,j,u2gi,j+N),(12)

where the superscript false in ri,jfalse denotes the second case. Pi,j,u2 denotes the transmit power received by BS1 when IoD2i uses the co-channel of CHj for the uplink information transmission; the superscript 2 denotes the communication range of BS2, the subscript i denotes the device IoD2i, the subscript j denotes the co-channel of CHj, and the subscript u indicates uplink information transmission. Additionally, gi,j denotes the channel gain for communication between IoD2i and BS1 performed using the co-channel of CHj.

The uplink interference generated by IoD1i and BS1 using CHj for the uplink information transmission to IoD2i within the communication range of BS2 using the co-channel of CHj can be computed as:

Ij,u=TuTpjfalse∑i=1Dli,jPi,j,u1ϕi,j,(13)

where li,j is the channel gain for communication between IoD1i to BS2 performed using the co-channel of CHj. ϕi,j={0,1} is used to indicate whether IoD1i uses CHj, and it is called the channel assignment factor. When IoD1i uses CHj, ϕi,j=1; conversely, ϕi,j=0.

The uplink throughput of IoD1i on CHj can be expressed as:

ri,j,u=TuT[pjtrueri,jtrue(Tu)+pjfalseri,jfalse(Tu)](14)

The rate of transmission for the downlink information transmission between IoD1i and BS1 performed using CHj is computed as:

ri,jfalse(Td)=Wlog2⁡(1+Pi,j,d1hi,jPi,j,d2li,j+N),(15)

where Pi,j,d2 denotes the transmit power of BS2 when the co-channel of CHj is used for the downlink information transmission.

The downlink interference generated by BS1 and IoD1i using CHj for the downlink information transmission to BS2 using the co-channel of CHj can be computed as:

Ij,d=TdTpjfalse∑i=1Dgi,jPi,j,d1ϕi,j(16)

The downlink throughput can be expressed as:

ri,j,d=TdT[pjtrueri,jtrue(Td)+pjfalseri,jfalse(Td)](17)

2.3 Energy Harvesting

For energy harvesting in the downlink, a linear energy harvesting model is used in this study. Thus, the energy harvested by IoD1i using CHj can be expressed as:

Ei,j=βTc,eT[hi,j(pjtrue+pjfalse)Pi,j,d1+N]ϕi,j,(18)

where β is the energy efficiency conversion constant, and β∈[0,1].

It is assumed in this study that the IoD can simultaneously use more than one channel to communicate with the base station. Therefore, the total energy harvested by IoD1i in a time slot can be expressed as:

Ei=∑j=1CEi,j(19)

3  SWIPT IoT ATS Resource Allocation Algorithm

In order to achieve an optimal system, a joint optimization is performed to maximize the sum of the uplink and downlink throughputs of all IoDs within the communication range of BS1. This optimization problem can be stated as follows:

max∑i=1D∑j=1Cϕi,j(ri,j,u+ri,j,d)(20a)

s.t.TdTpjfalse∑i=1Dgi,jPi,j,d1ϕi,j≤Imax,j,d,j∈C(20b)

TuTpjfalse∑i=1Dli,jPi,j,u1ϕi,j≤Imax,j,u,j∈C(20c)

TdT∑i=1D∑j=1C(pjtrue+pjfalse)Pi,j,d1ϕi,j≤Pmax,d(20d)

TuT∑j=1C(pjtrue+pjfalse)Pi,j,u1ϕi,j≤Pmax,i,u,i∈D(20e)

∑j=1CβTc,eTϕi,j[hi,j(pjtrue+pjfalse)Pi,j,d1+N]≥Emin,i,i∈D(20f)

∑i=1Dϕi,j=1,j∈C(20g)

0<ts≤T(20h)

0<α≤1(20i)

ϕi,j={0,1},i∈D,j∈C(20j)

Pi,j,u1≥0,i∈D,j∈C(20k)

Pi,j,d1≥0,i∈D,j∈C(20l)

Here, Imax,j,d denotes the maximum acceptable downlink interference in the communication range of BS2, Imax,j,u denotes the maximum acceptable uplink interference in the communication range of BS2, Pmax,d is the maximum value of the transmit power of BS1, Pmax,i,u is the maximum value of the transmit power of IoD1i and Emin,i is the minimum energy to be harvested by IoD1i in the time slot.

In this optimization problem, the constraint represented by Eq. (20b) indicates that the interference of the downlink information transmission within the communication range of BS1 to the downlink information transmission in BS2 using the co-channel is less than the maximum acceptable threshold for downlink interference. The constraint represented by Eq. (20c) indicates that the interference of the uplink information transmission within the communication range of BS1 to the uplink information transmission in BS2 using the co-channel is less than the maximum acceptable threshold for uplink interference. The constraint represented by Eq. (20d) indicates that the sum of the power transmitted by BS1 to the IoD within its communication range is less than the maximum transmit power of the base station. The constraint represented by Eq. (20e) indicates that the sum of the transmit power of each IoD for the uplink information transmission over the available channel is less than the maximum transmit power of the device. The constraint represented by Eq. (20f) indicates that the sum of the energy harvested by each IoD through its available channels is greater than the minimum energy harvesting requirement. The constraint represented by Eq. (20g) indicates that each channel cannot be used by more than one IoD. The constraint represented by Eq. (20h) provides the range of values for sensing time. The constraint represented by Eq. (20i) provides the range of values for the time allocation factor; The constraint in Eq. (20j) provides the range of values for the channel allocation factor. Finally, the constraints in Eqs. (20k) and (20l) ensure that neither the uplink nor downlink transmit power is less than zero.

To solve the optimization problem, a SIATS algorithm is proposed in this study that can be described as follows.

The channel assignment factor ϕi,j={0,1} in Eqs. (20a)–(20g) renders the original optimization problem a mixed-integer programming problem, which causes the original problem to become intractable. Therefore, the value of the channel assignment factor ϕi,j can be relaxed as ϕi,j=[0,1] in the derivation process. In this study, we introduce the auxiliary variables φi,j,d=ϕi,jPi,j,d1, φi,j,u=ϕi,jPi,j,u1, which can be incorporated into Eqs. (9), (10), (12), and (15) to obtain the following four equations:

ri,jtrue(Tu)=Wlog2⁡(1+φi,j,uhi,jϕi,jN)(21)

ri,jtrue(Td)=Wlog2⁡(1+φi,j,dhi,jϕi,jN)(22)

ri,jfalse(Tu)=Wlog2⁡(1+φi,j,uhi,jϕi,j(Pi,j,u2gi,j+N))(23)

ri,jfalse(Td)=Wlog2⁡(1+φi,j,dhi,jϕi,j(Pi,j,d2li,j+N))(24)

According to Eqs. (14) and (17), the optimization objective Eq. (20a) can be expressed as:

ψ=∑i=1D∑j=1Cϕi,j{TuT[pjtrueri,jtrue(Tu)+pjfalseri,jfalse(Tu)]+TdT[pjtrueri,jtrue(Td)+pjfalseri,jfalse(Td)]}(25)

Subsequently, upon incorporating Eqs. (21)–(24) into the optimization problem in Eq. (20), it can be transformed into the formulation expressed by Eqs. (26a)–(26n):

maxψ(26a)

s.t. TdTpjfalse∑i=1Dgi,jφi,j,d≤Imax,j,d,j∈C(26b)

TuTpjfalse∑i=1Dli,jφi,j,u≤Imax,j,u,j∈C(26c)

TdT∑i=1D∑j=1C(pjtrue+pjfalse)φi,j,d≤Pmax,d(26d)

TuT∑j=1C(pjtrue+pjfalse)φi,j,u≤Pmax,i,u,i∈D(26e)

∑j=1CβTc,eT[hi,j(pjtrue+pjfalse)φi,j,d+ϕi,jN]≥Emin,i,i∈D(26f)

∑i=1Dϕi,j=1,j∈C(26g)

0<ts≤T(26h)

0<α≤1(26i)

0≤ϕi,j≤1,i∈D,j∈C(26j)

Pi,j,u1≥0,i∈D,j∈C(26k)

Pi,j,d1≥0,i∈D,j∈C(26l)

φi,j,u≥0,i∈D,j∈C(26m)

φi,j,d≥0,i∈D,j∈C(26n)

In this optimization problem, the constraints in Eqs. (26m) and (26n) ensure that the auxiliary variables are not less than zero. The reason for adding these two constraints is that the auxiliary variables are calculated by multiplying the transmit power and the channel assignment factor, while the transmit power takes a value greater than or equal to 0 and the channel assignment factor takes a value between 0 and 1. Therefore, the auxiliary variables also need to comply with the condition that they are not less than zero. The optimization problem expressed by Eqs. (26a)–(26n) is a nonconvex problem, and a coupling exists between the sensing time ts, time allocation factor α, time slot allocation parameters m and n, and auxiliary variables φi,j,u and φi,j,d. Therefore, the optimal solutions ts∗, α∗, m∗, and n∗ can be solved by using the method of predetermined values, and the optimization problem presented in Eqs. (26a)–(26n) is transformed into a convex optimization problem using the four determined parameters ts, α, m, and n. This study uses the dual gradient descent method to address the convex optimization problem with the given parameters. The problem formulated by Eqs. (26a)–(26n) is transformed into a dual problem using a Lagrangian function, which is expressed as Eq. (27), where λj, χj, ω, θi, ϑi, and ξj are the Lagrangian variables.

L(φi,j,u,φi,j,d,ϕi,j;λj,χj,ω,θi,ϑi,ξj)=ψ+∑j=1Cλj(Imax,j,d−TdTpjfalse∑i=1Dgi,jφi,j,d)+∑j=1Cχj(Imax,j,u−TuTpjfalse∑i=1Dli,jφi,j,u)+ω[Pmax,d−TdT∑i=1D∑j=1C(pjtrue+pjfalse)φi,j,d]+∑j=1Cξj(1−∑i=1Dϕi,j)+∑i=1Dθi[Pmax,i,u−TuT∑j=1C(pjtrue+pjfalse)φi,j,u]+∑i=1Dϑi[∑j=1CβTc,eT(hi,j(pjtrue+pjfalse)φi,j,d+Nϕi,j)−Emin,i](27)

The problem, as described by Eqs. (26a)–(26n) can be transformed into a dual problem, which can be expressed as follows:

minλj,ω,θi,ϑi,ξj≥0f(λj,χj,ω,θi,ϑi,ξj),(28)

where

f(λj,χj,ω,θi,ϑi,ξj)=maxL(φi,j,d,φi,j,u,ϕi,j;λj,χj,ω,θi,ϑi,ξj)(29)

According to the principle of the Karush–Kuhn–Tucker (KKT) condition, the partial derivative of the Lagrangian function with respect to φi,j,u satisfies the following conditions:

∂L∂φi,j,u{=0φi,j,u∗>0<0φi,j,u∗<0,(30)

where x∗ represents the optimal solution for x.

ri,jtrue(Tu) and ri,jfalse(Tu) in the Lagrangian function are related to φi,j,u. The derivatives of the above two functions with respect to φi,j,u can be expressed as:

∂ri,jtrue(Tu)∂φi,j,u=Whi,jln⁡2(ϕi,jNi,j,u+φi,j,uhi,j,u)∂ri,jfalse(Tu)∂φi,j,u=Whi,jln⁡2(ϕi,j(Pi,j,u2gi,j+N)+φi,j,uhi,j)(31)

The partial derivative of Lagrangian function with respect to φi,j,u be expressed as:

∂L∂φi,j,u=ϕi,jTuT[pjtrueWhi,jln⁡2(ϕi,jN+φi,j,uhi,j)+pjfalseWhi,jln⁡2(ϕi,j(Pi,j,u2gi,j+N)+φi,j,uhi,j)]−θiTuT(pjtrue+pjfalse)−χjTuTpjfalseli,j(32)

For ∂L∂φi,j,u=0, the optimal solution φi,j,u∗ can be obtained by using the quadratic formula. According to the definition of the auxiliary variables, the optimal uplink transmit power can be obtained by using the condition: Pi,j,u1∗=φi,j,u∗/ϕi,j.

According to the principle of the KKT condition, the partial derivative of the Lagrangian function with respect to φi,j,d satisfies the following conditions:

∂L∂φi,j,d{=0φi,j,d∗>0<0φi,j,d∗<0(33)

Furthermore, ri,jtrue(Td) and ri,jfalse(Td) in the Lagrangian function are related to φi,j,d. Thus, the derivatives of the above two functions with respect to φi,j,d can be expressed as:

∂ri,jtrue(Td)∂φi,j,d=Whi,jln⁡2(ϕi,jN+φi,j,dhi,j)∂ri,jfalse(Td)∂φi,j,d=Whi,jln⁡2(ϕi,j(Pi,j,d2li,j+N)+φi,j,dhi,j)(34)

The partial derivative of L with respect to φi,j,d be expressed as:

∂L∂φi,j,d=ϕi,jTdT[pjtrueWhi,jln⁡2(ϕi,jN+φi,j,dhi,j)+pjfalseWhi,jln⁡2(ϕi,j(Pi,j,d2li,j+N)+φi,j,dhi,j)]+(pjtrue+pjfalse)(ϑiβhi,jTc,eT−ωTdT)−λjTdTpjfalsegi,j(35)

For ∂L∂φi,j,d=0, the optimal solution φi,j,d∗ can be obtained by using the quadratic formula. According to the definition of the auxiliary variables, the optimal uplink transmit power can be obtained by using the condition: Pi,j,d1∗=φi,j,d∗/ϕi,j.

According to the principle of the KKT condition, the partial derivative of the Lagrangian function with respect to ϕi,j satisfies the following conditions:

∂L∂ϕi,j{<0ϕi,j∗=0=00<ϕi,j∗<1>0ϕi,j∗=1(36)

ri,jtrue(Tu), ri,jfalse(Tu), ri,jtrue(Td), and ri,jfalse(Td) in the Lagrangian function are related to ϕi,j. Therefore, the derivatives of the above four functions with respect to ϕi,j can be expressed as:

∂ri,jtrue(Td)∂ϕi,j=−Wφi,j,dhi,jln⁡2(Nϕi,j2+φi,j,dhi,jϕi,j)∂ri,jfalse(Td)∂ϕi,j=−Wφi,j,dhi,jln⁡2((Pi,j,d2li,j+N)ϕi,j2+φi,j,dhi,jϕi,j)∂ri,jtrue(Tu)∂ϕi,j=−Wφi,j,uhi,jln⁡2(Ni,j,uϕi,j2+φi,j,uhi,j,uϕi,j)∂ri,jfalse(Tu)∂ϕi,j=−Wφi,j,uhi,jln⁡2((Pi,j,u2gi,j+N)ϕi,j2+φi,j,uhi,jϕi,j)(37)

The partial derivative of the L with respect to ϕi,j be expressed as:

∂L∂ϕi,j=TuT[pjtrue(ri,jtrue(Tu)+ϕi,j∂ri,jtrue(Tu)∂ϕi,j)+pjfalse(ri,jfalse(Tu)+ϕi,j∂ri,jfalse(Tu)∂ϕi,j)]+TdT[pjtrue(ri,jtrue(Td)+ϕi,j∂ri,jtrue(Td)∂ϕi,j)+pjfalse(ri,jfalse(Td)+ϕi,j∂ri,jfalse(Td)∂ϕi,j)]+ϑiβTc,eTN−ξj(38)

Let

ϖi,j=TuT[pjtrue(ri,jtrue(Tu)+ϕi,j∂ri,jtrue(Tu)∂ϕi,j)+pjfalse(ri,jfalse(Tu)+ϕi,j∂ri,jfalse(Tu)∂ϕi,j)]+TdT[pjtrue(ri,jtrue(Td)+ϕi,j∂ri,jtrue(Td)∂ϕi,j)+pjfalse(ri,jfalse(Td)+ϕi,j∂ri,jfalse(Td)∂ϕi,j)]+ϑiβTc,eTN(39)

In order to maximize the system throughput, different channel assignments are compared, and the result that maximizes ϖi,j is the optimal channel assignment; hence, the expression for ϕi,j∗ can be reformulated as:

ϕi,j∗={0other1argmax ϖi,j(40)

After determining the optimal φi,j,u∗, φi,j,d∗, Pi,j,u1∗, Pi,j,d1∗, and ϕi,j∗ using the above method, the values for the Lagrangian variable for the kth iteration can be obtained based on the principle of the gradient descent method, which is described as follows:

λjk+1=[λjk+α1k∇λjkf(λjk,χjk,ωk,θik,ϑik,ξjk)]+χjk+1=[χjk+α2k∇χjkf(λjk,χjk,ωk,θik,ϑik,ξjk)]+ωk+1=[ωk+α3k∇ωkf(λjk,χjk,ωk,θik,ϑik,ξjk)]+,θik+1=[θik+α4k∇θikf(λjk,χjk,ωk,θik,ϑik,ξjk)]+ϑik+1=[ϑik+α5k∇ϑikf(λjk,χjk,ωk,θik,ϑik,ξjk)]+(41)

where [x]+=max(x,0) and α1k, α2k, α3k, α4k, α5k denote the step size of the kth iteration, which are both set to constants less than zero, and the superscript k denotes the kth iteration. Additionally, ∇xf(x,y) represents the partial derivative of f(x,y) with respect to x.

The five partial derivatives in Eq. (41) can be computed as:

∇λjkf(λjk,χjk,ωk,θik,ϑik,ξjk)=Imax,j,d−TdTpjfalse∑i=1Dgi,jφi,j,d∗(42)

∇χjkf(λjk,χjk,ωk,θik,ϑik,ξjk)=Imax,j,u−TuTpjfalse∑i=1Dli,jφi,j,u∗(43)

∇ωkf(λjk,χjk,ωk,θik,ϑik,ξjk)=Pmax,d−TdT∑i=1D∑j=1C(pjtrue+pjfalse)φi,j,d∗(44)

∇θikf(λjk,χjk,ωk,θik,ϑik,ξjk)=Pmax,i,u−TuT∑j=1C(pjtrue+pjfalse)φi,j,u∗(45)

∇ϑikf(λjk,χjk,ωk,θik,ϑik,ξjk)=−Emin,i+∑j=1CβTc,eT(hi,j(pjtrue+pjfalse)φi,j,d∗+N)(46)

The new Lagrangian variables obtained by Eqs. (41)–(46) are compared with the old values, and the iteration ends when it satisfies Eq. (47). Otherwise, the gradient descent step is repeated. Notably, ||x||2 denotes the two-parametric number of x.

‖λjk+1−λjk‖2≤ε1‖χjk+1−χjk‖2≤ε2‖ωk+1−ωk‖2≤ε3,‖θik+1−θik‖2≤ε4‖ϑik+1−ϑik‖2≤ε5(47)