A novel approach to energy-aware resource management: Toward green NOMA heterogeneous networks

Abstract

The dense deployment of small cell networks is a key feature of next generation mobile networks aimed at providing the necessary capacity increase. In order to reach an acceptable performance in such ultra-dense networks, real-time resource management is of great importance. Therefore, self-optimization networking is proposed as the only viable solution to increase the networks’ utility. This paper proposed a self-optimizing model to enhance network performance and guarantee the users’ QoS requirements by considering limited resources and using effective user association, carrier scheduling and handover optimization algorithms. In order to maximize the network performance, we applied the smart backhauling technique in order to analyze the signaling to increase the validity of the decision making process. Based on the semantic information extracted from the access layer, the network decision-making center is able to adjust the network parameters and resource allocation effectively. The goal function is defined as maximizing the total energy efficiency by considering the transmission power, energy harvesting capability and the user QoS constraints so that the idle small cells are considered turned off temporarily to boost the power efficiency. Although the optimization problem is non-convex, a quadratic mixed-integer function is solved to obtain a global optimal solution. Since the actual implementation of the real-time algorithm has high computational complexity, two algorithms with different complexity levels are proposed. These algorithms use the carrier matching feature and optimal transmission power for problem-solving. The simulation results prove that, despite the increased computational complexity, effective resource allocation and optimal HO relations made the proposed approach capable to increase performance indices such as network throughput by up to 30%.

Keywords

Radio resource management convex optimization 5G NOMA green HetNet

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1. Introduction

The deployment of ultra-dense small cells will probably be a major component of next generation wireless networks, to manage the increasing traffic of mobile networks and to offload the traffic of high-utilized macro cells in order to improve users’ quality of service (QoS) [1]. However, despite the optimal resource supply, using these methods and attributes in heterogeneous networks (HetNets), exponentially increase the complexity of planning and optimization. Hence, using self-optimization technique is the only viable solution for increasing the efficiency in these networks [2, 3]. The goal of proposing a self-optimization model is to maximize the network efficiency and increase the quality of services provided to macrocell and femtocell users, considering the limited resources in radio access networks [4, 5]. The proposed self-optimization approach concerned the intelligent control of network resources so that the problem of carrier scheduling and resource allocation [6] is addressed and the subject is tackled in two layers [7]. This study can be distinguished from the common self-optimization technology framework with three main characteristics: 1-comprehensive information on the current network status, 2-capability of predicting user behavior, and 3-dynamic capability of relating the network response to the network parameters. Actually, the proposed self-optimizing model is aimed at maximizing the network performance in terms of capacity/coverage and increasing the QoS provided to macrocell and small cell users by taking into account limited resources for access networks. The basis of our proposed scheme is to introduce a self-optimization model based on the Resource Allocation (RA), User Association (UA), Carrier Scheduling (CS) and Handover Optimization (HO). Using this model, we can create the capability of controlling resources and neighboring parameters as real-time without the need of human manipulation and only based on the network’s intelligence.

The ever-increasing popularity of cellphones and the continuous use of communication equipment have now intensified the need for wireless networks. Heterogeneous networks evolved over time in line with the growing user demand rates. in this regard, some network features were introduced by researchers to increase the network capacity like massive MIMO [8], beamforming [9] and so on. The massive MIMO technology is now used in the new generations of mobile networks, the beamforming process can only reduce the strong path loss variations in cellular modes as well. In this case, cell-edge users can receive services with a coverage of 50 dB lower than the UEs located within a cell, also the massive MIMO system with 64 antennas is able to only increase the signal level nearly 20 dB. Some novel promising adaptive massive MIMO technologies are intended for the mediocre cell-edge performance with the distribution of small cells over the network and elimination of poor covered areas with common operations. The transmission process of each small cell is performed coherently in the downlink so that the received signals are processed coherently in the uplink. This leads to a higher signal-to-interference-plus-noise ratio (SINR) with no further power consumption. The concept of the coherent joint transmission in a massive MIMO system emerges from the design of classical coordinated multipoint beamforming with multiple antenna arrays. It then gradually develops into some scenarios with further multiple-antenna distributed small cells, as shown in Fig. 1. Previous network designs mainly considered multipath fading channels and spectral efficiency criteria, which are treated as a set of functions of fast Rayleigh fading channels. Subsequently, some novel feature of the adaptive massive MIMO technology is associated with considering practical Rayleigh/Rician fading channels, the performance of such schemes are evaluated through ergodic spectral efficiency used as a performance criterion [10]. In fact, spectral efficiency depends on the pilot pollution and imperfect channel State Information.

Figure 1.

HetNet model with distributed antennas and multiple backhauls connections.

When all cells and UEs are considered as a whole integrated system, due to the large amount of mobile traffic and massive deployment of small cells, energy efficiency becomes an important issue in ultra-dense networks. From the energy efficiency perspective, the main problem in the next generation HetNets will be relevant to the concerns over power consumption and the consequent pollution issues [11]. Cellular networks were developed to maximize spectral efficiency and coverage, resulting in the maximum transmission power in a downlink and leading directly to more power utilization in macro cells and small cells even under light traffic conditions. In order to maximize the long-term energy efficiency in millimeter-wave backhauling, the authors in [12] proposed a deep learning based UA strategy with a sub-optimal carrier matching algorithm. In [13], the authors introduced a novel RA strategy to maximize the system energy efficiency applying markov decision process theory in massive MIMO enabled HetNets. In [14], the authors propose an user-centric access framework in large-scale HetNets to minimize the network power consumption. Many researches and standardization work have been done on this issue, among which small cells on/off control has received a significant attention from both academic and industrial community [15]. The authors in [16, 17] proposed an link prediction on heterogeneous information network (HIN) as a challenge problem due to the complexity and diversity in types of nodes and links in multi-access HetNets. In large-scale networks the density of distributed small cells is comparable to the UEs density, which enables any user in close proximity to multiple serving cells. In order to maximize the performance of the whole system, when traffic load decreased, small cells with low or even no traffic can be turned off to alleviate inter-cell interference and energy efficiency can be improved. In [18], an inter-cell interference-aware cell on/off switching control algorithm was suggested to optimize power utility and HO relations [19, 20]. In [21], analytical work has been done on power utility with micro layer cells on/off control under two cases: high-rate requirements and low-rate requirements. Also, [22] applies deep learning algorithm to achieve the optimal energy efficient small base stations on/off control result.

Nevertheless a novel approach seems required to deploy in order to make a reasonable balance between network payload and the power utilization; therefore, power consumption decreases when users demand lower spectral utilization. The optimization of energy efficiency in such massive MIMO systems was analyzed in [23], the results of which showed that the fronthaul energy consumption reduced only by providing each UE with a limited number of active cells, whereas all small cells remained continuously ON. According to [24], the power utility of cloud or HetNets reduced sufficiently, and small cells were also turned on and off repetitively. Nevertheless, an effective and practical approach must consider the quality of service (QoS) constraints of the users. Therefore, load balancing was aimed at mapping the network payload on the accessible radio access network resources more efficiently. These papers assumed that each user could have a condition as an limited spectral efficiency including the transmission power and the power consumed by hardware in active cells as the system power consumption. They also consumed that the system should satisfy the condition by consuming minimum power. Accordingly, it is possible to perform resource allocation under load balancing and traffic analysis with only providing necessary services with low spectral utilization through minimum power consumption. Despite the previous studies which are appropriate only for specific scenarios where UEs are completely stationary, this model conducted optimization based on an activation channel. To the best of the authors’ knowledge, no studies have yet been published on the dynamic activation of cells joint UA, CS, HO and RA approaches in massive NOMA systems.

According to the described scenario which considered multiple users sending service requests with their specific QoS requirements, each user have a predefined condition of the spectrum utilization in the downlink and the network is required to guarantee all these conditions to avoid service disruption. Based on the problem assumption, users and cells are distributed randomly; hence, the spectral efficiency requirements were likely to be met with no cells. Since each mobile user employed only its adjacent cells, the goal function considered the probability of turning off the small cells which were not needed while serving a set of UEs. This is remarkable because adaptive massive MIMO systems have several serving cells, many of which can provide a consistent coverage; however, not all of them are necessarily required at all instants [25]. Therefore, the novel model was developed by using the concepts of closed-form ergodic spectral efficiency, linear pre-coding, imperfect channel state information, and pilot pollution. This interpretation allowed for the optimization of ultra-dense HetNets with effective cell activation patterns and the support of more users simultaneously.

According to the research literature, the proposed approach will be the first attempt to simultaneous execution of RA, UA, CS, and dynamic cell activation control in NOMA HetNets. For simplicity, we call the proposed approach as “NOMA-based Energy Efficient Radio Resource Optimization (NOMA-EERRO)”.

The paper is organized as follows: after briefly summarizing the related works and mentioning the motivation in the introduction part, Section 2 describes the system model and problem formulation for different energy efficiency strategies. This section includes downlink transmission, uplink pilot strategy and description of the energy efficiency model. In Section 3, solutions and algorithms to solve the proposed power optimization problems are investigated. This section also introduced the proposed CS, UA and HO optimization solutions. Also, investigation on the different transmission strategies is done in this section with focusing on NOMA-based radio resource management model for backhaul communications. In Section 4, simulation results are presented and interpreted in order to exhibit the effectiveness of different optimization approaches. In this section, we described the simulation scenarios and environments and we tried to evaluate the performance of the proposed algorithms accurately. Finally, the concluding remarks mentioned in Section 5.

2. System model and problem formulation

In the proposed two-tier NOMA HetNet, we considered a BS located at the center of each macro cell in addition to $K$ small cell randomly distributed within the macro coverage area [26]. Each base station is empowered with the energy harvesting capability. In each cluster, the set of macro base stations and small base stations are illustrated by $k\in\{1,2,\ldots,K+1\}$ , so that base station $k=K+1$ is assumed as the macro BS. $M_{k}\in\{M_{1},M_{2},\ldots,M_{K+1}\}$ shows the set of associated UEs to cell $k$ as well. $B W$ represents the network accessible bandwidth divided to $N$ separated subchannels $n\in\{1,2,\ldots,N\}$ , in such a way that the bandwidth of each channel is $B_{sc}={BW}/{N}$ . Based on the problem assumptions, $h_{k,j,m,n}$ is channel $n$ ’s transmission gain of user $m$ relevant to connection between cell $k$ to cell $j$ in which $m\in\{1,2,\ldots,M_{K}\}$ , [27]. The schematic of the proposed self-organized network is shown in Fig. 2.

Figure 2.

Self-organized system model: Two-tiered NOMA-heterogeneous network.

In this system model, ${\lambda}_{0}(k)$ and ${\lambda}_{s}(k)$ represent the average entry rate of the primary and secondary users respectively. This process has a Poisson distribution and $b_{0}(k)$ and $b_{s}(k)$ indicate the statistical distribution of service time for the primary and secondary users which their median values are equal to $\bm{E}[X^{k}_{0}]$ and $\bm{E}[X^{k}_{s}]$ respectively. The arrival rate to channel $K$ for the interrupted secondary users is shown by ${\lambda}^{(k)}_{i}$ . In this formulation, $t_{s}$ denotes the channel switch time and we have ${\rho}_{0}={\lambda}_{0}E[X_{0}]$ and ${\rho}_{s}={\lambda}_{s}E[X_{s}]$ . PU and SU show the primary and secondary users in which $\mathcal{N}=\{1,2,\dots,N\}$ is the players, $\mathcal{M}=\{1,2,\dots,M\}$ is the set of resources, $\mathcal{N}=\{{SU}_{1},{SU}_{2},\dots,{SU}_{N}\}$ demonstrates all of the secondary paired users. However, ${\mathcal{F}}_{\bm{i}}$ and $U_{\bm{i}}$ represent the accessible resources and the utility function of player $i$ . Based on the described system model, $n^{m}(\bm{\sigma})$ shows all players that use resource $m$ with the strategy $\bm{\sigma}$ and $c^{m}(n)$ illustrates the cost of utilizing channel $m$ . But the total cost of applying resource $m$ by user $i$ is obtainable via $c^{m}_{i}(n^{m}_{i}(\bm{\sigma})+1)$ in which $n^{m}_{i}(\bm{\sigma})=|\{j:m\in{\sigma}_{j},j\in{\mathcal{B}}_{i}\}|$ . Note that in this scenario, $\beta$ represents the channel quality and its inverse is represented by $\alpha$ .

2.1 Uplink pilot strategy

Based on the uplink transmission strategy, ${\Psi}=[{\psi}_{1},\dots,{\psi}_{{\tau}_{P}}]\in{\mathbb{C}}^{{\tau}_{P}\times% {\tau}_{P}}$ represents a matrix indicator of a set of ${\tau}_{P}$ orthonormal pilot signals (PS) allocated to UE $k$ . Particularly, UE $k$ which transmits the PS $\sqrt{{\tau}_{P}}{{\psi}_{i_{k}}\in\mathbb{C}}^{{\tau}_{P}}$ with $i_{k}\in\{1,\dots,{\tau}_{P}\}$ determines the pilot index. In this scenario, we consider hotspot and random pilot allocation patterns. Within each cluster, ${\mathcal{P}}_{k}$ demonstrates the subset of UEs allocated to a specific PS as UE $K$ , hence we have

$\displaystyle{\psi}^{H}_{i_{k}}{\psi}_{i_{\acute{k}}}=\left\{\begin{array}[]{% ll}1&\text{if }\acute{k}\in{\mathcal{P}}_{k},\\ 0&\text{if }\acute{k}\notin{\mathcal{P}}_{k}.\\ \end{array}\right.$ (1)

The data ${{\mathrm{Y}}_{m}\in\mathbb{C}}^{N\times{\tau}_{P}}$ delivered to small cell $m$ consists of a summation of all the transmitted PSs by the set of $K$ UEs:

$\displaystyle{\mathrm{Y}}_{mk}=\sum\limits^{K}_{k=1}{\sqrt{{\tau}_{P}P_{\acute% {k}}}}h_{m\acute{k}}{\psi}^{H}_{i_{k}}+N_{m},$ (2)

In which $P_{\acute{k}}$ denotes the transmit power of UE $k$ ’s PS and ${{{N}}_{m}\in\mathbb{C}}^{N\times{\tau}_{P}}$ represents the white noise with the distribution as $\mathcal{C}\mathcal{N}(0,o^{2}_{UL})$ . Estimation of $h_{mk}$ calculated by small cell $m$ with the probability ${\mathrm{y}}_{mk}={\mathrm{Y}}_{mk}{{\psi}_{i_{k}}\in\mathbb{C}}^{N}$ , is achieavable via

$\displaystyle{\mathrm{y}}_{mk}=\sum\limits_{\acute{k}\in{\mathcal{P}}_{k}}{% \sqrt{{\tau}_{P}P_{\acute{k}}}}h_{m\acute{k^{\prime}}}+N_{m}{\psi}_{i_{k}}.$ (3)

In this framework, the minimum mean square error relevant to the estimation of PS between UE $k$ and small cell $m$ is obtainable as

$\displaystyle{\widehat{\mathrm{h}}}_{mk}=\mathbb{E}\{{\mathrm{h}}_{mk}\mathrel% {|\vphantom{{\mathrm{h}}_{mk}y_{mk}}.\kern-1.2pt}y_{mk}\}=\frac{\sqrt{{\tau}_{% P}P_{k}}{\beta}_{mk}}{{\tau}_{P}\sum\nolimits_{\acute{k}\in{\mathcal{P}}_{k}}{% P_{\acute{k}}{\beta}_{m\acute{k}}+{\mathrm{\sigma}}^{\mathrm{2}}_{\mathrm{UL}}% }}{\mathrm{y}}_{mk}.$ (4)

The distribution of the PS channel estimation is ${\widehat{\mathrm{h}}}_{mk}\sim\mathcal{C}\mathcal{N}(0,{\gamma}_{mk}I_{N})$ , where the variance ${\gamma}_{mk}$ is achieved by Eq. (5).

$\displaystyle{\gamma}_{mk}=\frac{{\tau}_{P}p_{k}{\beta}^{2}_{mk}}{{\tau}_{P}% \sum\nolimits_{\acute{k}\in{\mathcal{P}}_{k}}{P_{\acute{k}}{\beta}_{m\acute{k}% }+{\mathrm{\sigma}}^{\mathrm{2}}_{\mathrm{UL}}}}.$ (5)

2.2 Downlink transmission performance

In the suggested framework for the downlink, each small cell applies a specific precoding vector (PV) which is determined based on the locally estimated PSs. In this notation, $s_{k}$ illustrates the data frame sent to UE $k$ from a set of small base stations where we suppose $\mathbb{E}\{{|s_{k}|}^{2}\}=1$ . Also, we can calculate the small cell $m$ ’s transmission signal ${\mathrm{x}}_{m}{\in\mathbb{C}}^{N}$ to UEs $K$ as Eq. (7)

$\displaystyle{\mathrm{x}}_{m}=\sum\limits^{K}_{k=1}{\sqrt{{\rho}_{mk}}{\mathrm% {w}}_{mk}s_{k},}$ (6)

In this equation, ${\rho}_{mk}\geqslant 0$ illustrates power of the transmitted data which is assigned to UE $k$ from small cell $m$ . So, the delivered signal to UE $k$ from the set of active cells can be expressed as the following

$\displaystyle r_{k}=\sum\limits_{m\in\mathcal{A}}{{\mathrm{h}}^{H}_{mk}{% \mathrm{x}}_{m}+{\tilde{w}}_{k},}$ (7)

Note that ${\tilde{w}}_{k}\sim\mathcal{C}\mathcal{N}(0,o^{2}_{DL})$ denotes the additive noise with the mean equal to 0 and the variance $o^{2}_{{DL}}$ . Also, we applied the capacity bounding method to obtain the minimum acceptable ergodic capacity of UE $k$ as Eq. (8)

$\displaystyle R_{k}=\left(1-\frac{{\tau}_{P}}{{\tau}_{c}}\right)\times{{% \mathrm{log}}_{2}\left(1+\frac{{|\mathrm{D}{\mathrm{S}}_{k}|}^{2}}{\mathbb{E}% \{{|\mathrm{B}{\mathrm{U}}_{k}|}^{2}\}+\sum\nolimits^{K}_{\acute{k}\neq k}{% \mathbb{E}\{{|\mathrm{U}{\mathrm{I}}_{\acute{kk}}|}^{2}\}+{\mathrm{\sigma}}^{% \mathrm{2}}_{\mathrm{DL}}}}\right)},$ (8)

In this formulation, where $\mathrm{U}{\mathrm{I}}_{\acute{kk}}$ , $\mathrm{B}{\mathrm{U}}_{k}$ , and $\mathrm{D}{\mathrm{S}}_{k}$ represent the inter-UE interference, the beamforming gain, and the UE $k$ ’s desired signal, respectively. These three parameters are defined as the following.

$\displaystyle\mathrm{D}{\mathrm{S}}_{k}=\mathbb{E}\left\{\sum\limits_{m\in% \mathcal{A}}{\sqrt{{\rho}_{mk}}}{\mathrm{h}}^{H}_{mk}{\mathrm{w}}_{mk}\right\},$ (9) $\displaystyle\mathrm{B}{\mathrm{U}}_{k}=\sum\limits_{m\in\mathcal{A}}{\sqrt{{% \rho}_{mk}}}{\mathrm{h}}^{H}_{mk}{\mathrm{w}}_{mk}-\mathbb{E}\left\{\sum% \limits_{m\in\mathcal{A}}{\sqrt{{\rho}_{mk}}}{\mathrm{h}}^{H}_{mk}{\mathrm{w}}% _{mk}\right\},$ (10) $\displaystyle\mathrm{U}{\mathrm{I}}_{\acute{kk}}=\sum\limits_{m\in\mathcal{A}}% {\sqrt{{\rho}_{m\acute{k}}}}{\mathrm{h}}^{H}_{mk}{\mathrm{w}}_{m\acute{k}}.$ (11)

It should be noted that the lower-bound capacity obtained by Eq. (8) does not depend on the small cells’ activity pattern or the applied precoding methods. Nevertheless, to achieve the closed form formulation of the goal function, we consider the active cells to either use maximum ratio transmission or full-pilot zero-forcing precoding approaches, which are described in micro layer network as the following.

$\displaystyle w_{mk}=\left\{\begin{array}[]{ll}\frac{{\widehat{\mathrm{h}}}_{% mk}}{\sqrt{\mathbb{E}\{{||{\widehat{\mathrm{h}}}_{mk}||}^{2}\}}}&\mathrm{if\ % MRT,}\\ \frac{{\widehat{\mathrm{H}}}_{m}{({\widehat{\mathrm{H}}}^{H}_{m}{\widehat{% \mathrm{H}}}_{m})}^{-1}{\mathrm{e}}_{i_{k}}}{\sqrt{\mathbb{E}\{{||{\widehat{% \mathrm{H}}}_{m}{({\widehat{\mathrm{H}}}^{H}_{m}{\widehat{\mathrm{H}}}_{m})}^{% -1}{\mathrm{e}}_{i_{k}}||}^{2}\}}}&\mathrm{if\ F-ZF,}\\ \end{array}\right.$ (12)

In which ${\widehat{\mathrm{H}}}_{m}={\mathrm{Y}}_{m}{{\psi}\in\mathbb{C}}^{N\times K}$ and ${\mathrm{e}}_{i_{k}}$ denotes the $i_{k}-th$ column of the matrix ${\mathrm{I}}_{{\tau}_{P}}$ . Also, the downlink statistical ergodic equilibrium of UE $k$ is

$\displaystyle R_{k}(\{{\rho}_{mk}\},\mathcal{A})=\left(1-\frac{{\tau}_{P}}{{% \tau}_{c}}\right){{\mathrm{log}}_{2}(1+{\mathrm{SINR}}_{k}(\{{\rho}_{mk}\},% \mathcal{A})),}$ (13)

In this equation, the effective signal to noise/interference ratio is given in Eq. (14). The variables $G$ and $z_{mk}$ are determined based on the applied precoding scheme. The maximum ratio transmission approach gives $G=N$ and $z_{mk}={\beta}_{mk}$ . For $N>{\tau}_{P}$ , The full-pilot zero-forcing precoding gives $G=N-{\tau}_{P}$ and $z_{mk}={\beta}_{mk}-{\gamma}_{mk}$ .

$\displaystyle{\mathrm{SINR}}_{k}(\{{\rho}_{mk}\},\mathcal{A})=\frac{G{(\sum% \nolimits_{m\in\mathcal{A}}{\sqrt{{\rho}_{mk}{\gamma}_{mk}}})}^{2}}{G\sum% \nolimits_{\acute{k}\in{\mathcal{P}}_{k}\backslash\{k\}}{{(\sum\nolimits_{m\in% \mathcal{A}}{\sqrt{{\rho}_{m\acute{k}}{\gamma}_{mk}}})}^{2}+\sum\nolimits^{K}_% {\acute{k}=1}{\sum\nolimits_{m\in\mathcal{A}}{{\rho}_{m\acute{k}}z_{mk}+{% \mathrm{\sigma}}^{\mathrm{2}}_{\mathrm{DL}}}}}}.$ (14)

2.3 Energy efficiency model

In order to formulate the energy efficiency, we define $s_{k,m,n}$ as an auxiliary factor of the CS process in such a way that $s_{k,m,n}=1$ if user equipment $m$ connected to cell $k$ on channel $n$ , and if user equipment $m$ is not associated to cell $k$ , we have $s_{k,m,n}=0$ . In this notation, $p_{k,m,n}$ denotes the transmit power of user equipment $m$ associated to cell $k$ on channel $n$ . Also, in this scenario we illustrate the channel matrix and the set of accessible resource blocks by $S=[s_{k,m,n}]$ and ${P=[p}_{k,m,n}]$ respectively. Based on the Shannon’s capacity proposition, the upper bound of the throughput of user $m$ connected to base station $k$ on channel $n$ is achievable as

$\displaystyle r_{k,m,n}=B_{SC}{\log}_{2}(1+\textit{SINR}_{k,m,n})$

where $\textit{SINR}_{k,m,n}$ denotes the SINR of base station $k$ on the $n^{\text{th}}$ channel of user $m$ [29].

As mentioned before, in NOMA-based communications, we are permitted to allocate one distinct channel to more than one user in each time slot. The receivers should use decoding and demodulation in addition to apply successive interference cancellation method in order to eliminate the interference’s impacts. But to decrease the computational complexity, we assumed that the maximum allowed number of users connected to a channel is two so that each receiver first decodes the payloads of the user with the higher power. Hence, the SINR level at the point of user $m$ on channel $n$ of cell $k$ is obtained as

$\displaystyle\textit{SINR}_{k,m,n}=\frac{s_{k,m,n}p_{k,m,n}{|h_{k,k,m,n}|}^{2}% }{{|h_{k,k,m,n}|}^{2}\sum\limits^{M_{k}}_{r=m+1}{s_{k,r,n}p_{k,r,n}+\sum% \limits^{K+1}_{j=1,j\neq k}{p_{j,n}{|h_{k,j,m,n}|}^{2}+{\sigma}^{2}}}}$ (15)

In Eq. (15) ${\sigma}^{2}$ is gaussian noise and $p_{j,n}=\sum\limits^{M_{j}}_{r=1}{s_{j,r,n}p_{j,r,n}}$ expresses the power of base station $j$ on the channel $n$ .

In this scenario, we try to achieve the optimal value of the energy efficiency index using UA, CS and HO algorithms in addition to guarantee the users’ QoS constraints. Therefore, we formulate the goal function based on maximizing the user throughput considering the total power constraints. In the CS and RA process, the macro and micro base stations are capable to harvest energy [30]. Based on the mentioned circumstances, the total system throughput can be expressed as the following

$\displaystyle R(S,P)=\sum\limits^{K+1}_{k=1}{\sum\limits^{M_{k}}_{m=1}{\sum% \limits^{N}_{n=1}{r_{k,m,n}}}}$ (16)

Also, the total transmit power is obtained as Eq. (17)

$\displaystyle Q(S,P)=\sum\limits^{K+1}_{k=1}{\sum\limits^{M_{k}}_{m=1}{\sum% \limits^{N}_{n=1}{s_{k,m,n}p_{k,m,n}.}}}$ (17)

If $n$ denotes the channel number, the energy harvested by user equipment $m$ from base station $k$ is illustrated by Eq. (18)

$\displaystyle H_{k,m,n}=\sum\limits^{K+1}_{j=1,j\neq k}{{\lambda}_{j,n}s_{k,m,% n}p_{k,m,n}}{|h_{k,j,m,n}|}^{2}$ (18)

which ${\lambda}_{j,n}$ shows the efficiency indicator of energy harvesting. Based on $H_{k,m,n}$ and the problem assumptions, the overall harvested energy is calculated as $H(S,P)$ .

$\displaystyle H(S,P)=\sum\limits^{K+1}_{k=1}\sum\limits^{M_{k}}_{m=1}\sum% \limits^{N}_{n=1}\sum\limits^{K+1}_{j=1,j\mathrm{\neq}\mathrm{k}}{{\lambda}_{j% ,n}}s_{k,m,n}p_{k,m,n}{|h_{k,j,m,n}|}^{2}.$ (19)

For simplicity, we can ignore the power consumed by hardware because the total harvested energy can be significantly higher. So, the overall utilized resources are obtained as Eq. (20)

$\displaystyle U(S,P)=Q(S,P)-H(S,P)=\sum\limits^{K+1}_{k=1}{\sum\limits^{M_{k}}% _{m=1}{\sum\limits^{N}_{n=1}{{s_{k,m,n}p}_{k,m,n}\left(1-\sum\limits^{K+1}_{j=% 1,j\neq k}{{\lambda}_{j,n}}{|h_{k,j,m,n}|}^{2}\right)}}}$ (20)

According to the total harvested energy and the total transmission power, we can compute the energy efficiency index as Eq. (21)

$\displaystyle EE(S,P)=\frac{R(S,P)}{Q(S,P)-H(S,P)}=\frac{R(S,P)}{U(S,P)}.$ (21)

As mentioned before, the target is maximizing the energy efficiency so, the goal function can be formulated as the following.

$\displaystyle{\mathop{\mathrm{max}}_{S,P}EE(S,P)=\mathop{\mathrm{max}}_{S,P}% \frac{R(S,P)}{U(S,P)}}$ (22) $\displaystyle\text{s.t.}C1:\sum\limits^{M_{k}}_{m=1}{\sum\limits^{N}_{n=1}{{s_% {k,m,n}p}_{k,m,n}\leqslant P_{k,\max,}\qquad\forall k}}C2:p_{k,m,n}\geqslant 0% ,\qquad\forall k,m,nC3:s_{k,m,n}\in\{0,1\},\qquad\forall k,m,nC4:\sum\limits^{% M_{k}}_{m=1}{s_{k,m,n}\leqslant 2,\qquad\forall k,n}C5:\sum\limits^{M_{k}}_{m=% 1}{\sum\limits^{N}_{n=1}{{s_{k,m,n}r}_{k,m,n}\leqslant R_{k,\min,}\qquad% \forall k}}C6:\sum\limits^{K}_{k=1}{\sum\limits^{M_{k}}_{m=1}{\sum\limits^{N}_% {n=1}{s_{k,m,n}p_{k,m,n}{|h_{k,k+1,n}|}^{2}\leqslant I_{\max}}}}$ (23)

3. Energy-aware resource management for noma HetNets

In this scenario the optimization problem of the energy-efficient approach is not a convex problem, and the described goal function is a non-linear fraction.

3.1 Hierarchical power optimization

As shown in the problem formulation, the power utility function depends on both transmission power utilization and total power loss. Based on the problem assumptions, we can formulate the overall energy consumption as Eq. (24)

$\displaystyle P_{\textit{total}}(\{{\rho}_{mk}\},\mathcal{A})=\sum\limits_{m% \in\mathcal{A}}{\sum\limits^{K}_{k=1}{{\Delta}_{m}{\rho}_{mk}+\sum\limits_{m% \in\mathcal{A}}{P_{m}+B\sum\limits_{m\in\mathcal{A}}{\sum\limits^{K}_{k=1}{P_{% bt,m}R_{k}(\{{\rho}_{mk}\},\mathcal{A}),}}}}}$ (24)

In this formulation, the consumed transmission power at small cell $m$ is calculated as ${\Delta}_{m}\mathbb{E}\{||\mathrm{x}_{m}||^{2}\}={\Delta}_{m}\sum\limits^{K}_{% k=1}{{\rho}_{mk}}$ , weighting factor $\mathrm{\Delta}m\geqslant 1$ is applied to determine the power efficiency level of the amplifiers. $\sum\limits_{m\in\mathcal{A}}{P_{m}}$ is relevant to the energy consumption of transmitter and receiver sets associated to active cells and the energy consumed through the baseband processors. As the third part of the total power consumption model $B\sum\limits_{m\in\mathcal{A}}{\sum\limits^{K}_{k=1}{P_{bt,m}R_{k}(\{{\rho}_{% mk}\},\mathcal{A})}}$ , represents the power consumption of the traffic-dependent fronthaul connections that is related to the network bandwidth and spectral efficiency. Consequently, we can formulate our target function as the following

$\displaystyle\begin{array}[]{ll}\text{minimize}&P_{\textit{total}}(\{{\rho}_{% mk}\},\mathcal{A})\\ \{{\rho}_{mk}\geqslant 0\},\mathcal{A}&R_{k}(\{{\rho}_{mk}\},\mathcal{A})% \geqslant{\xi}_{k},\forall k,\\ \text{subject to}&\sum\limits^{K}_{k=1}{{\rho}_{mk}\leqslant P_{\max,m},}% \forall m\in\mathcal{A},\\ \end{array}$ (25)

In which, $P_{\max,m}$ denotes the upper bound of the allowed transmit power of small cells. In this notation, the spectral efficiency constraint of UE $k$ is shown as ${\xi}_{k}$ [b/s/Hz] and it means the user spectral efficiency should not be less than this threshold. It is considerable that all of the power optimization parameters have significant impact on the spectral efficiency because mutual interference.

In order to determine the number of required small cells to provide the user’s spectral efficiency, we can consider ${\nu}_{k}=2^{{{\xi}_{k}{\tau}_{c}}/{({\tau}_{c}-{\tau}_{p})}}-1,\forall k$ . So, the goal function Eq. (25) will be reformulated with respect to the SINR constraints.

$\displaystyle\begin{array}[]{ll}\text{minimize}&\sum\limits_{m\in\mathcal{A}}{% \sum\limits^{K}_{k=1}{{\Delta}_{m}{\rho}_{mk}+\sum\limits_{m\in\mathcal{A}}{P_% {hw,m}}}}\\ \{{\rho}_{mk}\geqslant 0\},\mathcal{A}&{\mathrm{SINR}}_{k}(\{{\rho}_{mk}\},% \mathcal{A})\geqslant{\nu}_{k},\forall k,\\ \text{subject to}&\sum\limits^{K}_{k=1}{{\rho}_{mk}\leqslant P_{\max,m},}% \forall m\in\mathcal{A},\\ \end{array}$ (26)

In this formulation, $P_{hw,m}$ as the overall consumed hardware energy at small cell $m$ is simplified from Eq. (25) in accordance that all the signal to noise/interference ratio constraints is satisfied with achieving the optimal equality:

$\displaystyle P_{hw,m}=P_{m}+B\sum\limits^{K}_{k=1}{P_{bt,m}{\xi}_{k}.}$ (27)

By considering the hardware power utilization as a fixed parameter, the computational complexity of Eq. (26) will be significantly decreased compared to Eq. (25). This assumption leads to modify the non-linear goal function to a linear function. We have also defined some auxiliary factors to make the main problem simplified.

$\displaystyle r_{\mathcal{A}}={\left[\sqrt{{\Delta}_{m_{1}},{\rho}_{m_{1}},1},% \dots,\sqrt{{\Delta}_{m_{|\mathcal{A}|}}{\rho}_{m_{|\mathcal{A}|}}K},\sqrt{% \sum\limits_{m\in\mathcal{A}}{P_{hw,m}}}\right]}^{T}\in{\mathbb{C}}^{|\mathcal% {A}|K+1},$ (28) $\displaystyle z_{k\mathcal{A}}={[\sqrt{z_{m_{1},k}},\dots,\sqrt{z_{m_{|% \mathcal{A}|}k}}]}^{T}\in{\mathbb{C}}^{|\mathcal{A}|},$ (29) $\displaystyle{\mathrm{g}}_{k\mathcal{A}}={[\sqrt{{\mathrm{g}}_{1k}},\dots,% \sqrt{{\mathrm{g}}_{m_{\mathcal{A}}k}}]}^{T}\in{\mathbb{C}}^{|\mathcal{A}|},$ (30) $\displaystyle{\mathrm{U}}_{\mathcal{A}}={[{\mathrm{u}}_{1},\dots,{\mathrm{u}}_% {K}]}^{T}\in{\mathbb{C}}^{|\mathcal{A}|\times K},$ (31) $\displaystyle s_{k\mathcal{A}}={[\sqrt{{\nu}_{k}}({\mathrm{g}}^{T}_{k\mathcal{% A}}{\mathrm{u}}_{{\acute{t}}_{1}},\dots,{\mathrm{g}}^{T}_{k\mathcal{A}}{% \mathrm{u}}_{{\acute{t}}_{|{\mathcal{P}}_{k}\backslash\{k\}|}},.z_{k\mathcal{A% }}\circ{\mathrm{u}}_{1}||,\dots,||z_{k\mathcal{A}}\circ{\mathrm{u}}_{K}.,{% \mathrm{\sigma}}_{\mathrm{DL}})]}^{T}\in{\mathbb{C}}^{K+|{\mathcal{P}}_{k}|},$ (32)

In this formulation, $m_{1^{\prime}},\dots m_{|\mathcal{A}|}$ describe the number of active cells, $\mathcal{A}$ . In the constraint Eq. (30), ${\mathrm{g}}_{m_{\mathcal{A}}k}$ is calculated based on the precoding method we applied. So that, for maximum ratio transmission approach we have ${\mathrm{g}}_{m,k}=N{\gamma}_{mk}$ and the full-pilot zero-forcing precoding gives ${\mathrm{g}}_{m,k}=(N-{\tau}_{p}){\gamma}_{mk}$ . The user association matrix in Eq. (31) consists of $m-th$ row $u^{\prime}_{m}$ and its $k-th$ column is illustrated as ${\mathrm{u}}_{k}={[\sqrt{\rho 1k},\dots,\sqrt{{\rho}_{m_{\mathcal{A}k}}}]}^{T}$ . According to the constraint Eq. (32), ${\acute{t}}_{1},\dots,{\acute{t}}_{|{\mathcal{P}}_{k}\backslash\{k\}|}$ is an index which represents belonging of the UEs to the set ${\mathcal{P}}_{k}\backslash\{k\}$ . Based on these formulation and restrictions, the goal function Eq. (25) can be expressed as follows.

$\displaystyle\mathrm{minimize}\qquad s_{\mathcal{A}}$ (33) $\displaystyle\{{\rho}_{mk}\geqslant 0\},\mathcal{A},s_{\mathcal{A}}$ $\displaystyle\text{Subject to\ \ \ }||r_{\mathcal{A}}||\leqslant s_{\mathcal{A% }},$ (34) $\displaystyle\phantom{\text{Subject to\ \ \ }}||s_{k\mathcal{A}}||\leqslant{% \mathrm{g}}^{T}_{k\mathcal{A}}{\mathrm{u}}_{k\mathcal{A}},\forall k=1,\dots,K,$ (35) $\displaystyle\phantom{\text{Subject to\ \ \ }}||{\acute{\mathrm{u}}}_{m}||% \leqslant\sqrt{P_{\max,m}},\forall m\in\mathcal{A}.$ (36)

In this scenario, we define the activation pattern of the small cells by using a binary variable ${\alpha}_{m}\in\{0,1\}$ which indicates the on/off activity of small cell $m$ . For inactive small cells, ${\alpha}_{m}=0$ , we can either set their transmit power equal to zero, $\{{\rho}_{m1},{\rho}_{m2},\dots,{\rho}_{mK}\}=0$ or replace their maximum transmission power by ${{\alpha}^{2}_{m}P}_{\max,m}$ . Consider the mixed-integer second order cone program of the main problem as the following.

$\displaystyle\mathrm{minimize}\qquad S$ (37) $\displaystyle\{{\rho}_{mk}\geqslant 0\},\{{\alpha}_{m}\},s$ $\displaystyle\text{Subject to\ \ \ }||r||\leqslant s,$ (38) $\displaystyle\phantom{\text{Subject to\ \ \ }}||s_{k}||\leqslant{\mathrm{g}}^{% T}_{k}{\mathrm{u}}_{k},\forall k=1,\dots,K,$ (39) $\displaystyle\phantom{\text{Subject to\ \ \ }}||{\acute{\widetilde{\mathrm{u}}% }}_{m}||\leqslant{\alpha}_{m}\sqrt{P_{\max,m}},\forall m=1,\dots,M,$ (40) $\displaystyle\phantom{\text{Subject to\ \ \ }}{\alpha}_{m}\in\{0,1\},\forall m% =1,\dots,M,$ (41)

In which ${\acute{\widetilde{\mathrm{u}}}}_{m}$ denotes the $m-th$ row of the matrix $\widetilde{\mathrm{U}}=[{\widetilde{\mathrm{u}}}_{1},\dots,{\widetilde{\mathrm% {u}}}_{K}]\in{\mathbb{C}}^{M\times K}$ and it is obvious that ${\widetilde{\mathrm{u}}}_{k}={[\sqrt{\rho 1k},\dots,\sqrt{{\rho}_{MK}}]}^{T}% \in{\mathbb{C}}^{M},k=1,\dots,K$ . Subsequently, the vectors $r$ and $s_{k}$ can be expressed as

$\displaystyle r={[\sqrt{{\Delta}_{1}{\rho}_{11}},\dots,\sqrt{{\Delta}_{M}{\rho% }_{MK}},{\alpha}_{1}\sqrt{P_{hw,1}},\dots,{\alpha}_{M}\sqrt{P_{hw,M}}]}^{T}\in% {\mathbb{C}}^{MK+M},$ (42) $\displaystyle s_{k}={[\sqrt{{\nu}_{k}}({\mathrm{g}}^{T}_{k}{\mathrm{u}}_{{% \acute{t}}_{1}},\dots,{\mathrm{g}}^{T}_{k}{\mathrm{u}}_{{\acute{t}}_{|{% \mathcal{P}}_{k}\backslash\{k\}|}},||z_{k}\circ{\mathrm{u}}_{1}||,\dots,||z_{k% }\circ{\mathrm{u}}_{K}||,{\mathrm{\sigma}}_{\mathrm{DL}})]}^{T}\in{\mathbb{C}}% ^{K+|{\mathcal{P}}_{k}|},$ (43) $\displaystyle z_{k}={[\sqrt{z_{1k}},\dots,\sqrt{z_{Mk}}]}^{T}\in{\mathbb{C}}^{% M},$ (44) $\displaystyle{\mathrm{g}}_{k}={[\sqrt{{\mathrm{g}}_{1k}},\dots,\sqrt{{\mathrm{% g}}_{Mk}}]}^{T}\in{\mathbb{C}}^{M}.$ (45)

Based on the descriptions, the main problem Eq. (37) can be considered equivalent to an optimal transmit power problem. So that if $\{{\alpha}^{*}_{m}\}$ assumed to be the optimal solution for $\{{\alpha}_{m}\}$ , achieved by solving Eq. (26), we can determine the optimal set of active small cells as the following.

$\displaystyle\mathcal{A}=\{m:{\alpha}^{*}_{m}=1,m\in\{1,\dots,M\}\}.$ (46)

If we apply the standard interior-point (SIP) approach in order to obtain the optimal value of the mixed-integer second order cone program, we have the following time complexity of branch-and-bound (BAB) method for solving Eq. (37) as follows.

$\displaystyle\mathrm{ln}({\varepsilon}^{-1})\sum\limits^{N_{1}}_{n=1}{\sum% \limits_{i\in\{0,1\}}{\mathcal{O}(C^{(n),\bm{ub}}_{i})+\mathcal{O}(C^{(n),% \mathrm{lb}}_{i}),}}$ (47)

In this formulation, $\varepsilon>0$ shows the precision level of second order cone program’s solution in different iterations. $N_{1}(N_{1}\leqslant 2^{M}-1)$ represents the number of iterations required for the BAB algorithm to obtain the best solution. However, $C^{(n),\bm{ub}}_{i}$ and $C^{(n),\mathrm{lb}}_{i}$ illustrate the cost of calculation of the lower and upper bounds. These two indicators are achievable as the following.

$\displaystyle C^{(n),\mathrm{lb}}_{i}=\sqrt{L^{(n)}_{i2}+K^{2}+L^{(n)}_{i1}K+Z% ^{(n)}}{}\times\left({(L^{(n)}_{i2})}^{3}+L^{(n)}_{i2}\sum\limits^{K}_{k=1}{|{% \mathcal{P}}_{k}|+}L^{(n)}_{i2}L^{(n)}_{i1}K^{2}+Z^{(n)}K\right),$ (48) $\displaystyle C^{(n),\bm{ub}}_{i}={(U^{(n)}_{i})}^{3}K^{3}\sqrt{U^{(n)}_{i}K+K% ^{2}}.$ (49)

In which, $M^{(n-1)}_{0}$ and $M^{(n-1)}_{1}$ demonstrate the number of small cells which were active and inactive, respectively, these variables are obtained based on the former iteration. The initial state is $M^{(1)}_{0}=M^{(2)}_{1}=0$ . And, $T^{(n)}=M-M^{(n-1)}_{0},{\mathcal{Q}}^{(n)}=M-M^{(n-1)}_{0}-M^{(n-1)}_{1},Z^{(% n)}=K(Q^{(n)}-1)$ and $L^{(n)}_{i1},L^{(n)}_{i2},U^{(n)}_{i}$ are achieved based on the value of the binary function as Table 1.

Table 1

Parameters’ value based on the binary function

Parameter	$i=0$	$i=1$
$L^{(n)}_{i1}$	$M^{(n-1)}_{1}$	$M^{(n-1)}_{1}+1$
$L^{(n)}_{i2}$	$(T^{(n)}-1)K+{\mathcal{Q}}^{(n)}$	$T^{(n)}K+{\mathcal{Q}}^{(n)}$
$U^{(n)}_{i}$	$T^{(n)}-1$	$T^{(n)}$

3.2 RRM and HO optimization in NOMA HetNets

In this model the optimization problem of the handover optimization and channel selection approach is a non-convex problem. To find the optimal solution for this problem, we have to modify the formulation. At the beginning, the model presented in [31] was applied in order to estimate the convex alteration to find the best channel selection strategy.

Channel allocation and inter-layer handover optimization

In the reference [32], it is assumed that each primary user is assigned a separate channel and the primary user activity model is according to an ON-OFF process. This article also assumes that each secondary user has been equipped with two radios. One is used to send data and control messages and the other is used to scan all channels and obtain statistical information from them. Based on the statistical information, each secondary user measures two criteria using prediction.

1.
Probability of occupying or emptiness the current channel and the target channel,
2.
The expected duration for the channel emptiness or the mathematical hope of the emptiness time the channel.

To calculate the first criterion, it is assumed that $X^{k}_{i}$ is the time between arrival of the $k^{\text{th}}$ package and $L^{k}_{i}$ is the length of the $k^{\text{th}}$ package of the primary user in the $i^{\text{th}}$ channel. The probability that the $i^{\text{th}}$ channel is empty at moment $t$ is:

$\displaystyle{\mathrm{Pr}(N_{i}(t)=0)}={\mathrm{Pr}(X^{1}_{i}>t-t_{0}+L^{0}_{i% })}$ (50) $\displaystyle\quad+\sum\limits^{U}_{h-1}{\left[{\mathrm{Pr}\left(\sum\limits^{% h}_{k-1}{X^{k}_{i}+L^{h}_{i}<t-t_{0}+L^{0}_{i}}\right)}{\mathrm{Pr}\left(\sum% \limits^{h+1}_{k-1}{X^{k}_{i}>t-t_{0}+L^{0}_{i}}\right)}\right]}$

Where $U$ represents the maximum number of incoming packets from the primary user between $t_{0}$ to $t$ .

$\displaystyle{\mathrm{Pr}(t_{i,\textit{off}}>\eta|N_{i}(t)=0)}$ $\displaystyle\quad=\left\{\begin{array}[]{l}{\mathrm{Pr}(X^{1}_{i}>t-t_{0}+L^{% 0}_{i}+\eta)}\\ +\sum\limits^{U}_{h-1}{\left[{\mathrm{Pr}\left(\sum\limits^{h}_{k-1}{X^{k}_{i}% +L^{h}_{i}<t-t_{0}+L^{0}_{i}}\right)}{\mathrm{Pr}\left(\sum\limits^{h+1}_{k-1}% {X^{k}_{i}>t-t_{0}+L^{0}_{i}+\eta}\right)}\right]}\end{array}\right\}$ (51) $\displaystyle\qquad\left/\left\{\begin{array}[]{l}{\mathrm{Pr}(X^{1}_{i}>t-t_{% 0}+L^{0}_{i})}\\ +\sum\limits^{U}_{h-1}{\left[{\mathrm{Pr}\left(\sum\limits^{h}_{k-1}{X^{k}_{i}% +L^{h}_{i}<t-t_{0}+L^{0}_{i}}\right)}{\mathrm{Pr}\left(\sum\limits^{h+1}_{k-1}% {X^{k}_{i}>t-t_{0}+L^{0}_{i}}\right)}\right]}\end{array}\right\}\right.$

By applying the thresholds, we will have:

1.
Despite the following condition, the secondary user must switch to another channel

$\displaystyle{\mathrm{Pr}(N_{i}(t)=0)}<{\tau}_{L}$ (52)
2.
The condition for the fact that channel $j$ is one of the selected channels at moment $t$ :

$\displaystyle\left\{\begin{array}[]{l}{\mathrm{Pr}(N_{j}(t)=0)}\geqslant{\tau}% _{H}\\ {\mathrm{Pr}(t_{i,\textit{off}}>\eta|N_{j}(t)=0)\geqslant\theta}\end{array}\right.$ (53)

Therefore, the strategy space will be $S^{\textit{mixed}}_{i}=\{(\textit{switch}=p),(\textit{stay}=(1-p))\}$ and all players have the above strategy, in other words, the value of $p$ is the same for all players. To solve a special mode, all players are in the same band at the beginning of the game. So, there will be:

$\displaystyle E[C^{\textit{switch}}_{i}]=\sum\limits^{N-1}_{j=0}{Q_{j}\times c% ^{f(N,M)}}$ (54)

Where, $j$ indicates the number of players who choose the switch strategy and $Q_{j}$ represents the probability of switching $j$ players from $N-1$ players.

$\displaystyle Q_{j}=\left(\begin{array}[]{c}N-1\\ j\end{array}\right)p^{j}{(1-p)}^{N-1-j}$ (55)

So we will have.

$\displaystyle\left\{\begin{array}[]{l}E[C^{\textit{switch}}_{i}]=c^{f(N,M)}\\ E[C^{\textit{stay}}_{i}]=\sum\limits^{N-2}_{j=0}{Q_{j}(1+E[C_{i}({G}_{(N-j)}^{% \prime})])}+Q_{N-1}\times 0\end{array}\right.$ (56)

Where $E[C_{i}({G}_{(N-j)}^{\prime})]$ represents the expected cost of the subgame ${G}_{(N-j)}^{\prime}$ .

If $E[C^{\textit{switch}}_{i}]<E[C^{\textit{stay}}_{i}]$ or $E[C^{\textit{switch}}_{i}]>E[C^{\textit{stay}}_{i}]$ will never reach Nash equilibrium according to the superiority strategy and previous theorem. Therefore, for a Nash equilibrium strategy to exist, there must be $E[C^{\textit{switch}}_{i}]=E[C^{\textit{stay}}_{i}]$ , meaning that the $i^{\text{th}}$ player is indifferent to the stay or switch strategy, regardless of the strategy of the other players. In other words, multiples ( $p,1-p)$ help to choose a strategy that, for the $i^{\text{th}}$ player, no other user’s response will lead to the strategies being superior to each other.

$\displaystyle E[C^{\textit{nash}}_{i}]=c^{f(N,M)}=\sum\limits^{N-2}_{j=0}{Q_{j% }(1+E[C_{i}({G^{\prime}}_{(N-j)})])}$ (57)

According to the above equation, it can be said that the expected cost in the Nash equilibrium does not depend on $j$ but on $N$ and $M$ . Therefore, it is inferred that the expected cost for the $i^{\text{th}}$ player in the ${G}_{(N-j)}^{\prime}$ sub-game is the same as in the main game.

$\displaystyle\sum\limits^{N-2}_{j=0}{Q_{j}}=\frac{c^{f(N,M)}}{1+c^{f(N,M)}}{{{% {\xlongrightarrow{\mathrm{\ }binomial\ expansion}}}}}Q_{N-1}=\frac{1}{1+c^{f(N% ,M)}}$ (58)

Consequently, the value of $p$ is obtained as follows:

$\displaystyle p={\left(\frac{1}{1+c^{f(N,M)}}\right)}^{\frac{1}{N-1}}$ (59)

Two utility functions have been presented for the HO optimization problem:

1.
This utility function has been designed for selfish players and is defined as follows:

$\displaystyle{U1}_{i}(s_{i},s_{-i})=-\sum\limits^{N}_{j\neq i,j=1}{p_{j}G_{ij}% f(s_{j},s_{i})}\forall i=1,2,\dots,N$ (60)

Where $P=[p_{1},p_{2},\dots,p_{N}]$ is the secondary user’s $N$ sending powers, and $S=[s_{1},s_{2},\dots,s_{N}]$ is the strategies profile and $G_{ij}$ is the gain of the link between transmitter $i$ and receiver $j$ , and $f(s_{j},s_{i})$ is an interference function defined as follows:

$\displaystyle f(s_{j},s_{i})=\left\{\begin{array}[]{ll}1&\text{if }s_{j}=s_{i}% ,\text{transmitter }j\text{ and }i\text{ choose the same channel}\\ 0&\text{otherwise}\\ \end{array}\right.$
2.
In addition to the interference received by the $i^{\text{th}}$ user and the interference created by the $i^{\text{th}}$ user on neighboring nodes, this utility function is also considered in the utility function. In fact, this is the appropriate utility function for the players who can participate together.

$\displaystyle{U2}_{i}(s_{i},s_{-i})=-\sum\limits^{N}_{j\neq i,j=1}{p_{j}G_{ij}% f(s_{j},s_{i})}-\sum\limits^{N}_{j\neq i,j=1}{p_{i}G_{ji}f(s_{i},s_{j})}$ (61) $\displaystyle\forall i=1,2,\dots,N$

Formally, this game is defined according to Eq. (58).

$\displaystyle\mathit{\Gamma}=<\mathcal{N},{\{{\mathcal{F}}_{\bm{i}}\}}_{i\in% \mathcal{N}},{\{U_{\bm{i}}\}}_{i\in\mathcal{N}}>$ (62)

Where $\mathcal{N}=\{{SU}_{1},{SU}_{2},\dots,{SU}_{N}\}$ is a set of players or the pair of the secondary users and ${\mathcal{F}}_{\bm{i}}$ represents the set of resources available for the $i^{\text{th}}$ player, $U_{\bm{i}}$ also represents the utility of the $i^{\text{th}}$ player. In the game output analysis, as players make their decisions independently and are influenced by other players’ decisions, we are interested in identifying a convergence point for the game if possible. So we have:

$\displaystyle u_{i}({\sigma}_{i},{\bm{\sigma}}_{-i})={-W}_{1}\times\sum\limits% ^{|\mathcal{N}|}_{j=1}{{\alpha}^{{\sigma}_{i}}\times f^{\bm{\sigma}}(i,j)}-K% \times W_{2}\times{\textit{SCM}}^{{\sigma}_{i}}_{i}-W_{3}\times{\textit{price}% }^{{\sigma}_{i}}_{i}$ (63)

The variable $K$ is the effect of the cost of channel switching, and we will talk about weights $W_{1}$ , $W_{2}$ , and $W_{3}$ in the following sections. The utility functions Eq. (60) was presented. For the second state, the utility function is presented as follows:

$\displaystyle u_{i}({\sigma}_{i},{\sigma}_{-i})=-W^{i}_{1}\times\left(\left(% \left(\sum\limits^{|\mathcal{N}|}_{j=1}{f^{\bm{\sigma}}(i,j)}\right)+1\right)% \times\alpha^{\sigma_{i}}\right){}\!-K\times W^{i}_{2}\times\textit{SCM}(% \textit{psc}_{i},{\sigma}_{i})-W^{i}_{3}\times\textit{price}^{{\sigma}_{i}}_{i}A$ (64)

potential function for Eq. (64) is as follows:

$\displaystyle\mathit{\Phi}(\bm{\sigma})=\mathit{\Phi}({\sigma}_{i},{\sigma}_{-% i})=\left(\begin{array}[]{c}\left(\begin{array}[]{c}-\frac{1}{2}\times\left(% \sum\limits^{|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}_{\bm{r}}}{% \sum\limits_{l\in{\mathcal{N}}_{\bm{r}}}{{\alpha}^{{\sigma}_{j}}\times g^{% \sigma}_{r}(j,l)}}}\right)\\ +\frac{1}{2}\times\left(\sum\limits^{|\mathcal{N}|}_{k=1}{\sum\limits^{|% \mathcal{N}|}_{t=1}{(\textit{count}(k,t)-1)\times{\alpha}^{{\sigma}_{k}}}% \times f^{\sigma}(k,t)}\right)\end{array}\right)\\ -\sum\limits^{|\mathcal{N}|}_{d=1}{{\alpha}^{{\sigma}_{d}}}-K\times\sum\limits% ^{|\mathcal{N}|}_{d=1}{\frac{W^{d}_{2}}{W^{d}_{1}}\textit{SCM}(\textit{psc}_{d% },{\sigma}_{d})}-\sum\limits^{|\mathcal{N}|}_{d=1}{\frac{W^{d}_{3}}{W^{d}_{1}}% \textit{price}^{{\sigma}_{d}}_{d}}\end{array}\right)$ (65)

Note that the potential function is a weighted equation introduced in the previous section. Therefore Eq. (65) confirms the following equation.

$\displaystyle{\Delta{u}}_{{i}}=W^{i}_{1}\times\Delta\Phi$ (66)

Secondary users can use the new utility function to adjust the weights of bandwidth importance, price preference, and channel switching according to their demands. For example, the users who have a high coding rate either use advanced modulation or do not need real-time and instead, the spectrum trading cost is important to them, they can achieve their goal by losing weight W1 and increasing weight W3. Considering the primary problem, we have $\Delta u_{i}=\Delta{\Phi}\forall i\in\mathcal{N}$ . Based on the problem assumption, we obtain $\bm{\Delta}{\bm{u}}_{\bm{i}}$ as the following.

$\displaystyle u_{i}({\sigma}_{i},{\sigma}_{-i})=W_{1}\times\left(-\sum\limits^% {|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}_{\bm{r}}}{{\alpha}^{{% \sigma}_{i}}\times g^{\sigma}_{r}(i,j)}}\right){}\!+W_{1}\times\left(\sum% \limits^{|\mathcal{N}|}_{l=1}((\textit{count}(i,l)-1)\times{\alpha}^{{\sigma}_% {i}}\times f^{\sigma}(i,l)\right){}\!-K\times W_{2}\times{\textit{SCM}}^{{% \sigma}_{i}}_{i}-W_{3}\times{\textit{price}}^{{\sigma}_{i}}_{i}$ (67)

By substituting $\textit{count}(i,l)$ in the previous relationship we will have:

$\displaystyle u_{i}({\sigma}_{i},{\sigma}_{-i})=-W_{1}\times\left(\sum\limits^% {|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}_{\bm{r}}}{{\alpha}^{{% \sigma}_{i}}\times g^{\sigma}_{r}(i,j)}}\right){}\!+W_{1}\times\left(\sum% \limits^{|\mathcal{N}|}_{l=1}{\left(\left(\sum\limits^{|\mathcal{R}|}_{r=1}{g^% {\sigma}_{r}(i,l)}\right)-1\right)\times{\alpha}^{{\sigma}_{i}}\times f^{% \sigma}(i,l)}\right){}\!-K\times W_{2}\times{\textit{SCM}}^{{\sigma}_{i}}_{i}-% W_{3}\times{\textit{price}}^{{\sigma}_{i}}_{i}$ (68)

As we know, in the strategy $\bm{\sigma}=({\sigma}_{i},{\sigma}_{-i})$ , player $i$ can only interact with players ${\mathcal{N}}^{1}$ , therefore, the term $g^{\sigma}_{r}(i,j)$ in the first parenthesis is equal to zero for ${\mathcal{N}}^{\bm{2}}_{\bm{r}}$ and ${\mathcal{N}}^{\bm{3}}_{\bm{r}}$ . Also, the term $f^{\sigma}(i,l)$ in the second parenthesis for ${\mathcal{N}}^{2}$ and ${\mathcal{N}}^{3}$ is zero. So, we will have:

$\displaystyle u_{i}({\sigma}_{i},{\sigma}_{-i})=-W_{1}\times\left(\sum\limits^% {|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}^{\bm{1}}_{\bm{r}}}{{% \alpha}^{{\sigma}_{i}}\times g^{\sigma}_{r}(i,j)}}\right){}\!+W_{1}\times\left% (\sum\limits_{l\in{\mathcal{N}}^{1}}{\left(\left(\sum\limits^{|\mathcal{R}|}_{% r=1}{g^{\sigma}_{r}(i,l)}\right)-1\right)\times{\alpha}^{{\sigma}_{i}}\times f% ^{\sigma}(i,l)}\right){}\!-K\times W_{2}\times{\textit{SCM}}^{{\sigma}_{i}}_{i% }-W_{3}\times{\textit{price}}^{{\sigma}_{i}}_{i}$ (69)

There is exactly the same simplification for the term $u_{i}({\sigma}^{\prime}_{i},{\sigma}_{-i})$ as we have:

$\displaystyle u_{i}({\sigma}^{\prime}_{i},{\sigma}_{-i})=-W_{1}\times\left(% \sum\limits^{|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}^{\bm{2}}_{\bm% {r}}}{{\alpha}^{{\sigma}^{\prime}_{i}}g^{{\sigma}^{\prime}}_{r}(i,j)}}\right){% }\!+W_{1}\times\left(\sum\limits_{l\in{\mathcal{N}}^{2}}{\left(\left(\sum% \limits^{|\mathcal{R}|}_{r=1}{g^{{\sigma}^{\prime}}_{r}(i,l)}\right)-1\right)% \times{\alpha}^{{\sigma}^{\prime}_{i}}\times f^{{\sigma}^{\prime}}(i,l)}\right% ){}\!-K\times W_{2}\times{\textit{SCM}}^{\sigma^{\prime}_{i}}_{i}-W_{3}\times{% \textit{price}}^{\sigma^{\prime}_{i}}_{i}$ (70)

To further simplify the proof, we use the following two term substitutions:

$\displaystyle\left\{\begin{array}[]{c}A(k,t)=\left(\left(\sum\limits^{|% \mathcal{R}|}_{r=1}{g^{\sigma}_{r}(k,t)}\right)-1\right){\alpha}^{{\sigma}_{i}% }f^{\sigma}(k,t)\\ B(k,t)=\left(\left(\sum\limits^{|\mathcal{R}|}_{r=1}{g^{{\sigma}^{\prime}}_{r}% (k,t)}\right)-1\right){\alpha}^{{\sigma}^{\prime}_{i}}f^{{\sigma}^{\prime}}(k,% t)\end{array}\right.$ (71)

Therefore, for two utility functions $u_{i}(\bm{\sigma})$ and $u_{i}({\bm{\sigma}}^{\bm{{}^{\prime}}})$ we will have:

$\displaystyle u_{i}({\sigma}_{i},{\sigma}_{-i})=-W_{1}\times\left(\sum\limits^% {|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}^{\bm{1}}_{\bm{r}}}{{% \alpha}^{{\sigma}_{i}}\times g^{\sigma}_{r}(i,j)}}\right){}\!+W_{1}\times\sum% \limits_{l\in{\mathcal{N}}^{1}}{A(i,l)}-K\times W_{2}\times{\textit{SCM}}^{{% \sigma}_{i}}_{i}-W_{3}\times{\textit{price}}^{{\sigma}_{i}}_{i}$

And, we have

$\displaystyle u_{i}({\sigma}^{\prime}_{i},{\sigma}_{-i})=-W_{1}\times\left(% \sum\limits^{|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}^{\bm{2}}_{\bm% {r}}}{{\alpha}^{{\sigma}^{\prime}_{i}}g^{{\sigma}^{\prime}}_{r}(i,j)}}\right){% }\!+W_{1}\times\sum\limits_{l\in{\mathcal{N}}^{2}}{B(i,l)}-K\times W_{2}\times% {\textit{SCM}}^{\sigma^{\prime}_{i}}_{i}-W_{3}\times{\textit{price}}^{\sigma^{% \prime}_{i}}_{i}$

We get the following relationship:

$\displaystyle\Delta u_{i}=\left(\begin{array}[]{c}-W_{1}\times\left(\sum% \limits^{|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}^{\bm{1}}_{\bm{r}}% }{{\alpha}^{{\sigma}_{i}}\times g^{\sigma}_{r}(i,j)}}\right)+W_{1}\times\sum% \limits_{l\in{\mathcal{N}}^{1}}{A(i,l)}\\ -K\times W_{2}\times{\textit{SCM}}^{{\sigma}_{i}}_{i}-W_{3}\times{\textit{% price}}^{{\sigma}_{i}}_{i}\end{array}\right){}\!-\left(\begin{array}[]{c}-W_{1% }\times\left(\sum\limits^{|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}^% {\bm{2}}_{\bm{r}}}{\alpha^{\sigma^{\prime}_{i}}g^{\sigma^{\prime}}_{r}(i,j)}}% \right)+W_{1}\times\sum\limits_{l\in{\mathcal{N}}^{2}}{B(i,l)}\\ -K\times W_{2}\times{\textit{SCM}}^{\sigma^{\prime}_{i}}_{i}-W_{3}\times{% \textit{price}}^{\sigma^{\prime}_{i}}_{i}\end{array}\right)$ (72)

Considering the potential function $\mathit{\Phi}(\bm{\sigma})$ in the former expression, we divide this proposed potential function into four parts as follows:

$\displaystyle\left\{\begin{array}[]{l}{{\mathit{\Phi}}_{1}(\bm{\sigma})=\sum% \limits^{|\mathcal{R}|}_{r=1}{\sum\limits_{j\in{\mathcal{N}}_{\bm{r}}}{\sum% \limits_{l\in{\mathcal{N}}_{\bm{r}}}{{\alpha}^{{\sigma}_{j}}\times g^{\sigma}_% {r}(j,l)}}}}\\ {{\mathit{\Phi}}_{2}(\bm{\sigma})=\sum\limits^{|\mathcal{N}|}_{k=1}{\sum% \limits^{|\mathcal{N}|}_{t=1}{(\textit{count}(k,t)-1)\times{\alpha}^{{\sigma}_% {k}}}\times f^{\sigma}(k,t)}}\\ {{\mathit{\Phi}}_{3}(\bm{\sigma})=\sum\limits^{|\mathcal{N}|}_{d=1}{{\textit{% SCM}}^{{\sigma}_{d}}_{d}}}\\ {{\mathit{\Phi}}_{4}(\bm{\sigma})=\sum\limits^{|\mathcal{N}|}_{d=1}{{\textit{% price}}^{{\sigma}_{d}}_{d}}}\end{array}\right.$ (73)

Hence, we will have:

$\displaystyle\mathit{\Phi}(\bm{\sigma})=\mathit{\Phi}({\sigma}_{i},{\sigma}_{-% i})=W_{1}\times\left(-\frac{1}{2}\times{\mathit{\Phi}}_{1}(\bm{\sigma})+\frac{% 1}{2}\times{\mathit{\Phi}}_{2}(\bm{\sigma})\right)-K\times W_{2}\times{\mathit% {\Phi}}_{3}(\bm{\sigma}){}\!-W_{3}\times{\mathit{\Phi}}_{4}(\bm{\sigma})$ (74)

There is a similar division for $\mathit{\Phi}(\bm{\sigma}^{\prime})$ :

$\displaystyle\mathit{\Phi}({\bm{\sigma}}^{\prime})=\mathit{\Phi}({\sigma}^{% \prime}_{i},{\sigma}_{-i})=W_{1}\times\left(-\frac{1}{2}\times{\mathit{\Phi}}_% {1}(\bm{\sigma}^{\prime})+\frac{1}{2}\times{\mathit{\Phi}}_{2}(\bm{\sigma}^{% \prime})\right)-K\times W_{2}\times{\mathit{\Phi}}_{3}({\bm{\sigma}}^{\prime})% {}\!-W_{3}\times{\mathit{\Phi}}_{4}({\bm{\sigma}}^{\prime})$ (75)

By separating the difference of the potential, there is:

$\displaystyle\left\{\begin{array}[]{c}{{\Delta\mathit{\Phi}}_{1}={\mathit{\Phi% }}_{1}(\bm{\sigma})-{\mathit{\Phi}}_{1}({\bm{\sigma}}^{\prime})}\\ {{\Delta\mathit{\Phi}}_{2}={\mathit{\Phi}}_{2}(\bm{\sigma})-{\mathit{\Phi}}_{2% }({\bm{\sigma}}^{\prime})}\\ {{\Delta\mathit{\Phi}}_{3}={\mathit{\Phi}}_{3}(\bm{\sigma})-{\mathit{\Phi}}_{3% }({\bm{\sigma}}^{\prime})}\\ {{\Delta\mathit{\Phi}}_{4}={\mathit{\Phi}}_{4}(\bm{\sigma})-{\mathit{\Phi}}_{4% }({\bm{\sigma}}^{\prime})}\end{array}\right.$ (76)

Consequently,

$\displaystyle\Delta\mathit{\Phi}=W_{1}\times\left(-\frac{1}{2}\times({\Delta% \mathit{\Phi}}_{1})+\frac{1}{2}\times({\Delta\mathit{\Phi}}_{2})\right)-K% \times W_{2}\times({\Delta\mathit{\Phi}}_{3})-W_{3}\times({\Delta\mathit{\Phi}% }_{4})$ (77)

Therefore, it is necessary to perform four calculations ${\Delta\mathit{\Phi}}_{1}$ , ${\Delta\mathit{\Phi}}_{2}$ , ${\Delta\mathit{\Phi}}_{3}$ and ${\Delta\mathit{\Phi}}_{4}$ and replace them in $\Phi_{1}(\sigma)=\Phi_{1}(\sigma_{i},\sigma_{-i})$ . Also note that $g^{\sigma}_{r}(i,i)=0$ because we assume that no player is its neighbor. We also have ${\mathit{\Phi}}_{1}(\bm{\sigma}^{\prime}):{\mathit{\Phi}}_{1}(\bm{\sigma}^{% \prime})={\mathit{\Phi}}_{1}({\sigma}^{\prime}_{i},{\sigma}_{-i})$

In this equation, according to the value of the $g^{\sigma}_{r}(j,l)$ term, some terms become zero. Since the two equations the equalities $A=A^{\prime}$ , $B=B^{\prime}$ and $C=C^{\prime}$ are established, so they can be neglected in the equation ${\mathit{\Phi}}_{1}(\bm{\sigma})-{\mathit{\Phi}}_{1}(\bm{\sigma}^{\prime})$ and so we have:

$\displaystyle{\Delta\mathit{\Phi}}_{1}={\mathit{\Phi}}_{1}(\sigma)-{\mathit{% \Phi}}_{1}({\sigma}^{\prime})=\sum\limits^{|\mathcal{R}|}_{r=1}{\left(\sum% \limits_{j\in{\mathcal{N}}^{\bm{1}}_{\bm{r}}}{{\alpha}^{{\sigma}_{i}}g^{\sigma% }_{r}(j,i)}+\sum\limits_{l\in{\mathcal{N}}^{\bm{1}}_{\bm{r}}}{{\alpha}^{{% \sigma}_{i}}g^{\sigma}_{r}(i,l)}\right)}{}\!-\sum\limits^{|\mathcal{R}|}_{r=1}% {\left(\sum\limits_{j\in{\mathcal{N}}^{\bm{2}}_{\bm{r}}}{({\alpha}^{{\sigma}^{% \prime}_{i}}g^{{\sigma}^{\prime}}_{r}(j,i))}+\sum\limits_{l\in{\mathcal{N}}^{% \bm{2}}_{\bm{r}}}{{\alpha}^{{\sigma}^{\prime}_{i}}g^{{\sigma}^{\prime}}_{r}(i,% l)}\right)}=\sum\limits^{|\mathcal{R}|}_{r=1}{\left(2\times\sum\limits_{j\in{% \mathcal{N}}^{\bm{1}}_{\bm{r}}}{{\alpha}^{{\sigma}_{i}}g^{\sigma}_{r}(j,i)}% \right)}-\sum\limits^{|\mathcal{R}|}_{r=1}{\left(2\times\sum\limits_{j\in{% \mathcal{N}}^{\bm{2}}_{\bm{r}}}{({\alpha}^{{\sigma}^{\prime}_{i}}g^{{\sigma}^{% \prime}}_{r}(j,i))}\right)}=2\times\sum\limits^{|\mathcal{R}|}_{r=1}{\left(% \sum\limits_{j\in{\mathcal{N}}^{\bm{1}}_{\bm{r}}}{{\alpha}^{{\sigma}_{i}}g^{% \sigma}_{r}(j,i)}-\sum\limits_{j\in{\mathcal{N}}^{\bm{2}}_{\bm{r}}}{({\alpha}^% {{\sigma}^{\prime}_{i}}g^{{\sigma}^{\prime}}_{r}(j,i))}\right)}$ (78)

A similar approach as to the calculation of ${\Delta\mathit{\Phi}}_{1}$ was considered in the calculation of ${\Delta\mathit{\Phi}}_{2}$ as follows:

$\displaystyle{\Delta\mathit{\Phi}}_{2}={\mathit{\Phi}}_{2}(\sigma)-{\mathit{% \Phi}}_{2}({\sigma}^{\prime})=\sum\limits^{|\mathcal{N}|}_{k=1}{\sum\limits^{|% \mathcal{N}|}_{t=1}{(\textit{count}(k,t)-1)\times{\alpha}^{{\sigma}_{k}}\times f% ^{\sigma}(k,t)}}{}\!-\sum\limits^{|\mathcal{N}|}_{k=1}{\sum\limits^{|\mathcal{% N}|}_{t=1}{(\textit{count}(k,t)-1)\times{\alpha}^{{\sigma}^{\prime}_{k}}\times f% ^{{\sigma}^{\prime}}(k,t)}}$ (79)

By substituting $\textit{count}(k,t)$ in the formulation, we will have:

$\displaystyle{\Delta\mathit{\Phi}}_{2}=\sum\limits^{|\mathcal{N}|}_{k=1}{\sum% \limits^{|\mathcal{N}|}_{t=1}{\left(\left(\sum\limits^{|\mathcal{R}|}_{r=1}{g^% {\sigma}_{r}(k,t)}\right)-1\right)\times{\alpha}^{{\sigma}_{k}}\times f^{% \sigma}(k,t)}}{}\!-\sum\limits^{|\mathcal{N}|}_{k=1}{\sum\limits^{|\mathcal{N}% |}_{t=1}{\left(\left(\sum\limits^{|\mathcal{R}|}_{r=1}{g^{{\sigma}^{\prime}}_{% r}(k,t)}\right)-1\right)\times{\alpha}^{{\sigma}^{\prime}_{k}}\times f^{{% \sigma}^{\prime}}(k,t)}}$ (80)

By substitution we will have: ${\Delta\mathit{\Phi}}_{2}=\sum\limits^{|\mathcal{N}|}_{k=1}{\sum\limits^{|% \mathcal{N}|}_{t=1}{A(k,t)}}-\sum\limits^{|\mathcal{N}|}_{k=1}{\sum\limits^{|% \mathcal{N}|}_{t=1}{B(k,t)}}$ . In which, due to the presence of expression $g^{\sigma}_{r}(j,l)$ , some terms become zero and given the equalities $A=A^{\prime}$ , $B=B^{\prime}$ and $C=C^{\prime}$ in the formulation cancel each other out and can be ignored. It is also recalled that the reason the fact that the expressions $A(i,i)$ and $B(i,i)$ are zero is that we have assumed that no player is their neighbor and therefore we have.

$\displaystyle{\Delta\mathit{\Phi}}_{2}={\mathit{\Phi}}_{2}(\sigma)-{\mathit{% \Phi}}_{2}({\sigma}^{\prime})=\left(\sum\limits_{k\in{\mathcal{N}}^{1}}{A(k,i)% }+\sum\limits_{t\in{\mathcal{N}}^{1}}{A(i,t)}\right)-\left(\sum\limits_{k\in{% \mathcal{N}}^{2}}{B(k,i)}+\sum\limits_{t\in{\mathcal{N}}^{2}}{B(i,t)}\right)=2% \times\left(\sum\limits_{k\in{\mathcal{N}}^{1}}{A(k,i)}-\sum\limits_{k\in{% \mathcal{N}}^{2}}{B(k,i)}\right)$

So there is:

$\displaystyle{\Delta\mathit{\Phi}}_{2}==2\times\left(\begin{array}[]{c}\sum% \limits_{k\in{\mathcal{N}}^{1}}{\left(\left(\left(\sum\limits^{|\mathcal{R}|}_% {r=1}{g^{\sigma}_{r}(k,i)}\right)-1\right)\times{\alpha}^{{\sigma}_{k}}\times f% ^{\sigma}(k,i)\right)}\\ -\sum\limits_{k\in{\mathcal{N}}^{2}}{\left(\left(\left(\sum\limits^{|\mathcal{% R}|}_{r=1}{g^{{\sigma}^{\prime}}_{r}(k,i)}\right)-1\right)\times{\alpha}^{{% \sigma}^{\prime}_{k}}\times f^{{\sigma}^{\prime}}(k,i)\right)}\end{array}\right)$ (81)

For the calculation of the third part, we have;

$\displaystyle{\Delta\mathit{\Phi}}_{3}=({\mathit{\Phi}}_{3}(\bm{\sigma})-{% \mathit{\Phi}}_{3}({\bm{\sigma}}^{\prime}))\to{\Delta\mathit{\Phi}}_{3}=\sum% \limits^{|\mathcal{N}|}_{d=1}{\textit{SCM}^{{\sigma}_{d}}_{d}}-\sum\limits^{|% \mathcal{N}|}_{d=1}{\textit{SCM}^{{\sigma}^{\prime}_{d}}_{d}}$ (82)

Since for all players except player $i$ , ${\sigma}_{d}$ and ${\sigma}^{\prime}_{d}$ are equal: ${\Delta\mathit{\Phi}}_{3}=\textit{SCM}^{{\sigma}_{i}}_{i}-\textit{SCM}^{{% \sigma}^{\prime}_{i}}_{i}$ To calculate the fourth part, we also have: ${\Delta\mathit{\Phi}}_{4}=({\mathit{\Phi}}_{4}(\bm{\sigma})-{\mathit{\Phi}}_{4% }({\bm{\sigma}}^{\prime}))\to{\Delta\mathit{\Phi}}_{4}=\sum\limits^{|\mathcal{% N}|}_{d=1}{\textit{price}^{{\sigma}_{d}}_{d}}-\sum\limits^{|\mathcal{N}|}_{d=1% }{\textit{price}^{{\sigma}^{\prime}_{d}}_{d}}$ . Since for all players except player i, ${\sigma}_{d}$ and ${\sigma}^{\prime}_{d}$ are equal: ${\Delta\mathit{\Phi}}_{4}=\textit{price}^{{\sigma}_{i}}_{i}-\textit{price}^{{% \sigma}^{\prime}_{i}}_{i}.$ As observed, since user selection does not affect the cost and switching cost of other users, the two relationships ${\Delta\mathit{\Phi}}_{3}$ and ${\Delta\mathit{\Phi}}_{4}$ were easily obtained. Consequently, by substituting four mentioned expressions in this equation, we will have:

$\displaystyle\Delta\mathit{\Phi}=W_{1}\times\left(-\frac{1}{2}\times({\Delta% \mathit{\Phi}}_{1})+\frac{1}{2}\times({\Delta\mathit{\Phi}}_{2})\right)-W_{2}% \times({\Delta\mathit{\Phi}}_{3})-W_{3}\times({\Delta\mathit{\Phi}}_{4})=-W_{1% }\times\left(\sum\limits^{|\mathcal{R}|}_{r=1}{\left(\sum\limits_{j\in{% \mathcal{N}}^{\bm{1}}_{\bm{r}}}{{\alpha}^{{\sigma}_{i}}g^{\sigma}_{r}(j,i)}-% \sum\limits_{j\in{\mathcal{N}}^{\bm{2}}_{\bm{r}}}{({\alpha}^{{\sigma}^{\prime}% _{i}}g^{{\sigma}^{\prime}}_{r}(j,i))}\right)}\right){}\!+W_{1}\times\left(\sum% \limits_{k\in{\mathcal{N}}^{1}}{A(k,i)}-\sum\limits_{k\in{\mathcal{N}}^{2}}{B(% k,i)}\right){}\!-K\times W_{2}\times(\textit{SCM}({\textit{psc}}_{i},{\sigma}_% {i})-\textit{SCM}({\textit{psc}}_{i},{\sigma}^{\prime}_{i}))-W_{3}\times({% \textit{price}}^{{\sigma}_{i}}_{i}-{\textit{price}}^{{\sigma}^{\prime}_{i}}_{i})$ (83)
4. Numerical results

In this section the simulation scenarios are presented and the numerical results are interpreted to assess the performance of the presented algorithms and to show the appropriateness of different optimization approaches.

4.1 Simulation scenarios

In this paper, we considered a two-tier NOMA heterogeneous network scenario with 10 macro base stations and network bandwidth equal to 20 MHz. The detailed simulation parameters are shown as the following. To compare the performance of the proposed algorithm, NOMA-based Energy Efficient Radio Resource Optimization (NOMA-EERRO), we implement it in addition to some other novel energy-efficient algorithms in this NOMA-based scenario. Based on the described model, NOMA-EERRO algorithm determines the activity mode of the small cells considering the initial user distribution pattern and their mobility model, whereas each small cell efficiently select its transmission power based on the power constraints and spectral efficiency. According to the proposed approach, if a certain small base station has no associated UE, then the small cell is considered as a sleep cell. This algorithm considers all state and reward information of all small cells in the entire network. Moreover, “On-Off” implies that this algorithm contains only two actions, i.e., “On” ( $i=1$ ) and “Off” ( $i=0$ ).

In this simulation result we consider a square network of area 800 m ${\times}$ 800 m. 50 mobile nodes are randomly moving in the coverage area with the Random Walk (RW) mobility model. Based on the problem assumption, about two-third of the users are connected to small cell layer and their average bit rate demand is 0.5 Mbps also other one-third of users are directly served by macro cells. The maximum transmission power is 1 W. The maximum allowed user throughput is $\textit{Rmax}=$ 5/Mbit/s and the maximum transmission rate at all the nodes is $h\max=\max i\in Rh\max i=$ 5 Mbit/s. The maximum tolerable delay of connection is assumed 5 ms. The operational frequency band of the access layer is 3.5 GHz and 6 GHz for the backhauling. Similar with [33], the applied propagation model is standard Okumura-hata. The noise power spectral density is $\eta=$ 10–20 W/Hz at all nodes. The acceptable lower bound of the signal to noise/interference ratio is $\Gamma=$ 52.9832 to make the target block error ratio to be 8-2 with the modulation order 64 [34].

MATLAB and CPLEX as two powerful flexible optimizers, have been used for the simulation execution and analyzing the results. The users’ preferences are assumed the same in other words, all users have the same needs for bandwidth, spectrum efficiency, and channel switching rates. To evaluate the NOMA-based Energy Efficient Radio Resource Optimization (NOMA-EERRO), we tried to compare its performance with some of the prominent energy-efficient algorithms. We selected Handover Detection Self-Organizing-HO Parameters (HD-SOHP) [35] and SDN-based Mobility and Handover Management (SB-MHM) [36], which are two common methods in the field of RRM for the comparison. The HD-SOHP algorithm selects the most appropriate radio access technology for entry and in this scheme user association is based on network load and UE mobility information. The main purpose of this method is to improve the users’ mobility performance by achieving effective session and channel selection thereby reducing call drop, improving load balancing and power utility at the cell level over the entire network. The SB-MHM algorithm estimates the neighbor eNB transition probabilities of the mobile node and their available resource probabilities by using a Markov chain formulation. This allows a mathematically elegant framework to select the optimal eNBs and then assign these to mobile nodes virtually, with all connections completed through the use of OpenFlow tables. The NFC/RA method has also considered information about the user’s mobility and UE future position in the channel allocation and RRM decision process.

4.2 Simulation results

Simulations show that the ideal situation for one policy may not work well in all other situations. For example although analytical studies demonstrates that SB-MHM for maximizing the fairness index and the use of network resources has the acceptable situation, but as shown in Fig. 3, this scheme does not have acceptable total spectral and energy efficiency. Also, as analytical results prove, while HD-SOHP is able to satisfy user satisfaction in a appropriate timely manner but this approach has limited state for the spectral efficiency and cannot work in an acceptable way for other scenarios of energy consumption and use of the network bandwidth.

Figure 3.

Overall utilized power per node vs Total number of users.

As shown in the achieved results, Fig. 3 exhibits NOMA-EERRO outperforms SB-MHM and HD-SOHP from the overall consumed energy viewpoint. Although limited, NOMA-EERRO has the lowest energy consumption. This figure illustrates the power utilization factor versus the number of small cells per mobile edge server with the channel gain estimation error variance 0.03. as it is obvious from the plot, the total power consumption of all of the optimization approaches increases with the increasing number of user equipment. On the other hand, HD-SOHP has limited state for the user satisfaction and it cannot provide acceptable utility in addition it does not have acceptable user satisfaction. SB-MHM has appropriate power consumption level for a smaller number of nodes but it cannot work in an acceptable way for other scenarios of energy consumption and use of network bandwidth resources.

Figure 4 illustrates the system’s energy efficiency for different network configurations with 10 to 50 number of small cells. The upper-bound transmission power of each small base station was 25 dB/mWatt and 20 UEs were considered for every small cell in average. The energy efficiency of the proposed approach is compared with the SDN-based Mobility and Handover Management, Orthogonal-frequency division multiple access and the Handover Detection Self-Organizing-HO Parameters algorithm, simultaneously. In SD_SOHP and the OFDMA, the carriers are dedicated to UEs based on the sub-optimal carrier-matching method considering the carrier status or the other channel quality conditions. So, we cannot expect superior energy efficiency for these two schemes. According to this figure, the proposed power-optimized scheme (NOMA-EERRO) has significantly premier energy-efficiency compared to the SB-MHM and HD-SOHP algorithms. This superiority is evident especially with increasing the maximum transmission power (MTP).

Figure 4.

Network energy efficiency based on the MTP constraints.

Figure 5 shows the system throughput for the four radio resource management methods with respect to the maximum allowed transmission power (MTP). Throughput increases in all methods with increasing MTP. As a result, with increasing the maximum power constraint, incoming traffic to macro layer increases, which leads to increased network throughput. Because the HD-SOHP method has higher blocking probability than the adaptive SB-MHM also, because this method uses all real-time services of the macro channels with low bit rate $r_{c}$ , it will have less throughput than SB-MHM. On the other hand, the NOMA-EERRO method has the highest throughput compared to the other three methods because, as mentioned earlier, all users in the hotspot area occupy the macro radio access technology. Therefore, it’s reasonable that all requests made within the hotspot area will use high-rate $r_{w}$ small cells’ channels, which will increase the system throughput. In the adaptive SB-MHM method, because the number of hotspot users will occupy macro channels and a number of users will occupy small cells’ channels, the throughput will be less than the EERRO approach. It should be noted that in the executed simulations, the effect of the parameter $\alpha$ with equal value for the vector $\omega$ , which is twice as important as the other parameters, has significant impact on the average sum data rate of users.

Figure 5.

Average throughput per maximum power constraint.

5. Conclusion

This paper proposed a self-optimizing model aims to enhance network performance and guarantee the users’ QoS requirements by considering limited resources and using effective user association, carrier scheduling and handover optimization algorithms. In order to maximize the network performance, we applied the smart backhauling technique in order to analyze the signaling to increase the validity of the decision making process. Based on the semantic information extracted from the uplink, the network’s decision-making center is able to effectively adjust the network parameters and resource allocation. In this paper, the adopted approach to the optimization of network energy efficiency in multi-tier HetNets was to simultaneously optimize the transmission power in the downlink and control cell activation patterns to meet the users QoS constraints. A global optimal solution was obtained by formulating the problem as a convex linear separation of concerns program with respect to the adoption of BAB approach. The goal function was defined as maximizing the total energy efficiency by considering the transmission power, energy harvesting capability and satisfying the user QoS constraints so that the idle cells are considered turned off temporarily to boost the power efficiency. Adopting this communication optimization framework resulted in a significant improvement in the system energy efficiency by more than 30% compared to the conventional transmission power minimization algorithms. The simulation results also prove that, despite the increased computational complexity, effective resource allocation and optimal HO relations made the proposed approach capable to increase performance indices such as network throughput by up to 25%.

Footnotes

Acknowledgments

This work was supported by Naghsh aval Keifiat (NAK).

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