Smart interference management methods are required to enhance the throughput, coverage, and energy efficiency of a dense small cell network. In this paper, we propose a transmit power control for energy efficient operation of a dense small cell network. We cast the power control problem as a noncooperative game to satisfy the design requirement that small cells do not need any information exchange among them. We analyze the sufficient condition for the existence of a Nash equilibrium (NE) state of the proposed game. We also analyze that the NE state is unique by transforming the original nonlinear fractional programming problem into a nonlinear parametric programming problem. Through simulation studies, we verify our analysis results. In addition, we show that the proposed method achieves higher energy efficiency of a network and balances the energy efficiency among cells more evenly than the methods based on the AIMD (additive increase and multiplicative decrease) algorithm.

1. Introduction

The amount of data traffic transferred through wireless networks has been increased exponentially and the growth rate is expected to increase in the future [1]. Various efforts have been made to accommodate the ever increasing traffic demand [2]. Massive MIMO (Multiple Input and Multiple Output) and CoMP (Coordinated Multipoint/Reception) increase the spectral efficiency of a wireless link [3, 4]. New spectrums for wireless networks and new radio access technologies have been sought [5, 6]. Small cells are deployed to increase the coverage and capacity of a network through spatial reuse [7]. Among those, densifying small cells are regarded as a promising way to enhance the spectral efficiency of a network in a cost-effective way. However, the interference among cells also increases as the cell density increases. Thus, efficient interference management methods have been devised to fully exploit the advantage of a dense small cell network (DSCN) [8, 9]. In the meantime, not only the interference but also the energy efficiency (EE) of a DSCN becomes one of the key design requirements to reduce the power consumption of a DSCN [10, 11].

Energy efficient DSCN operation methods can be classified into two categories. The methods belonging to the first category utilize that the density of small cells is high. Since a user equipment (UE) can be served by many cells, a DSCN can accommodate the traffic demands in an energy efficient way if some cells are running in a sleep mode [1215]. Thus, the schemes in this category attempt to find an optimal set of active cells that maximizes the EE of a DSCN for a given spatiotemporal traffic distribution. The methods in the second category consider a set of active cells and try to use network resources in an energy efficient manner. Radio resource scheduling [16], association control [17], cell clustering [18], and transmit power control belong to this category. Usually, scheduling, clustering and association management operate in a large timescale while the transmit power control works in a small timescale. Thus, following the timescale separation approach [18, 19], we focus on the power control problem for energy efficient operation of a DSCN by assuming that the clusters of active cells and the service order of UEs in each cell are determined.

For a single cell network, iterative algorithms are proposed in [20, 21] that determine an optimal transmit power maximizing EE. In [22, 23], a Markovian approach is taken for energy efficient power control. The authors in [24] use the parameter free fractional programming to derive an energy efficient power allocation to maximize the system energy efficiency and propose a power adaptation algorithm based on the analysis result. In [25], authors formulate a nonconvex optimization problem to maximize the energy efficiency of a cell. By reforming the problem into a convex optimization problem with the property of the parameter-free fractional programming and the concept of perspective function, they devise a distributed power control algorithm that requires the minimum amount of information to be exchanged among cells. A Nash bargaining cooperative game-theoretic framework is proposed in [26]. After showing the relationship between the energy efficiency and the spectral efficiency, they formulate the energy efficiency maximization problem. Then, they propose a distributed algorithm that gives a suboptimal solution and guarantees the efficiency and fairness. The authors in [27] propose an energy-efficient power allocation and wireless backhaul bandwidth allocation for a small cell network using OFDMA. They devise algorithms for the original non-convex problem so that each small cell can jointly determine the transmit power for serving UEs and the bandwidth for backhauling. In [28], a Stackelberg game model is adopted to increase the energy efficiency of small cell networks. The authors propose a pricing scheme between a macrocell and small cells and devise a power control method by transforming the original nonlinear fractional programming problem into a subtractive form.

However, we note that the nature of a DSCN is amorphous [29] because small cells are deployed in an unplanned manner by different entities. Thus, it is difficult to expect that small cells exchange information for cooperative control. In addition, each small cell is selfish and rational in that it attempts to increase its own EE in response to the environments. Therefore, in this paper, we design a power control method by which each cell determines its transmit power autonomously without any message exchange among cells. To achieve the goal, we model the power control problem for enhancing EE of a DSCN as a noncooperative game and propose a power control algorithm based on the best response function. Furthermore, we provide a sufficient condition for the existence of Nash equilibrium (NE) of the game. We also prove that the NE of the proposed power control game is unique.

The rest of the paper is organized as follows. In Section 2, we describe the system model and formally define the energy efficiency of a DSCN. In Section 3, we present the power control game and analyze its property. We especially prove that the proposed power control game has a unique NE. After we verify that the proposed method is superior to the other distributed method based on the additive increase and multiplicative decrease algorithm in Section 4, we conclude the paper in Section 5.

2. System Model

We consider the downlinks of a small cell network where the spectrum reuse factor is one. We assume that radio resources are divided into resource blocks which is the smallest resource unit that can be allocated to an UE. We assume the time is divided into a time slot with equal size.

We denote by the set of cells interfering with each other. We also denote the set of UEs served by a cell by . Let us denote by the transmit power of a cell . If we denote by the set of transmit power that each cell can choose from (i.e., ), the SINR at an UE can be given aswhere is the channel gain between a cell and a UE and is the noise power. Let us denote by the number of resource blocks that allocates to . We also denote by the maximum number of resource blocks that a small cell has. If we assume that small cells use a round-robin scheduler [30], becomes , where is the cardinality of a set . If the bandwidth of a resource block is and we denote by , from the Shannon’s capacity formula, the downlink data rate provided to a UE by a cell is expressed asTherefore, the throughput of a cell can be given as

From the measurement studies, the authors in [31] proposes a power consumption model for various cell types which is composed of a load-independent part and a load-dependent part. The resulting power consumption model of a cell is expressed as follows. where is the number of transmit antenna, is the power consumption at the minimum non-zero output power, and is the slope of the load-dependent power consumption.

Then, the energy efficiency of a cell is defined as

An optimization problem that attempts to find a transmit power vector maximizing the total energy efficiency of a network (i.e., ) can be formulated and a central controller can find a globally optimal solution. In terms of implementation, all the channel conditions between all UEs and cells are needed to solve the optimization problem. However, the signaling overhead increases exponentially with the number of cells. Thus, the signaling overhead for a central controller to obtain the required channel information will be prohibitive in a DSCN where the number of cells is very large. In addition, since a DSCN is assumed to be composed of autonomous cells, it is unlikely that there is a central controller gathering necessary information, solving the optimization problem, and distributing the optimal transmit power for each cell. Therefore, a distributed algorithm that enables each cell to determine an optimal transmit power using its local information is needed. To tackle the issue, in this paper, we propose a noncooperative power control game model for energy efficient operation of a DSCN even if each cell behaves in a self-interested way without any message exchange with other cells.

3. Power Control Game

The distributed power control game is defined as follows.where is the set of game players, is the set of power allocation profiles, and is the utility function of a cell . Since the power control game is played among small cells, the set of players is the set of small cells. For each cell , is the set of strategies (i.e., transmit powers) that can choose. Thus, a power allocation profile is a combination of strategies of all players . A utility function of a player maps each strategy profile to a real number. Since the purpose of the game is to determine the transmit power of a cell to optimize the energy efficiency, we set the utility function of a cell as . We also denote the strategy profile except by .

Then, in the noncooperative power control game , each cell attempting to maximize its utility faces the problem of determining its best response when other cells commit to play . The best response of a cell to the strategy profile is a strategy such that for all . Since a cell measures the influence of on during each time slot, it determines its transmit power for the -th time slot at the end of the -th time slot as

For the noncooperative game, we analyze the existence and the uniqueness of the Nash equilibrium (NE) as Propositions 1 and 2.

Proposition 1. There exists an NE in the game if , where

Proof. We prove the existence of a NE in the game by showing that is nonempty, convex, and compact subset in an Euclidean space and is continuous and quasi-concave in [32]. is the set of transmit power that a cell can choose. Since the transmit power of a cell is bounded by the minimum transmit power and the maximum power , is not empty, convex, and compact subset in an Euclidean space . In addition, is a continuous function of .
To show the concavity of , we consider a UE . Then,Therefore, We letThen,Sinceandthus (12) becomes Therefore, the condition that makes becomesBy rearranging (16), we obtainIf we let becomes negative ifSince , if for all and . Since is a strict concave function in if , there exits an NE in the game .

Proposition 2. An NE in the game is unique.

Proof. The transmit power of a cell is in which is a compact and connected subset of an Euclidean space . Both and are a continuous function of and produce real numbers. In addition, since , the nonlinear fractional programming problem that maximizes (i.e., the problem in (7)) can be transformed into the following nonlinear parametric programming problem.Then, from the Lemma 4 in [33], has a unique solution . In addition, from the theorem in [33], is if and only if . Therefore, the NE of the game is unique.

The transmission power control algorithm based on the proposed noncooperative game model can be summarized as in Algorithm 1.

1: at the end of each time slot, each cell :
2: : , ,
3: while do
7: if then
9: else
10:  if then
12:  else

4. Performance Evaluation

In this section, we compare the performance of the proposed transmit power control method with the other method that do not require a cell to exchange messages with other cells when it determines its transmit power. One of the most popular methods that determines a control variable in a distributed manner is the AIMD (additive increase and multiplicative decrease) adopted in TCP for the congestion control. The basic idea of the AIMD is to increase the current control variable linearly if a node does not detect performance degradation. When a node detects performance degradation, it considers that the competition for the shared resources in a network is severe. Thus, the node decreases its control variable multiplicatively to reduce the competition level fast. Therefore, AIMD increases the performance of each node opportunistically while stabilizing a network by inducing an implicit cooperation among the competing nodes without explicit message exchanges among them.

In case of the transmit power control, the control variable of a node is the transmit power of a cell and the performance metric is the energy efficiency of a cell. Thus, when AIMD is used, each cell determines the transmit power for the -th time slot denoted by based on , and , where is the energy efficiency of a cell at the beginning of the time slot when uses . Specifically, if , . If , . We consider two types of . One is , and the other is . Henceforth, the AIMD using the formal will be referred as a static AIMD and the AIMD using the latter will be called a dynamic AIMD.

We deployed cells in area according to the homogeneous Poisson point process (HPPP) to reflect the uncoordinated cell deployments. In the same area, we positioned UEs following the HPPP and associate each of them to its nearest cell. According to the 3GPP Pico base station system specification [30], we configure the system parameters of each cell as follows. The system bandwidth is , the bandwidth of a resource block is , an antenna height is set to be , and the antenna gain is configured as . The maximum and the minimum transmit power are set to and , respectively. The path loss model of is used, where is the distance between a sender and a receiver in meters. The parameters related to the cell power consumption are configured according to the Pico cell parameters in [31]. Specifically, we set , , and . The initial transmit power of each cell is randomly selected from according to the uniform distribution.

In Figure 1, we compare the energy efficiency of a network over time with different . The proposed method represented as NonCoGame in the figure makes to the steady state fast. In case of AIMD, also converges to a steady point which is smaller than that of NonCoGame case. However, s obtained by AIMD increases in the beginning of the simulation and decreases to the steady state.

To scrutinize the phenomenon, we analyze the dynamics of the total throughput of a network (i.e., ) and the total power consumption of a network (i.e., ) in Figures 2 and 3, respectively. We choose randomly at the start of the simulation. is determined not only by the received signal power at the UEs in but also by the total interference imposed on these UEs. If and are not large enough, both the received signal power at the UEs in and the total interference on these UEs are small. Thus, is relatively smaller than its maximum value achievable. In case of AIMD, each cell gradually increases its power whenever . By increasing , increases and increases if is bigger than . If the amount of the increment in is larger than that in for a given and , increases. If a cell detects , it decreases by half, which decreases both and . However, is a linear function of while involves not only but also . We observe in Figure 3 that does not change much while decreases sharply during the round 1 to 15. The result indicates that the amount of increments in by the cells increasing transmit powers is close to the amount of decrements in by the cells who reduce their transmit powers. Since AIMD decreases a transmit power multiplicatively, the received signal powers at the UEs in the cell that decreases transmit power also reduces while the total interference experienced by these UEs increases because of the other cells that increase transmit powers. Thus, reduces abruptly which leads to sharp decrease in during this period.

To inspect the effects of the fairness in terms of , we observe the Jain’s fairness index in Figure 4, where is given as As is closer to , s are more similar to each other. We can see that the proposed method can increase . Since cells and UEs are uniformly distributed, each cell can obtain similar energy efficiency when the proposed method is used.

5. Conclusions

In this paper, we propose a noncooperative game model for a cell to determine its transmit power autonomously to optimize its energy efficiency. We provide a sufficient condition for the existence of Nash equilibrium of the proposed game. We also prove that the game has unique Nash equilibrium by transforming the nonlinear fractional programming problem into a parametric programming problem. Through simulation studies that compare the performance of the proposed method with those of the AIMD method, we show that the proposed power control method can stabilize a system at higher total energy efficiency and balance cell energy efficiency more evenly than the AIMD methods.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflict of interest regarding the publication of this paper.


This work was supported by the National Research Foundation of Korea (NRF) funded by the Korea government under Grant NRF-2018R1D1A1B07050893.