Optimal power control and coverage management in twotier femtocell networks
 Georgios Aristomenopoulos^{1}Email author,
 Timotheos Kastrinogiannis^{1},
 Stamatina Lamprinakou^{1} and
 Symeon Papavassiliou^{1}
https://doi.org/10.1186/168714992012329
© Aristomenopoulos et al.; licensee Springer. 2012
Received: 24 November 2011
Accepted: 23 September 2012
Published: 30 October 2012
Abstract
In this article, we jointly consider the problem of efficient power control and coverage (PCC) management over an integrated twotier macrocell/femtocell network towards maximizing the expected throughput of the system subject to appropriate power constraints, under the existence of both cotier and crosstier interferences. Optimal network design amounts to joint optimization of users’ allocated power levels and cell’s maximum aggregated downlink transmitted power, i.e., coverage area management. This problem is inherently difficult because it is in fact a nonconvex optimization problem. A novel approach to address the latter is performed that entails a suitable transformation, which allows the use of convex optimization and also forms the basis for the design of a distributed PCC algorithm via performing twolevel primaldual decomposition. PCC algorithm’s convergence to optimality is established. We demonstrate that for realistic macrocell/femtocell deployment scenarios, overall system throughput increase up to approximately 50% can be achieved while guaranteeing 70% of power savings.
Keywords
Introduction
In recent years, the increasing demand for efficient indoor coverage coupled with the need for high data rates and quality of service (QoS) has highlighted the inflexibilities of the socalled traditional macrocell layer of a cellular network, calling for new indoor coverage/capacity management solutions. In response to this problem, the use of femtocell access points or home base stations [1] has been proposed. Femtocell networks are created by lowpower, lowcost, userdeployed base stations that are able of providing highquality cellular service in residential or enterprise environments while operating in licensed spectrum.
As femtocells share spectrum with macrocellnetwork, efficient crosstier interference mitigation between femto and macrocells is essential to facilitate both coverage and capacity enhancements. Nevertheless, efficient twotier power control and resource allocation faces several additional design challenges compared to their singletier cellular counterparts. First, cotier interference, in terms of intra and intercell interference (e.g., between neighboring femtocells), along with crosstier interference need to be jointly considered in the computation of users’ transmission environment (i.e., the achieved signaltointerference plus noise ratio, SINR), resulting in nonconvex power control optimization settings. Second, the network operator agnostic deployment of femtocells demands their ability to dynamically sense the radio environment and adjust their size accordingly (via tuning their transmission power level) towards minimizing the coverage area leak into neighboring cells [2]. Therefore, along with power control, efficient, distributed, and dynamic coverage management (CM) need to be considered.
This study deals with optimal joint power control and coverage (PCC) management in the presence of cotier and crosstier interferences in twotier codedivision multipleaccess (CDMA) macrocell/femtocell wireless networks. The goal is to find the optimal operation point that maximizes networkwide throughput performance, in terms of users’ downlink transmission power and cells’ maximum cumulative power assigned to users (i.e., cells’ coverage size). The large number of variables involved and the nonconvexity of the optimization problem associated to the latter goal reveal the difficulty in obtaining such an operation point.
Limitations of existing literature work
Such issues are typically addressed in the literature through the isolation and individual treatment of specific goals of the overall twotier network optimization problem, leading to fewer degrees of freedom. To that end, for treating femtocells’ CM problem only, a large variety of dynamic cell sizing schemes have been developed towards improving overall system’s capacity [3–10], when compared to that of fixed cell sizing schemes. This is achieved by the dynamic adaptation of femtocells’ maximum transmission power [3–5] or pilot signal power [7–9] in a stepwise heuristic manner, while aiming at one of the following goals: (a) assure an upper bound on femtocell user’s SINR [3], (b) minimize femtocells’ coverage leakage probability (i.e., the probability that a femtocell user’s SINR is larger than a threshold, towards minimizing femtocells’ coverage area leak into an outdoor macrocell) [5], (c) minimize the amount of handovers between the two tiers [7], or (d) minimize power consumption [9].
Furthermore, a vast number of twotier power control and interference mitigation problems have been proposed and studied towards serving a variety of alternative goals, i.e., from overall system interference minimization [11–13] to overall system throughput optimization [14], and from overall power minimization under SINR [15], QoS [16], and power [17] constraints, to integrated system outage probability minimization [18] and utilitybased optimization [19, 20].
Recently in [21], a distributed and selforganizing femtocell management architecture has been proposed to jointly mitigate twotier network interference. The proposed architecture consists of three control loops to determine (1) the maximum transmit power of femtocell users, (2) a target SINRs of femtocell users to reach a Nash equilibrium, and (3) instantaneous transmit power of femtocell users to achieve that target SINRs. Although both power control and CM are considered in [21], their correlation is limited to a specific periodic information exchange between the control loops that enable the latter mechanisms, leading to suboptimal heurist solutions.
Article’s contributions and methodology
The previous analysis highlights the lack of an efficient theoretically sound framework to jointly incorporate PCC management over a macrocell/femtocell system. This would provide enhanced flexibility on combining both mechanisms’ interference mitigation attributes towards optimizing overall users’ performance in the twotier system. This article develops an optimization framework that fills the above gap.
In brief, the proposed PCC management framework and corresponding PCC algorithm claims the following design attributes.

Explicit CM. In the proposed optimization framework, the maximum transmission power of each femtocell is considered as an optimization variable (properly constrained by a lower and upper bound) incorporated to the overall power control optimization problem. This enables explicit femtocells’ CM towards networkwide throughput optimality.

Networkwide optimality. Due to crosstier and cotier interferences, the initial PCC management problems is nonconvex and thus, hard to solve, especially in a distributed manner. To that end, a novel approach to address that latter is performed that entails a suitable transformation, which allows the use of convex optimization and also forms the basis for the design of a distributed PCC algorithm via performing twolevel primaldual decomposition.

Distributed implementation and convergence analysis. The resulting architectural framework is enabled via a distributed, lowcomplexity subgradient algorithm. This attribute reassures low excess signaling overhead and favors scalability. PCC algorithm’s convergence to the networkwide global optimal is proven, while its convergence speed and imposed signaling overhead are discussed.
The rest of the article is organized as follows. “System model” section describes the adopted system model, while in “Problem formulation and transformations” section the problem formulation and its transformation are detailed. In “PCC management for a twotier femtocell/macrocell system” section, problem’s primaldual decomposition and its solution via a subgradient method are provided, while in “Distributed PCC algorithm” section PCC algorithm’s optimality and convergence attributes are discussed. Finally, in “Numerical results” section extensive numerical results, demonstrating the efficacy of the proposed approach, are presented and discussed, while “Concluding remarks and future work” section concludes this article.
System model
In this article, we consider a twotier femtocell/macrocell network with a set C of C cells. The system consists of a single central macrocell C_{0} and C – 1 underlay cochannel femtocells C_{ c }, where c = 1,…, C – 1. We further consider a set J of J active mobile users located in the above area. On every occasion, a user j ∈ J can be attached to only one cell c ∈ C. This study assumes closed access (CA) [1], which means only licensed home users, within radio range, can communicate with their own femtocell. We denote as S_{ c } the set of S_{ c } users served by base station c thus, J = S_{0} × ⋅⋅⋅ × S_{C−1} is a Cartesian product.
The system is assumed to be timeslotted, while orthogonal CDMA downlink is assumed in each slot. All cells use the same frequency channel [i.e., share the same spreading bandwidth W (Hz)]. Therefore, to exploit multiuser diversity, at each timeslot active mobile users’ downlink transmission power control and cells’ coverage (PCC) management algorithm is executed.
The channel between cells’ BSs/APs transmitters and mobile nodes’ receivers is modeled as an additive white Gaussian noise multiaccess channel. Let us denote by G_{c,j} the path gain between cell c and user j, reflecting the longtime behavior of the channel gain. Fastfading can be considered via smoothing out by appropriate averaging [22]. We focus on one timeslot assuming that the path gain, background noise, and intercell interference for each mobile do not change during this timeslot [23, 24]. Hence, the notion of time is omitted in the definition of the variables introduced in the rest of the article.
where the denominator incorporates intracell, co and crosstier interference, and background noise, respectively.
This approximation is reasonable either when the signal level is much higher than the interference level (as in the case of a femtocell system, due to the proximity of the femtocell user’s at their base stations) or when the spreading gain is large. In (2), for simplicity in the presentation K has been incorporated and absorbed into G_{c,j}. Finally, it should be highlighted that while U_{ j } is a nonlinear nonconcave function of $\overline{P}$, it can be converted into a concave function through an appropriate log transformation, leading to a critical convexity property that establishes global optimality (see “PCC management for a twotier femtocell/macrocell system” section).
Problem formulation and transformations
where the computation of both vectors $\overline{P}=\left({\overline{P}}_{0},{\overline{P}}_{1},\dots ,{\overline{P}}_{\leftC\right1}\right)$ and ${\overline{P}}_{\text{var}}=\left({P}_{0,\text{var}},{P}_{1,\text{var}},\dots ,{P}_{\leftC\right1.\text{var}}\right)$ enables the integration of system’s CM into overall system’s power control and interference management optimization problem. Therefore, our goal is to jointly maximize the total system utility (e.g., the total expected system throughput) and to obtain the optimal cell’s coverage, in terms of maximum intracell transmission power, with constraints on cells’ cumulative transmission power to the attached users (i.e., constraint (4)) and base stations’ maximum intracell transmission power due to physical limitations (i.e., constraint (5)).
A necessary condition to optimality (first problem transformation)
To solve problem (P1) in an efficient and distributed manner via optimization decomposition, we first need to overcome the complexity imposed by the coupling of the optimization variables in problem’s objective function, due to the nature of the twotier SINR in (1). To this end, we provide the following proposition.
Proposition 1: A necessary condition to achieve maximum twotier overall system utility is that all base stations c ∈ C must assign all their resources to their attached users and thus transmit at their maximum intracell available cumulative transmission power P_{c,var}, for any given feasible ${\overline{P}}_{\text{var}}$ (i.e., when equalities in (4) hold).
Proof: See Appendix.
Therefore, our goal in this article is to establish a joint PCC management algorithm that distributively attains the jointly globally optimal solution ($\left(\overline{P}*,{\overline{P}}_{\text{var}}*\right)$) to problem (P2) and thus, the equivalent (P1). However, we first need to overcome two major challenges imposed by the global dependencies in (P2), that is

Nonconvexity. Although problem’s (P2) constraints are convex and preserve a decoupled attribute with respect to system cells, variables $\overline{P}$ and ${\overline{P}}_{\text{var}}$ are globally coupled across the twotier system, as reflected in the range of summations in the objective function (7), due to (6). As a consequence, problem (P2) is generally nonconvex and hence challenging to solve.
Treatment: Via a log transformation [29, 30], we convert (P2) into an equivalent convex optimization problem (P3) (see “Pursuing convexity (second problem transformation)” section). This enables the proposed PCC algorithm with the following desirable properties: (a) global convergence to optimality $\left(\overline{P}*,{\overline{P}}_{\text{var}}*\right)$; (b) elegant tradeoff between complexity and performance, and (c) geometric rate of convergence.

Deriving a distributed solution. The nonlinear optimization problem (P2) may be solved by centralized computation using the interiorpoint method for convex optimization [31], i.e., geometric programming [32]. However, in the context of wireless femtocell networks a distributive algorithm is required towards minimizing imposed signaling overhead and improving accuracy.
Treatment: The convex equivalent formulation of problem (P2) [that is problem (P3)] is further solved via twolevel primaldual optimization decomposition (see “PCC management for a twotier femtocell/macrocell system” section). Hence, two distributed algorithms for jointly attaining optimal (a) CM and (b) intracell’s power allocation are proposed and analyzed.
Pursuing convexity (second problem transformation)
The first term in the square bracket is linear in $\overline{p}$ and ${\overline{p}}_{\text{var}}$, since it is linear in p_{c,j}. The second term is a logarithmic summation of exponentials of linear functions of $\overline{p}$ and ${\overline{p}}_{\text{var}}$, which is concave in the latter domain, as proven in [25, 29]. Finally, since the objective function of (P3) is a sum of concave terms, we conclude that (P3) is a convex optimization problem.
PCC management for a twotier femtocell/macrocell system
The convex optimization problem (P3) has two features which facilitate a distributed solution. First, the objective in (10) is a sum of C intracell S_{ c } users’ utilities summations, that depend only on the variables ${\overline{p}}_{c}$ and ${\overline{p}}_{\text{var}}$. Second, the constraints (11)–(13) also depend only on ${\overline{p}}_{c}$ and ${\overline{p}}_{\text{var}}$. Based on the latter features, in this section an algorithm based on Lagrangian techniques for obtaining the solution of problem (P1) by solving problem (P3) is developed (i.e., deriving ${{\overline{p}}^{*}}_{c}$ and ${{\overline{p}}^{*}}_{\text{var}}$).
 I.
C – 1 independent IPC subproblems. Considering first primal decomposition of (P3) by fixing ${\overline{p}}_{\text{var}}$ (i.e., for a given cells’ coverage assignment) problem (P3) breaks in C – 1 independent IPC subproblems, each one responsible for computing the optimal intracell power vector ${{\overline{p}}^{*}}_{c}$ for a given ${\overline{p}}_{\text{var}}$ allocation. IPC’ solution is provided in “Treating IPC” section.
 II.
A master problem, responsible for updating the value of ${\overline{p}}_{\text{var}}$ towards obtaining ${{\overline{p}}^{*}}_{\text{var}}$. This master problem eventually performs CM [therefore denoted as problem (CM)] via controlling the maximum intracell cumulative power vector, and thus the coverage range, of each cell in the integrated system. To solve (CM), we use a subgradient method exploiting the information of IPCs’ Lagrange multipliers λ _{ c } * ∀ c ∈ C associated to the constraint (10), as detailed in “Deriving the lagrange multiplier of an (IPCs)” section.
It is important to note that since (P3) is a convex optimization problem, both master problem (CM) and subproblems (IPCs) are also convex optimization problems [34]. In the following, we provide the solutions of IPCs and CM, respectively.
Treating IPC
Problem (IPC_{ c }) obtains optimal intracell power allocation vector ${{\overline{p}}^{*}}_{c}$ for any given ${\overline{p}}_{\text{var}}$. To efficiently solve (IPC_{ c }) we use the methodology provided for the solution of problem (11) in [35]. Problem (11) in [35] is also a logarithmically transformed IPC problem under power constraints. Furthermore, the adopted utilities in our case [as defined in (2) and (7)] are subcases of the utilities considered in [35]. Therefore, for obtaining the solutions of (IPCs) we can directly utilize the gradient projection algorithm with constant step size proposed in [35] (Section IV), which is characterized by provable convergence and optimality attributes. We further refer to the latter algorithm as IPC_{ c }, which takes as input any ${\overline{p}}_{\text{var}}$ and provides as output the corresponding ${{\overline{p}}^{*}}_{c}$.
Deriving the Lagrange multiplier of an (IPCs)
where $k={\displaystyle \sum _{c\in C}{G}_{c,j}\text{exp}\left({p}_{c,\text{var}}\right)+{n}_{0}}$
Concluding this section’s analysis, let us underline that (IPC_{ c }) algorithm for cell c ∈ C takes as input any ${\overline{p}}_{\text{var}}$ and obtains (a) the corresponding ${{\overline{p}}^{*}}_{c}$, as analyzed in “Treating IPC” section, as well as (b) the optimal objective value of (ICP_{c}), ${U}_{c}*\left({\overline{p}}_{\text{var}}\right)$ for any given ${\overline{p}}_{\text{var}}$ (i.e., ${U}_{c}^{*}\left({\overline{p}}_{\text{var}}\right)\equiv {\text{U}}_{c}\left({{\overline{p}}^{*}}_{c},{\overline{p}}_{\text{var}}\right)={\displaystyle {\sum}_{j\in {S}_{c}}{U}_{j}\left({\gamma}_{c,j}\left({{p}^{*}}_{c,j}\right)\right)}$), with a subgradient given by the multiplier λ_{ c } *.
Treating CM (the master problem)
where cells’ optimal power vectors ${{\overline{p}}^{*}}_{c}$ are known, log(P_{ c }^{Max}) and log(P_{ c }^{Min}) are constants and only the optimal transmit power vector of all cells’ BSs ${\overline{p}}_{\text{var}}^{*}$ (i.e., cells’ coverage) is left to be computed.
 a.
the gradient of the objective function of CM is hard to compute,
 b.
it is shown in [33, 34] that the subgradient of each ${U}_{c}^{*}\left({\overline{p}}_{\text{var}}\right)$ is equal to the optimal Lagrange multiplier corresponding to the constraint (14) in (ICP_{ c }). Therefore, ${\nabla}_{c}U{}_{c}{}^{*}\left({\overline{p}}_{\text{var}}\right)={\lambda}_{c}^{*}\left({p}_{c,\text{var}}\right)$, which is computed in (16). Finally, the global subgradient is ${\nabla}_{{\overline{p}}_{\text{var}}}U{}_{c}{}^{*}\left({\overline{p}}_{\text{var}}\right)={\displaystyle {\sum}_{c\in C}{\lambda}_{c}^{*}\left({\overline{p}}_{\text{var}}\right)}$.
Nicely enough this feasible set enjoys the property of naturally decomposing into a Cartesian product for each cells: P = P_{0} × P_{1} × ⋅⋅⋅ × P_{c−1}. Therefore, the predescribed subgradient update can be performed independently by each cell, simply with the knowledge of its correspondent (IPC) problem Lagrange multiplier λ_{ c }*, which in turn is also independently computed and updated by each cell’s c (ICP_{ c }).
Therefore, all algorithms that constitute PCC, i.e., ICPs and CM, are distributively solved by each system cell. The only required information that needs to be disseminated among the cells of the twotier system is ${\overline{p}}_{\text{var}}\left(t\right)$ and a(t), imposing minimal overhead over the system.
Distributed PCC algorithm
In this section, we analyze the operation and justify the designing attributes of the proposed PCC algorithm. Furthermore, it’s converge to system’s global optimal operation point is proved, while it’s speed of convergence and the imposed signaling overhead are discussed.
Design and operation
Towards implementing the joint CM and power control algorithm, the cooperation of the ICPs and CM algorithms is required. Specifically, ICP algorithms, residing on every cell c ∈ C, are responsible for solving the corresponding intracell (macro or femto) power control and resource allocation problem. The CM algorithm in each femtocell dynamically updates cell’s maximum transmission power, in line with (23), while the CM one at the macro base station guarantees the synchronization among all cells via gathering and distributing ${\overline{p}}_{\text{var}}\left(t\right)$ and a(t). It is important to note that both ICP and CM algorithms are located in each cell’s base station and thus all information exchanged between them is local and does not impose any signaling overhead to the wireless medium.
PCC algorithm
Primaldual algorithm to solve PCC (P3):  

Initialization  Set t=0 and ${\overline{p}}_{\text{var}}\left(0\right)$ equal to a nonnegative value.  
Step  Operation  Input  Output 
1  Solve ICPs at each cell considering the specific users’ utilities as well as the power vector ${\overline{p}}_{\text{var}}\left(t\right)$, via the canonical dual algorithm in Section IV.A, deriving cell’s optimal utility vector, the normalized Lagrangian multiplier and users’ power vector for the specific ${\overline{p}}_{\text{var}}\left(t\right)$. This implies a iterative algorithm.  U_{ j } ∀ j ∈ S_{ j }${\overline{p}}_{\text{var}}\left(t\right)$  ${\overline{U}}_{c}^{*}\left({\overline{p}}_{c}^{*},{\overline{p}}_{\text{var}}\right)$ ${\lambda}_{c}^{*}\left({\overline{p}}_{\text{var}}\left(t\right)\right)$ ${\overline{p}}_{c}^{*}\left(t\right)$ ∀ c ∈ C 
2  Each cell c dynamically updates its maximum transmission power p_{c,var}(t + 1) in accordance to the gradient algorithm (23). This information is sent to the macro cell using existing 3GPP signaling.  ${\lambda}_{c}^{*}\left({\overline{p}}_{\text{var}}\left(t\right)\right)$a(t)  p_{c,var}(t + 1) ∀ c ∈ C 
3  Set t ← t + 1, disseminate ${\overline{p}}_{\text{var}}\left(t\right)$, a(t) and go to step 1 (until satisfying termination criterion). Upon convergence, optimal CM and power control vector $\left({\overline{p}}_{c}^{*},{{\overline{p}}^{*}}_{\text{var}}\right)$ have been derived.  $\left({\overline{p}}_{c}^{*},{{\overline{p}}^{*}}_{\text{var}}\right)$ 
On addressing convergence and optimality
In this section, we argue on the optimality and the convergence properties of the proposed PCC management algorithm.
Initially, we argue on the selected update step for the proposed subgradient method (22), which is a diminishing step size rule a(t) with the following properties: a(t) > 0, $\underset{t\to \infty}{lim}a\left(t\right)=0$ and ∑_{t=0}^{∞}a(t) = ∞. In accordance to [36], since ℂ is compact and not empty, and CM is a convex optimization problem, then via a diminishing step size rule, for example $a\left(t\right)=\frac{t}{\beta +t}$ where β > 1 is a fixed constant, the subgradient algorithm converges, i.e., ${\overline{p}}_{\text{var}}\left(t\right)\to {\overline{p}}_{\text{var}}^{*}$ as t → ∞ and thus $limsup\left\{{\displaystyle \sum _{c\in C}{U}_{c}^{*}\left({\overline{p}}_{\text{var}}\right)}\right\}\to {U}_{\mathit{TOTAL}}^{*}\triangleq {\displaystyle \sum _{c\in C}{U}_{c}^{*}\left({\overline{p}}_{\text{var}}\right)}$. In addition to that, since via (ICP_{ c }) we can compute ${\overline{p}}^{*}\left({\overline{p}}_{\text{var}}\right)\forall {\overline{p}}_{\text{var}}$ the convergence of ${\overline{p}}_{\text{var}}\left(t\right)$ to the global optimum implies the global optimality of ${\overline{p}}^{*}\left({\overline{p}}_{\text{var}}^{*}\right)$. Therefore, we conclude that both variable vectors $\left({\overline{p}}_{c},{\overline{p}}_{\text{var}}\right)$ converge to system’s global optimal values $\left({\overline{p}}_{c}^{*},{{\overline{p}}^{*}}_{\text{var}}\right)$ leading to overall twotier system throughput maximization.
Discussions and design features

Synchronization: The existence of the diminishing step size a(t) demands the synchronization among cells in the twotier system when computing ${\overline{p}}_{\text{var}}\left(t+1\right)$. Thus, all cells are required to maintain the value of the current iteration, as well as cells’ maximum power vector ${\overline{p}}_{\text{var}}\left(t\right)$. In other words, it is required to exchange the values of a(t) and p_{c,var}(t) per iteration t. This is feasible, especially considering the nature of the two tier system since (a) CDMA is slotted by nature, thus synchronization is inherent, and (b) the macrocell, as the overlay cell, can broadcast theses values using its broadcast channel, a common practice in 3GPP LTE [37].

Signaling overhead: In line with the previous analysis, the imposed signaling overhead is minimal and comprises of only C + 1 real numbers per iteration [a(t) and ${\overline{p}}_{\text{var}}$] which can easily be carried out via existing signaling.

Convergence speed: Although the convergence to optimality is proven, the time of the convergence cannot easily be determined or even bounded. Practically, it depends on the tradeoff between accuracy and chosen timecomplexity, which further depends on the termination threshold ε we set, i.e., if $\Vert {\overline{p}}_{\text{var}}\left(t+1\right){\overline{p}}_{\text{var}}\left(t\right)\Vert \le \epsilon $ and ε < a(t) then stop. However, as shown in the following section, the experimental results indicate that under all cases the algorithm converges in less than 30 iterations.
Numerical results
In this section, we provide indicative numerical results demonstrating the effectiveness of the proposed PCC management algorithm, as well as providing a proof of concept of its applicability in a twotier cellular environment. The results reveal that PCC achieves significant performance gains under various femtocell/macrocell network deployment topologies, in terms of overall achieved utilitybased performance (i.e., system throughput) and power consumption, compared to a power control (PC) policy that aims only at overall integrated system performance optimization (i.e., via enabling optimal IPC fall all cells) without CM.
Initially, we consider a twotier system with 14 femtocells assembling 2 clusters, as depicted in Figure 3a. System performance, in terms of (a) overall aggregated utility (i.e., throughput) increment, (b) macrocell’s aggregated utility (i.e., throughput) increment, and (c) overall power consumption decrement under PCC compared to PC is illustrated in Figure 3b–d, respectively, for increasing active users’ population (i.e., J = 20, 30, …, 80), arbitrarily placed in the system, and femtocell users’ percentages (i.e., FUP = 30, 40, …, 80).
The results show that under realistic (i.e., most expected [38, 39]) user population distributions, i.e., number of femtocells (14), number of overall users (from 40 to 60) and femtocell users’ percentage of 40–60%, which corresponds to an average of 2 to 3 users per femtocell (most common case over a practical femtocell deployment), overall system utilization (i.e., twotier system throughput) is increased up to 50% under the proposed PCC algorithm, as revealed in Figure 3b.
The main reason for attaining such a highly desirable system performance increase under PCC is related to the high improvement of overall macrocell users’ performance, which in many cases exceeds 60%, i.e., from 30 to 48 Mbps excess achieved twotier capacity in some scenarios (Figure 3c), as a result of PCC algorithm’s optimal CM control in the twotier network. Specifically under PCC, femtocells’ coverage areas are optimally fine tuned, therefore their overall maximum power is decreased (while maintaining optimal intracell utilitybased performance). This allows macrocell users to experience a highly reduced crosstier intercell interference and thus, establish high throughput performance gains, compared to the case where only optimal PC is achieved.
The latter behavior is also confirmed by the results presented in Figure 3d, where the percentage of overall system power reduction under PCC is up to 70%, with respect to the corresponding one under PC (i.e., from 55 to 17 W). This is due to femtocells’ efficient coverage control, under which femtocells' maximum downlink power allowed to be allocated to their corresponding attached users, is reduced up to the point where their utilitybased performance remains high. This, not only avoids high femtocell resource over provisioning, but also leads to femtocells’ coverage leakage reduction and thus, to overall twotier interference mitigation.
Nevertheless, it should be pointed out that the benefits of joint PCC management under PCC are condensed, when moving at the boundaries of users’ population distributions among the two tiers. Specifically, for very low or very large percentages of femtocell users (i.e., less than 30% or more than 70%, corresponding to less than 2 or more than 5 users per femtocell on average), overall system utilitybased performance improvement under PCC decreases, even though macrocell users still experience significant performance gains. This latter behavior is expected, since in the first case femtocells become underloaded (i.e., there are no active femtocell users to serve), therefore no CM can be performed, while on the second case, most femtocells are overloaded and thus, towards efficiently serving all attached users maintain high downlink transmission powers, resulting to high crosstier interference.
The same patterns can be observed even if we increase the complexity of the system, i.e., adding two more clusters in the system as illustrated in Figure 4a–d. Specifically, reductions of up to 60% in the overall cumulative power consumption are observed, saving up to 50 W, while the macrocell users enjoy up to 50% utility increase, implying a 15Mbps boost in their aggregated performance. Moreover, we should note that low values of performance gains observed do not imply that the algorithm performs poorly, but that the system is already close to its optimal state, leaving little room for improvement.
Concluding remarks and future work
An optimal joint PCC management framework for twotier CDMA macrocell/femtocell wireless networks in the presence of cotier and crosstier interference has been presented, aiming at maximizing users’ downlink throughput performance, while minimizing cells’ excess cumulative power consumption and thus, overall interference. Towards this goal, the integrated network is viewed as a global optimizer and the corresponding networkwide NUM problem is treated via a twolevel primaldual decomposition, allowing us to devise both optimal and distributed solution. A low complexity algorithm with provable convergence attributes was realized, that exploits cells’ intracell resource allocation process using a subgradient Lagrangian scheme. Finally, initial numerical results are presented, demonstrating up to 60% improvement in both system’s performance increase and overall power consumption savings.
It is important to note that even though the proposed twolevel primaldual decomposition architecture presented and evaluated in this article was applied on CDMA macrocell/femtocell wireless systems, it can be further extended accordingly to support several other wireless access schemes, given the proven separability and convexity of both the objective function and constraints. Thus, at the lower decomposition level, IPC problems will achieve optimal intracell resource allocation (e.g., for OFDM both transmit power and subcarrier vectors needs to be derived). At a higher level, a CM master problem will be responsible for dynamically determining optimal cells’ coverage area towards networkwide performance optimality.
Moreover, the proposed framework assumes CA for femtocell users, meaning that only subscribed users can access the femtocell infrastructure. An interesting future direction would be the treatment of the PCC problem assuming Hybrid or Open access. This would add an extra binary cell assignment constraint to the overall problem formulation, amplifying its complexity and nonconvex nature. Another interesting future extension would be the incorporation of (i) minimum SINR requirements for mobile users, towards minimum QoSrequirement incorporation or (ii) femtocells’ maximum available bandwidth limitations, imposed by their DSL backbone architecture. A common direction towards addressing all these problems would be either a complex triple level primal–dual–dual decomposition, or a sophisticated twolevel dual decomposition.
Appendix
Proof of Proposition 1
Our proof is founded on (a) the monotonic property of a user’s utility as a function of the achieved SINR and (b) on the nature of the twotier system downlink SINR, for any given ${\overline{P}}_{max}$. To that end, we follow and extend the thread of analysis in [23], Proposition 1, where similaroptimization problem is treated in the case of one cell power control only.
For any given feasible ${\overline{P}}_{\text{var}}$ and for any cell w ∈ C, let $\overline{P}=\left({\overline{P}}_{0},\dots ,{\overline{P}}_{w},\dots ,{\overline{P}}_{\leftC\right1}\right)$ be a power allocation vector such that $\sum}_{j\in {S}_{w}}{P}_{w,j}}<{P}_{w,\text{var}}\phantom{\rule{0.25em}{0ex$. Let us underline that for the rest of the cells’ power allocation no assumption is made (i.e., the following hold for any potential power allocation vector values for the rest of the cells in the system, within feasibility bounds, including the optimal one). It suffices to show that there always exists another power allocation ${\overline{P}}^{\text{'}}=\left({\overline{P}}_{0},\dots ,{{\overline{P}}^{\text{'}}}_{w},\dots ,{\overline{P}}_{\leftC\right1}\right)$ such that $\sum}_{j\in {S}_{w}}{{P}^{\text{'}}}_{w,j}}={P}_{w,\text{var$, that improves the intracell utilitybased power allocation.
Thus, ∀ j ∈ S_{ w } it holds that ${U}_{j}\left({\gamma}_{w,j}\left({\overline{P}}^{\text{'}}\right)\right)>{U}_{j}\left({\gamma}_{w,j}\left(\overline{P}\right)\right)$ due to the monotonicity of U as a function of γ, and thus, ${\sum}_{j\in {S}_{w}}{U}_{j}\left({\gamma}_{c,j}\left({\overline{P}}^{\text{'}}\right)\right)}>{\displaystyle {\sum}_{j\in {S}_{w}}{U}_{j}\left({\gamma}_{c,j}\left(\overline{P}\right)\right)$.
Therefore, each cell must transmit at its maximum feasible power ${\overline{P}}_{w,\text{var}}$ regardless of state of its neighbors, in order to achieve maximum intracell utilitybased performance. Generalizing the above, and towards maximizing the cumulative system utility, every cell must transmit at its own maximum feasible power, or in reverse, vector ${\overline{P}}_{\text{var}}=\left({\overline{P}}_{1,\text{var}},\dots ,{\overline{P}}_{w,\text{var}},\dots ,{\overline{P}}_{\leftC\right1,\text{var}}\right)$ guarantees maximum total twotier system cumulative utility. This completes the proof.
Declarations
Authors’ Affiliations
References
 Chandrasekhar V, Andrews JG: Femtocell networks: a survey. IEEE Commun. Mag. 2008, 46(9):5967.View ArticleGoogle Scholar
 Calin D, Claussen H, Uzunalioglu H: On femto deployment architectures and macrocell offloading benefits in joint macrofemto deployments. IEEE Commun. Mag. 2010, 48(1):2632.View ArticleGoogle Scholar
 FMhiri KS, Reguiga B, Bouallegue R, Pujolle G: A power management algorithm for green femtocell networks. In Proceedings of the 10th IFIP Annual Mediterranean Ad Hoc Networking Workshop (MedHocNet). Sicily, Italy; 2011:4549.Google Scholar
 Mach P, Becvar Z: QoSguaranteed power control mechanism based on the frame utilization for femtocells. EURASIP J. Wirel. Commun. Netw. 2011. 10.1155/2011/259253Google Scholar
 Jo H, Mun C, Moon J, Yook J: Selfoptimized coverage coordination in femtocell networks. IEEE Trans. Wirel. Commun. 2010, 9(10):29772982.View ArticleGoogle Scholar
 Jo H, Mun C, Moon J, Yook J: Interference mitigation using uplink power control for twotier femtocell networks. IEEE Trans. Wirel. Commun. 2009, 8(10):49064910.View ArticleGoogle Scholar
 Claussen H, Ho LTW, Samuel LG: Selfoptimization of coverage for femtocell deployments. In Proceedings of the Wireless Telecommunications Symposium. Pomona; 2008:278285.Google Scholar
 Choi SY, Lee T, Chung MY, Choo H: Adaptive coverage adjustment for femtocell management in a residential scenario. In Proceedings of the APNOMS. Jeju, Korea; 2009:221230.Google Scholar
 Ashraf I, Claussen H, Ho LTW: Distributed radio coverage optimization in enterprise femtocell networks. In Proceedings of the 2010 IEEE International Conference on Communications (ICC). Cape Town, South Africa; 2010:16.View ArticleGoogle Scholar
 Yalin Z, Yang Yang L, Sousa ES, Qinyu Z: Pilot power minimization in HSDPA femtocells. In Proceedings of the 2010 IEEE Global Telecom. Conference. Miami, Florida; 2010:15.Google Scholar
 Morita M, Matsunaga Y, Hamabe K: Adaptive power level setting of femtocell base stations for mitigating interference with macrocells. In Proceedings of the 2010 IEEE 72nd Vehicular Technology Conference Fall (VTC 2010Fall). Ottawa, Canada; 2010:15.View ArticleGoogle Scholar
 MinSung K, Hui Won J, Tobagi FA: Crosstier interference mitigation for twotier OFDMA femtocell networks with limited macrocell information. In Proceedings of the 2010 IEEE Global Telecommunications Conference, GLOBECOM. Miami, Florida; 2010:15.Google Scholar
 Shuping Y, Talwar S, Himayat N, Johnsson K: Power control based interference mitigation in multitier networks. In Proceedings of 2010 IEEE GLOBECOM (FEMnet Workshop). Miami, Florida; 2010:701705.Google Scholar
 Kisong L, Ohyun J, Cho DH: Cooperative resource allocation for guaranteeing intercell fairness in femtocell networks. IEEE Commu. Lett. 2011, 15(2):214216.View ArticleGoogle Scholar
 Akbudak T, Czylwik A: Distributed power control and scheduling for decentralized OFDMA networks. In Proceedings of the 2010 Int. ITG Workshop on Smart Antennas. Bremen, Germany; 2010:5965.View ArticleGoogle Scholar
 Li X, Lijun Q, Kataria D: Downlink power control in cochannel macrocell femtocell overlay. In Proceedings of the 43rd Annual Conf. on Inf. Sciences and Systems. Baltimore, Maryland; 2009:383388.Google Scholar
 Kwanghun H, Youngkyu C, Dongmyoung K, Minsoo N, Sunghyun C, Kiyoung H: Optimization of femtocell network configuration under interference constraints. In Proceedings of the WiOPT. Seoul, Korea; 2009:17.Google Scholar
 Chee Wei T: Optimal power control in rayleighfading heterogeneous networks. In Proceedings of INFOCOM. Shanghai, China; 2011:25522560.Google Scholar
 Chandrasekhar V, Andrews JG, Muharemovic T, Zukang S, Gatherer A: Power control in twotier femtocell networks. IEEE Trans. Wirel. Commun. 2009, 8(8):43164328.View ArticleGoogle Scholar
 Jianmin Z, Zhaoyang Z, Kedi W, Aiping H: Optimal distributed subchannel, rate and power allocation algorithm in OFDMbased twotier femtocell networks. In Proceedings of 2010 IEEE 71st Vehicular Technology Conference (VTC 2010Spring). Taipei, Taiwan; 2010:15.Google Scholar
 Yun JH, Shin KG: Adaptive interference management of OFDMA femtocells for cochannel deployment. IEEE J. Sel. Areas Commun. 2011, 29(6):12251241.View ArticleGoogle Scholar
 Ganti A, Klein TE, Haner M: Base station assignment and power control algorithms for data users in a wireless multiaccess framework. IEEE Trans. Wirel. Commun. 2006, 5(9):24932503.View ArticleGoogle Scholar
 Lee J, Mazumdar R, Shroff N: Joint resource allocation and basestation assignment for the downlink in CDMA networks. IEEE/ACM Trans. Netw. 2006, 14(1):114.View ArticleGoogle Scholar
 Hande P, Rangan S, Chiang M, Wu X: Distributed uplink power control for optimal sir assignment in cellular data networks. IEEE/ACM Trans. Netw. 2008, 16(6):14201433.View ArticleGoogle Scholar
 Chiang M: Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power control. IEEE J. Sel. Areas Commun. 2005, 23(1):104116.View ArticleGoogle Scholar
 Goldsmith A: Wireless Communications. Cambridge University Press, Cambridge, UK; 2004.Google Scholar
 Chiang M, Hande P, Lan T, Tan CW: Power control in wireless cellular networks. Found. Trends Netw. 2008, 2(4):381533.View ArticleGoogle Scholar
 Gatsis N, Marques A, Giannakis G: Power control for cooperative dynamic spectrum access networks with diverse QoS constraints. IEEE Trans. Commun. 2010, 58(3):933944.View ArticleGoogle Scholar
 Chiang M: Nonconvex optimization of communication systems. In Advances in Mechanics and Mathematics Special Volume on Strang's 70th Birthday. Edited by: Gao D, Sherali H. Springer, New York; 2007:137.Google Scholar
 Julian D, Chiang M, O’Neill D, Boyd S: QoS and fairness constrained convex optimization of resource allocation for wireless cellular and ad hoc networks. In Proceedings of the IEEE TwentyFirst Annual Joint Conf. of the IEEE Computer and Com. Societies, INFOCOM. 2nd edition. New York; 2002:477486.View ArticleGoogle Scholar
 Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, Cambridge, UK; 2004.View ArticleMATHGoogle Scholar
 Chiang M, Boyd S: Geometric programming duals of channel capacity and rate distortion. IEEE Trans. Inf. Theory 2004, 50(2):245258. 10.1109/TIT.2003.822581MathSciNetView ArticleMATHGoogle Scholar
 Chiang M, Low SH, Calderbank AR, Doyle JC: Layering as optimization decomposition: a mathematical theory of network architectures. IEEE Proc. 2007, 95(1):255312.View ArticleGoogle Scholar
 Palomar DP, Chiang M: A tutorial on decomposition methods for network utility maximization. IEEE J. Sel. Areas Commun. 2006, 24(8):14391451.View ArticleGoogle Scholar
 Stanczak S, Wiczanowski M, Boche H: Distributed utilitybased power control: objectives and algorithms. IEEE Trans. Signal Process. 2007, 55(10):50585068.MathSciNetView ArticleGoogle Scholar
 Bertsekas DP, NediÂ´c A, Ozdaglar A: Convex Analysis and Optimization. Athena Scientific, Belmont, MA; 2003.MATHGoogle Scholar
 3GPP TSG SA, 3GPP TS 23.402 V9.2.0: Architecture enhancements for non3GPP accesses. Release 9Google Scholar
 3GPP TSG RAN WG4 R4092042: Simulation Assumptions and Parameters for FDD HeNB RF Requirements. 2008.Google Scholar
 3GPP TS 36.213 V 8.8.0: Evolved Universal Terrestrial Radio Access (EUTRA); Physical Layer Procedures. 2009.Google Scholar
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