 Research
 Open Access
Fair greening of broadband access: spectrum management for energyefficient DSL networks
 Paschalis Tsiaflakis^{1}Email author,
 Yung Yi^{2},
 Mung Chiang^{3} and
 Marc Moonen^{1}
https://doi.org/10.1186/168714992011140
© Tsiaflakis et al; licensee Springer. 2011
 Received: 3 February 2011
 Accepted: 26 October 2011
 Published: 26 October 2011
Abstract
Dynamic spectrum management (DSM) is recognized as a promising technology to reduce power consumption in DSL access networks. However, the correct formulation of poweraware DSM problem statements requires a proper understanding of greening, i.e., reducing power consumption. In this paper, we, therefore, investigate greening and show that it can be decomposed into two dimensions: the price of greening and the fairness of greening. We first analyze the price of greening, providing theoretical bounds on the powerrate tradeoff and identifying the typical trends that can be expected in practice, with some particularly promising results. Then, we introduce the fairness dimension, where we show that fairness becomes crucial when reducing power consumption. We propose four different fair greening policies that can be used to obtain a favorable tradeoff between fast, fair and green DSL operation. Finally, we evaluate and quantify the corresponding tradeoffs for realistic DSL access networks.
Keywords
 Power Usage
 Digital Subscriber Line
 Normalize Data Rate
 Fairness Term
 Dynamic Spectrum Management
1. Introduction
Digital subscriber line (DSL) refers to a family of technologies that enable digital broadband Internet access over the local telephone network. It is currently the dominating wireline broadband access technology with a global market share of 65% [1]. The main reason for its popularity is its low deployment cost, as DSL reuses the twisted pairs of the existing telephone network infrastructure to connect subscribers to the Internet backbone.
To cope with the increasing demands of the users (endusers as well as service providers) and to stay competitive with other broadband access technologies, DSL technology is continuously innovated to further improve its broadband performance and so as to extend its life span. One of the major impairments that limits further improvement of DSL performance is crosstalk, i.e., the electromagnetic interference among different lines in the same cable bundle. The presence of crosstalk turns the DSL access network into a very challenging interference environment where the transmission of one line can significantly degrade the performance of the other lines. Proper network management of these interfering lines is, therefore, crucial to prevent huge performance degradations.
Dynamic spectrum management (DSM) is recognized as a key technology for tackling this crosstalk problem [2, 3] by jointly coordinating the transmission of the users, i.e., lines, in the network. There exist two types of multiuser coordination: spectrum level coordination[4–12] and signal level coordination[13–16]. In this paper, the focus will be on spectrum level coordination, also referred to as spectrum management, spectrum balancing or multicarrier power control. More specifically, spectrum level coordination comes down to allocating the transmit spectrum, i.e., transmit powers over all frequencies, to the different users so as to prevent the destructive impact of crosstalk. It is already shown in literature that this level of access network coordination can spectacularly boost data rates [2, 3, 5]. Note that in this paper, DSM will refer to spectrum coordination.
Significant research efforts have been spent on DSM to increase the data rates in DSL access networks, i.e., rateadaptive DSM. However, only little attention has been devoted to true power minimization. Recently, power consumption has gained importance (e.g., ITUT Study Group 15). Information and communication technologies (ICTs) have been identified as significant contributors to global warming [17]. Broadband equipment contributes to the electricity consumption, where a total European consumption of up to 50 TWh per year can be estimated for the year 2015 [18]. Therefore, the European Code of Conduct for Broadband Equipment takes initiative in setting up general principles and actions, and targets to limit the (maximum) electricity consumption to 25 TWh per year, which is equivalent to 5.5 million ton of oil equivalent and to a total saving of about € 7.5 billion per year. DSL, as the most widely deployed wireline broadband access technology today, plays an important role in this initiative [19]. The Broadband Forum encourages international standard bodies to develop techniques for power reduction within the scope of their activities and to maximize power savings while preserving and enhancing quality of service [19].
One of the technologies that fits well in this framework is DSM. More specifically, a large portion of the power consumption of today's stateoftheart VDSL2 chipsets is due to the line driver [20]. The line driver consumption depends on the transmit powers. Now, DSM can be used to reduce the transmit powers, which is reflected into an overall power reduction.
In this paper, we, therefore, revisit DSM with a specific focus on how to approach greening, i.e., reducing power consumption, in DSL systems. Although some powerminimizing approaches have been proposed in literature recently, there is no thorough investigation into the concept of greening. What is greening and how should we analyze it properly? What are good greening strategies for (possibly very asymmetric) interferencelimited systems? It is important to investigate the tradeoff between both contradictory objectives, i.e., data rate maximization (fast) versus power minimization (green). Reducing power consumption results in a loss of data rate performance, which can be seen as the price of greening. The relation between greening and the price of greening depends on the crosstalk among the users and should be investigated for different practical settings so as to assess the potential of greening DSL broadband Internet access. Furthermore, we identify that there is a second crucial dimension in greening, namely fairness. We observe that straightforwardly reducing total system power may result in very unfair allocations. Therefore, the focus is also on providing concrete greening policies that strive toward obtaining fairness among the users, resulting in a desirable tradeoff between fast, fair and green DSL operation. Note that the fairness part has a connection with the operator's service plans.
Powerefficient DSM is only a very recent topic in the field of DSL broadband access. However, the main focus up to now has been on algorithm design, where a number of parallel developments have been presented. In [4] iterative distributed DSM algorithms are proposed for minimizing the sum power for each user. In [7, 20–24] the problem of minimizing the (weighted) sum transmit powers of the multiuser DSL system has been considered, where different solution procedures are proposed: in [20, 24] procedures are proposed based on iterative geometric programming (GP) approximation; in [21, 23], power backoffbased procedures are proposed; in [22, 25, 26], solutions are proposed that make use of existing rateadaptive DSM algorithms; in [27], power consumption is reduced and stability is improved by using a bandpreference method. All the above research contributions on improving the power efficiency of DSL systems by the usage of DSM are understood to fall under a common term referred to as 'Green DSL'. We particularly also refer to [25, 26] in which a general Green DSL framework was proposed. Besides the field of DSL systems, powerefficient resource allocation has also been considered in different contexts such as for beamforming [28] and also for fading multipleaccess and broadcast channels [29] in wireless contexts.
 (i)
First, we show how existing work on powerminimizing DSM can be framed into a general Green DSL framework that covers all existing DSM problem formulations. We then propose a number of novel poweraware DSM problem formulations, next to a first energyaware DSM problem formulation that focuses on energies instead of on transmit powers. This general Green DSL framework is then explained using the duality between rate regions and power regions, which results in a simple but powerful geometric interpretation of poweraware DSM.
 (ii)
Within this geometric setting, we identify that greening can be understood and decomposed in two different dimensions: (1) the price of greening and (2) the fairness of greening. Both dimensions reflect different design objectives in a multiuser setting and should be properly taken into account when designing greening strategies.
 (iii)
The 'price of greening dimension' is then investigated for DSL scenarios. Next to a theoretical worstcase analysis, we quantify the powerrate tradeoff for different practical settings. We also investigate the properties that determine these promising powerrate tradeoffs and identify the trends that can be expected for practical DSL systems.
 (iv)
We then investigate the second dimension of greening, which is the 'fairness dimension', and identify that fairness becomes a crucial design objective when applying greening. We introduce the concept of fair greening, which allows to consider fairness when allocating the price of greening over the DSL users. Four concrete greening policies are proposed to obtain fair greening in future DSL networks, and their practical impact and potential for realistic DSL scenarios is evaluated. Note that fairness has been studied thoroughly in the context of data rates [30, 31]. However, fairness in the context of reducing energy is a novel concept that has, as far as we know, not been studied before.
The rest of the paper is organized as follows. In Section 2, the system model is introduced with some background information on 'DSM design'. In Section 3, the 'Green DSL' framework from [26] is briefly summarized, and novel power and energyaware DSM formulations are proposed. This is then framed within a rate and power region setting. In Section 4, the concept of greening is analyzed in which the two dimensions 'the price of greening' and 'the fairness of greening' are introduced. These dimensions are further investigated and evaluated with realistic DSL systems in Sections 4B and 4C, respectively. We would like to highlight here that although the concepts and ideas will be explained for DSL wireline communications, they can also be applied to OFDMbased wireless greening.
2. System model and background: DSM design
A. System model
The vector ${\mathbf{x}}_{k}={\left[{x}_{k}^{1},\dots ,{x}_{k}^{N}\right]}^{\text{T}}$ contains the transmitted signals on tone k, where ${x}_{k}^{n}$ refers to the signal transmitted by user n on tone k. Vectors z_{ k } and y_{ k } have similar structures; z_{ k } refers to the additive noise on tone k, containing thermal noise, alien crosstalk and radio frequency interference (RFI); y_{ k } refers to the received signals on tone k. H_{ k } is an N × N matrix with ${\left[{\mathbf{H}}_{k}\right]}_{n,m}={h}_{k}^{n,m}$ referring to the channel gains from transmitter m to receiver n on tone k. The diagonal elements are the direct channels, and the offdiagonal elements are the crosstalk channels.
The transmit power of user n on tone k, also referred to as transmit power spectral density, is denoted as ${s}_{k}^{n}\triangleq {\Delta}_{f}E\left\{\mid {x}_{k}^{n}{\mid}^{2}\right\}$, where Δ _{ f } refers to the tone spacing. The vector ${\mathbf{s}}_{k}\triangleq \left\{{s}_{k}^{n},n\in \mathcal{N}\right\}$ denotes the transmit powers of all users on tone k. The vector ${\mathbf{s}}^{n}\triangleq \left\{{s}_{k}^{n},k\in \mathcal{K}\right\}$ denotes the transmit powers of user n on all tones, i.e., the transmit spectrum of user n. The vector $\mathbf{s}\triangleq \left\{{s}_{k}^{n},k\in \mathcal{K},\phantom{\rule{2.77695pt}{0ex}}n\in \mathcal{N}\right\}$ denotes all transmit powers in the network, i.e., of all users and over all tones. The received noise power by user n on tone k, also referred to as noise spectral density, is denoted as ${\sigma}_{k}^{n}\triangleq {\Delta}_{f}E\left\{\mid {z}_{k}^{n}{\mid}^{2}\right\}$.
We would like to highlight that R^{ n } corresponds to a challenging nonconvex function of the transmit powers ${s}_{k}^{n},n\in \mathcal{N},\phantom{\rule{2.77695pt}{0ex}}k\in \mathcal{K},$.
B. DSM design
in which ${s}_{k}^{n,\text{mask}}$ refers to the spectral mask constraint for user n on tone k.
Note that this rate region can be assumed to be convex for the considered multitone interference channel as the number of tones K is typically very large [5, 33].
A typical design objective [4–9, 12], is to optimize the transmit powers so as to achieve some Paretooptimal allocation of data rates R^{ n } , $n\in \mathcal{N}$, somewhere on the boundary of the achievable rate region. However, other design objectives are also possible, considering other performance measures like transmit power, fairness, delay, etc. In general, we will refer to a DSM design as an optimization problem where the transmit powers correspond to the optimization variables, and where the objectives and/or constraints are functions of these transmit powers and reflect the quality of service (QoS) requirements for the corresponding DSL network. Note that the transmit power constraints (3) and (4) are required in all DSM designs in order to be compliant with DSL standards.
3. GREEN DSL
A. Green DSL framework
in which the weights w = [w_{1}, ..., w_{ N } ] are used to give more priority to some users with respect to the other users. This formulation also plays a central role in many other network control and optimization methods, such as, for instance, crosslayer control policies in wireless networks [34, 35].
in which t = [t_{1}, ..., t_{ N } ] are weights.
with the objective function $U\left(\mathbf{R},\mathbf{P}\right):{\mathbb{R}}_{+}^{N}\times {\mathbb{R}}_{+}^{N}\to \mathbb{R}$, the inequality constraints $\mathbf{I}\left(\mathbf{R},\mathbf{P}\right):{\mathbb{R}}_{+}^{N}\times {\mathbb{R}}_{+}^{N}\to {\mathbb{R}}^{M}$, the equality constraints $\mathbf{E}\left(\mathbf{R},\mathbf{P}\right):{\mathbb{R}}_{+}^{N}\times {\mathbb{R}}_{+}^{N}\to {\mathbb{R}}^{L}$, and in which M denotes the number of inequality constraints and L the number of equality constraints. Although typical DSM designs are generally NPhard nonconvex problems, efficient dualdecompositionbased procedures have been proposed in [25, 26, 36] for tackling Green DSL DSM designs.
B. Novel poweraware DSM designs
Using the general Green DSL framework of [26], we can now define a whole range of novel DSM designs that model different tradeoffs between data rates and transmit power consumption.
where 0 < α ≤ 1 is a chosen constant and denotes the desired power reduction with respect to full transmit power usage. This formulation can be relevant when a service provider aims to reduce the network transmit power consumption by a factor α.
where the $\left\{{\gamma}_{n}:n\in \mathcal{N}\right\}$ indicate the proportions with respect to a base power P. One interesting objective would, for instance, be to minimize the maximum of all transmit powers P^{ n } , which corresponds to (11) with γ_{ n } = 1, $n\in \mathcal{N}$.
with a nonlinear third constraint.
C. Energyaware DSM design
in which the first set of constraints corresponds to time deadlines T^{n,target}, before which the jobs need to be finished. In "Appendix A", we prove that this problem statement can be reformulated as (8) with weights t_{ n } = T^{n, target}and constraints R^{ n } = R^{n,target}= J^{ n }/T^{n,target}and can thus be readily solved using the Green DSL algorithms described in [25, 26, 36]. The main message for energy minimization is to transmit as slow as possible tightly satisfying the timing constraints.
D. A geometric view of green DSL by achievable rate and power regions
The working points of the different DSM designs (6)(13) are indicated and explained in Figure 1. Note that the data rate driven DSM designs (6)(7) can only characterize a working point on the boundary of the rate region, not a working point inside the rate region. These boundary points mostly correspond to transmit power allocations where all users fully use their available transmit power. Obviously, this is not the best strategy in terms of energy efficiency. In contrast, the poweraware DSM designs (8)(13) also allow to achieve working points strictly inside the rate region, corresponding to more powerefficient working points inside the power region.
4. Analysis of greening concept in green DSL
A. Greening dimensions
with η and ζ scalar multiples, and vector v orthogonal to the rate vector before greening R^{ o } .
Dimension g lies along the direction of the weight w (of (7)) in R^{ o } and can be seen as the negative gradient of the rate region, i.e., the direction in which the weighted rate sum decreases the fastest. This dimension indicates the 'price of greening'.
Dimension f is orthogonal to the direction in which the data rates are reduced proportionally in R^{ o } . This dimension gives the direction in which data rates are distributed unfairly, and it thus indicates the 'fairness' dimension.
which are also shown in Figure 2.
These two dimensions are essential in understanding greening. For instance, data rate reduction ΔR^{ A } has a small component along g and thus corresponds to a small price of greening, i.e., a good powerrate tradeoff. In other words, the distance to the boundary of the rate region before greening is small. However, the data rate distribution over the two users is very unfairly, i.e., user 2 reduces its data rate a lot, whereas user 1 even increases its data rate. In contrary, data rate reduction ΔR^{ B } has a large price of greening, but a very fair (proportional) distribution of the data rates. In the ideal case, we should have small components along both dimensions.
A good greening formulation should thus focus on both dimensions, to obtain a good powerrate tradeoff and also to prevent that weak users have to pay all data rate performance loss. Current DSM formulations do not consider these dimensions and consequently can result in very unfair or bad powerrate tradeoffs, depending on the considered DSL scenario.
The previous twouser case can be straightforwardly extended to a general Nuser case, for which dimension g is unchanged (onedimensional) and dimension f becomes an N  1dimensional subspace orthogonal to vector R^{ o } .
In the following two sections, we will analyze DSL systems along the two dimensions g and f. This will allow to identify good greening strategies and also to assess the potential of greening DSL broadband Internet access.
B. Powerrate tradeoff in green DSL
The price of greening dimension, i.e., the powerrate tradeoff, is an important greening dimension as it reflects the impact of saving power. The relation between transmit powers and data rates is, however, governed by very a nonconvex function, given by (1) and (2), which significantly complicates the theoretical analysis of the powerrate tradeoff. Furthermore, the channel parameters depend heavily on the considered DSL setting. Therefore, a theoretical analysis that holds for all possible channel parameters does not allow to capture the particular structure of different practical settings.
In this section, we will, therefore, first provide theoretical lower and upper bounds on the optimal powerrate tradeoff for a network without interference. It is then discussed how these bounds extend to the interference case. Based on these insights, we assess the price of greening for different practical settings by numerical simulations using practical channel models. This will allow to identify the typical trends that can be expected in DSL access systems under different practical settings.
1) Theoretical analysis of powerrate tradeoff
We start with a multitone singleuser bound analysis:
with R^{ n } (P) being the optimal data rate for given total power P and ${\text{CNR}}_{k}^{n}=\mid {h}_{k}^{n,n}{\mid}^{2}\u2215\left(\Gamma {\sigma}_{k}^{n}\right)$.
Proof: See "Appendix B". ■
Note that for the multiuser setting, we will use CNR to refer to the channel to interference and noise ratio, i.e., ${\text{CNR}}_{k}^{n}=\mid {h}_{k}^{n,n}{\mid}^{2}(\Gamma ({\sum}_{m\ne n}\mid {h}_{k}^{n,m}{\mid}^{2}{s}_{k}^{m}+{\sigma}_{k}^{n}))$.
Lemma 4.1 shows how the powerrate tradeoff, i.e., f (c) in function of c, depends on the CNRs, the transmit power budget and the power saving factor c. Using this lemma, we can obtain the following theorem describing the theoretical bounds on the powerrate tradeoff.
Proof: The proof follows trivially from Lemma 4.1 by taking a limit operation of (16). ■
Lemma 4.1 and Theorem 4.1 provide lower and upperbounds on the optimal powerrate tradeoff that hold for any singleuser DSL system setting. Furthermore, they show that the tradeoff depends on the CNRs and the transmit power budget: larger CNRs and a larger transmit power budget result in a more favorable powerrate tradeoff, as more power can be saved for a certain data rate decrease.
We can give a very reasonable intuitive explanation as follows: in an access network with a lot of crosstalk, it requires a relatively larger amount of power to increase the data rate by a factor two, e.g., from 1 to 2 Mbps. This also means that in this network, reducing the data rate from 2 to 1 Mbps, will result in a larger relative power decrease than for a network with less crosstalk. Another explanation can be obtained from [33] in which it is shown that for the multiuser case and when the number of tones is large, the (un)weighted data rates show a socalled time sharing property in function of transmit powers. This means that the relation between the data rates and the total transmit powers is concave and thus in the worst case linear.
Given the above arguments, we can conclude that the upper and lower bounds of Theorem 4.1 also hold for the multiuser case, and we furthermore understand the factors that determine the powerrate tradeoff.
2) Simulationbased analysis of the powerrate tradeoff
The bounds of Section 4B1 are valid for general channel parameters. They, however, do not capture the particular characteristics of typical practical settings. In this section, we will, therefore, assess these bounds for concrete DSL settings and investigate trends that can be expected for practical DSL systems using a simulationbased approach.
Simulation Setup
The following realistic parameters settings are assumed for the DSL scenarios. The twisted pair lines have a diameter of 0.5 mm (24 AWG). The maximum transmit power is 20.4 dBm for the ADSL scenarios and 11.5 dBm for the VDSL scenarios. The SNR gap Γ is 12.9 dB, corresponding to a coding gain of 3 dB, a noise margin of 6 dB and a target symbol error probability of 10^{7}. The tone spacing Δ _{ f } is 4.3125 kHz. The DMT symbol rate f_{ s } is 4 kHz. The DSM algorithms ((I)DSB, (I)MSDSB), discussed in [12, 36], are used to solve the Green DSL problem formulations.
Impact of ChanneltoNoise Ratio
The second scenario is a symmetric downstream ADSL scenario and is depicted in Figure 5b. Its corresponding powerrate tradeoff is shown as the red curve in Figure 5d. One can observe a data rate performance of 72% for a 50% power saving.
The evolutions of the corresponding optimal bit loadings are shown for a linearly increasing power budget ranging from 10 to 100% (i.e., full power) in steps of 10% for the two scenarios in Figure 5e, f). One can observe a law of diminishing marginal returns, i.e., a linear increase in power leads to a diminishing data rate increase. Note that this effect is less obvious in Figure 5f, i.e., for the symmetric DSL scenario. The tradeoff between data rate performance and power usage thus depends on the type of scenario.
From Theorem 4.1, we know that large CNRs result in a good powerrate tradeoff, whereas small CNRs result in a linear powerrate tradeoff. From Figure 5e, it can be seen that the bit loadings range from 0 to 13 bits, with an average of 6 bits. This corresponds to a rather large CNR and thus also explaining the favorable powerrate tradeoff, i.e., by reducing transmit powers up to 50%, only a small decrease in data rate performance of 15% is observed. In Figure 5f, the bit loadings, in contrary, range only between 0 to 4 bits, with an average of 2 bits. This corresponds to smaller CNRs and thus leads to a larger impact on the data rate performance, i.e., 72 for 50% power usage.
The impact on the data rate performance thus depends on the CNR, which in turn depends on the line attenuation, i.e., $\mid {h}_{k}^{n,n}{\mid}^{2}$. Note that longer lines have larger attenuations, i.e., smaller values for $\mid {h}_{k}^{n,n}{\mid}^{2}$. Thus, in scenarios with long line lengths, we will observe larger decreases in data rates for given power savings. Note, however, that the considered symmetric DSL scenario of Figure 5b has a maximum line length of 5.5 km and is characterized by a 5072% powerrate tradeoff. For most practical DSL scenarios, which generally have much shorter line lengths, CNRs are typically much larger, and thus, we can expect even more favorable powerrate tradeoffs. This will be confirmed by further simulations in this section.
Impact of Network Size
We would like to highlight here that the law of diminishing marginal returns is not just stating the concavity of the logarithmic relation between bits and powers, but there is also an additional influence by the characteristics of the crosstalk.
Impact of DSL technology
It is not so easy to make a general statement of which technology (ADSL(2) versus VDSL(2)) has a more favorable powerrate tradeoff as it depends on multiple factors. Simulations, however, show that VDSL(2) systems typically have a better powerrate tradeoff compared to ADSL(2)(+) systems. This can be explained by (i) the shorter line lengths that result in higher average CNRs and (ii) larger crosstalk levels as higher frequencies are used, which both contribute to obtain a more favorable powerrate tradeoff.
Summary for price of greening dimension
We can summarize that the price of greening is very small for practical DSL access networks, i.e., much better than linear. A first explanation is the typical very high CNR values of DSL copper wires. Furthermore, the presence of crosstalk in DSL systems results in an improved powerrate tradeoff. In addition, newer DSL technologies such as VDSL(2) that are currently widely being deployed have better powerrate tradeoffs. This means that there is a huge potential of saving a lot of transmit power in future DSL access networks by the usage of poweraware DSM.
C. Fairness in green DSL
It is important to define a number of greening policies, i.e., policies to green the DSL access network, which follow some fairness notion so as to prevent that some users are treated unfairly, especially for asymmetric DSL scenarios. This will be referred to as fair greening.
In this section, we propose and study different fair greening policies that have different powerrate tradeoffs. These policies fit within the Green DSL framework of [26] and lead to a threeway tradeoff between fast, green and fair operation, i.e., a tradeoff between the price of greening, greening and the fairness of greening, respectively. This tradeoff will be quantified for some concrete practical DSL scenarios.
1) Fair greening policies
There is no universally agreed notion of fairness, especially when applied in the context of an emerging topic like Green ICT. Based on a range of reasonable views on what fairness is, we develop four different greening policies for green DSL, each parametrized by a parameter β that is used to vary the degree of greening. Throughout this subsection, for a given fair greening formulation, we denote by {R^{ o }, P^{ o }} the ratepower operating point somewhere on the boundary of the rate region $\mathcal{R}$, i.e., the operating point before greening and that corresponds to the operating point obtained for β = 1. We distinguish two strategies of integrating fairness: (1) fair greening by constraints and (2) fair greening by regularization in the objective function.
Fair greening by constraints
Here, fairness definitions are incorporated into the constraints.
(1) Greening 1: Powerfair greening
with greening parameter β ∈ [0, 1], where β = 1 corresponds to no greening and β = 0 to 100% greening. This fair greening policy can be summarized as "the degree of greening should be proportional for all users". Note that this approach does not necessarily enforce that the data rate reduction ΔR in Figure 2 lies parallel with R^{ o } , i.e., not fair in the reduction in the rates.
(2) Greening 2: Ratefair greening
with greening parameter β ∈ [0, 1], where β = 1 corresponds to no greening and β = 0 to 100% greening. This fair greening policy can be summarized as "the price of greening should be proportional for all users". It enforces that the data rate reduction ΔR in Figure 2 is parallel with R^{ o } , i.e., fair in the reduction in the rates.
(3) Greening 3: Power/rate proportional greening
Note that this approach jointly considers fairness in data rates as well as in total transmit powers. This fair greening policy can be summarized as "the ratio of the price of greening to the degree of greening should be proportional for all users". Note that β = 1 corresponds to no greening, and increasing β corresponds to greening. One can easily extend (20) with an extra bisection search to find the value of β that corresponds to a particular system power usage. This is because system power usage is monotonically decreasing with increasing β.
Fair greening by regularization in objective function
This type of fair greening policies implements fairness by adding a weighted fairness term to the objective.
(4) Greening 4: Weighted rate sum greening with fair regularization
where δ is a weighting factor that can be tuned to emphasize the importance of greening fairness relative to the maximum weighted data rate performance. The function G^{ n } (P^{ n } ) may take different forms depending on the desired power fairness. One possibility is αfairness [31]: G^{ n } (·) = (·)^{1α}/(1  α), for α > 0, and G^{ n } (·) = log (·), for α = 1, which includes maxmin (α → ∞) and proportional fairness (α = 1) as special cases. Another possibility is to use a second moment as a measure of fairness: G^{ n } (.) =  (.)^{2}. We define the following instances of this fair greening policy, which will be evaluated in Section 4C2:
• Greening 4A: (21) with δ = 0;
• Greening 4B: (21) with δ > 0 and G^{ n } (.) = (.)^{2};
• Greening 4C: (21) with δ > 0 and G^{ n } (.) = log(.).
These fair greening policies can be summarized as "green the DSL network with or without peruser green fairness objectives installed". Note that by tuning δ, we can tradeoff power fairness versus weighted rate maximization. Also, note that instead of putting a fairness term in transmit powers, we can also choose a fairness term in data rates, i.e., G^{ n } (R^{ n } ) to enforce fairness over data rates if desired.
2) Fair greening simulations and analysis
The four proposed fair greening policies (Greening 1/2/3/4) fit within the Green DSL framework (9) and can thus be solved using the efficient algorithms as proposed in [25, 26, 36]. To demonstrate the importance of considering fairness when reducing power consumption, we have simulated the performance of the different fair greening policies for two different DSL scenarios. The first scenario is the standard nearfar ADSL downstream scenario, as depicted in Figure 5a. The second scenario is an asymmetric 6user ADSL downstream scenario, which is depicted in Figure 6a with 'modem 7' inactive. The same realistic system parameter settings are used as in Section 4B.
Figure 8b shows the distribution of the normalized data rates (w.r.t. the fullpower data rate R^{n,o}for each user n) and normalized transmit powers (w.r.t. the fullpower P^{n,tot}budget of each user n) across the two users for the scenario of Figure 5a for the different greening policies when 50% greening is applied, i.e., ∑_{ n }P^{ n }/∑_{ n }P^{n, tot}= 0.5.
It can be seen that Greening 4A is the best in terms of rate sum performance, i.e., 50% of system transmit power is saved while still achieving 93% of fullpower data rate performance. However, as can be seen from Figure 8b, Greening 4A allocates the data rates and the transmit powers very unfairly over the users, i.e., user 2 dominates over user 1 in terms of transmit power as well as data rate. This unfair behavior is a result of the asymmetric interference of the considered DSL scenario and also because Greening 4A does not implement any fairness regularization, lacking any mechanism to steer toward fair transmit power and/or data rate allocations. So, in terms of Figure 2, Greening 4A has a small projected component along the price of greening dimension g, i.e., ${\mathcal{P}}_{\mathbf{g}}\left(\Delta \mathbf{R}\right)$ is small, but a large projected component along the fairness dimension f, i.e., ${\mathcal{P}}_{\mathbf{f}}\left(\Delta \mathbf{R}\right)$ is large. This very unfair behaviour should be prevented, and it demonstrates that taking fairness into account is essential, especially for asymmetric DSL scenarios.
Greening 1 proportionally allocates the transmit powers over the users. Greening 2 equalizes the normalized data rates, but results in an uneven allocation of transmit powers. Greening 4B and 4C succeed in obtaining relatively better fairness in terms of transmit powers as well as data rates than Greening 4A. This is due to the addition of the fairness term into their objective functions. Note that for Greening 4B and 4C, only one simulation point is shown rather than a parametric curve, due to the dependence on the chosen weight δ in (21). Different values for this weight lead to different tradeoffs between data rate performance and fairness in transmit powers, where we have tuned the weight to obtain a good tradeoff between both objectives. Finally, Greening 3 results in a ratio of the normalized data rates to the normalized transmit powers that is proportional for all users.
The inclusion of fairness thus results in a fairer allocation of data rates and/or transmit powers, but the price of adding fairness is a reduced data rate sum performance compared to Greening 4A, as can be seen in Figure 8a.
Fairness metric
with x_{ n } = (R^{ n }/R^{n, o})/(P^{ n } /P^{n, o}) and {R^{ o } , P^{ o } } denotes the point on the boundary of the rate region without greening. $\mathcal{F}=1$ when all users have the same ratio between data rate decrease and power usage decrease and approaches zero as these ratios start to deviate from each other. Figure 8c shows the tradeoff between the greening fairness index $\mathcal{F}$ and the power usage for the proposed fair greening policies. The key messages are as follows:

Power/Rate Proportional Greening 3 is 100% fair w.r.t. $\mathcal{F}$, since it was constructed by considering both power usage and data rate in the first place.

Powerfair Greening 1 is also quite fair, whereas ratefair Greening 2 is unfair.

Weighted rate sum Greening 4A is very unfair. However, by adding the fairness terms to Greening 4A, i.e., Greening 4B and 4C, the fairness behavior, w.r.t. $\mathcal{F}$, becomes much better, i.e., from 67% to more than 94%.
The key messages can thus be summarized as follows:

Fairness should be considered carefully so as to prevent that some users are treated unfairly or even put outofservice.

Greening 1, i.e., proportionally decreasing all users' transmit powers, generally leads to a good fairness behavior, which is much better than Greening 2, i.e., proportionally decreasing the data rates for all modems.

Greening 3 results in a behavior that is jointly fair in data rate and transmit power allocations.

The inclusion of fairness comes with a price in data rate performance.
Tradeoff Fast versus Fair versus Green
We have used the following definitions to quantify the threeway tradeoff for the proposed fair greening policies:

FAST: $\frac{{\sum}_{n\in \mathcal{N}}{w}_{n}{R}^{n}}{{\sum}_{n\in \mathcal{N}}{w}_{n}{R}^{n,o}}$

FAIR: $\mathcal{F}$ (= fair greening index (22))

GREEN: $1\frac{{\sum}_{n\in \mathcal{N}}{P}^{n}}{{\sum}_{n\in \mathcal{N}}{P}^{n,o}}$
Note that, with these definitions, it is impossible to achieve a triangle that is both 100% fast and 100% green. In Figure 10a, b, we have plotted the FFG triangles for 'No greening', Greening 4A and Greening 3 for 50% greening for the 2user DSL scenario of Figure 5a and the 6user DSL scenario of Figure 6a, respectively. 'No greening' corresponds to the working point on the boundary of the rate region. It can be seen that Greening 3 achieves a considerable better threeway tradeoff w.r.t Greening 4A and 'No greening'.
FFG tradeoff performances for different fair greening policies for 50% greening.
2user ADSL (Fig. 5a) (%)  6user ADSL (Fig. 6a) (%)  

No greening  33.33  33.33 
Greening 1  60.65  60.69 
Greening 2  51.30  40.57 
Greening 4A  47.80  36.08 
Greening 4B  60.80  48.85 
Greening 4C  61.23  54.24 
Greening 3  61.37  60.17 
One can see that Greening 3 results in a very good FFG tradeoff performance. For the 2user case, it performs 84.13% better than 'No greening' and 28.39% better than Greening 4A. For the 6user case, Greening 3 performs 80.53% better than 'No greening' and 66.77% better than Greening 4A. One can notice that Greening 1 also results in a very good FFG tradeoff performance. This means that the proportional reduction in the available transmit powers results in a quite reasonable proportional data rate reduction, and so Greening 1 can be seen as a good fair greening policy. In contrast, Greening 2 results in a much less favorable FFG tradeoff. Greening 4B and 4C have a better FFG tradeoff performance with respect to Greening 4A, which is a result of the inclusion of the peruser fairness objectives G^{ n } (.).
5. Conclusion
Reducing the power consumption of broadband access networks has gained quite some momentum over the last few years. Dynamic spectrum management (DSM) is a very promising technique to significantly improve the power efficiency of DSL broadband access networks by optimizing the transmit powers. In this paper, we have first proposed some novel power and energyaware DSM problem formulations that fit within our recently proposed 'Green DSL' framework [25, 26] and that improve the power efficiency of DSL transmission. However, the formulation of these poweraware DSM formulations requires a proper understanding of greening in interferencelimited networks. We have, therefore, investigated the concept of greening and have identified that greening can be decomposed into two essential greening dimensions: (1) the price of greening, i.e., the powerrate tradeoff, and (2) the fairness of greening. Only by addressing both dimensions, one can obtain proper greening strategies. Along both dimensions, we have conducted a thorough analysis consisting of theoretical results and also simulationbased results so as to identify the typical trends that can be expected for practical DSL scenarios under different realistic settings. We have shown that the powerrate tradeoff depends on the CNRs, the available power budgets, the DSL technology, as well as the interference characteristics. Furthermore, we have shown that very large power savings can be obtained with only a minor impact on the data rate performances. For instance, we have demonstrated that for some VDSL scenarios transmit power savings of up to 50% can be achieved while preserving 95% of fullpower data rate performance. Finally, we have identified that fairness is a very important dimension that has to be taken into account to prevent unfair allocation when greening, especially in asymmetric DSL scenarios. Therefore, we have proposed four different fair greening policies to incorporate fairness when greening, of which Greening 3 achieves a good tradeoff between fast, fair and green operation.
Appendix
A. Proof of (13) being a special case of (8)
As R^{ n } (P^{ n } ) is concave in P^{ n } for the multiuser case (see Section 4B1), R^{ n } (P^{ n } )/P^{ n } is maximized when P^{ n } → 0 or equivalently R^{ n } (P^{ n } ) → 0. Reducing P^{ n } , and the corresponding R^{ n } (P^{ n } ), results in a decrease of ∑_{m ≠ n}P^{ m }J^{ m }/R^{ m }as the data rates of the other users are increased because less crosstalk is radiated to them. The optimal solution thus corresponds to minimizing powers and rates for all users, or equivalently maximizing the times to finish the jobs, which results in the equality T^{ n } = T^{n,target}. This proofs that (13) is special case of (8) with t_{ n } = T^{n,target}and R^{n,target}= J^{ n }/T^{n,target}. ■
B. Proof of Lemma 4.1
■
Declarations
Acknowledgements
Paschalis Tsiaflakis is a postdoctoral fellow funded by the Fonds Wetenschappelijk Onderzoek (FWO)  Vlaanderen. This research work was carried out at the ESAT Laboratory of Katholieke Universiteit Leuven, in the frame of K.U. Leuven Research Council CoE EF/05/006 Optimization in Engineering (OPTEC), Concerted Research Action GOAMaNet, the Belgian Programme on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy Office IUAP P6/04 (DYSCO, Dynamical systems, control and optimization, 20072011), Research Project FWO nr.G.0235.07 (Design and evaluation of DSL systems with common mode signal exploitation), and is supported in part by Princeton Grand Challenge Grant, NSF NetSE grant on SocioTechnical Networking and a Google grant, and also by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20110015042). A part of this paper has appeared in the Proceedings of the IEEE International Conference on Communications (ICC), Dresden, Germany, June 2009 [25] and in the Proceedings of the ACM GreenMetrics Workshop, Seattle, WA, June 2009 [26].
Authors’ Affiliations
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