Fair Adaptive Bandwidth and Subchannel Allocation in the WiMAX Uplink

In some modern communication systems, as it is the case of WiMAX, it has been decided to implement Demand Assignment Multiple Access (DAMA) solutions. End-users request transmission opportunities before accessing the system, which provides an e ﬃ cient way to share system resources. In this paper, we brieﬂy review the PHY and MAC layers of an OFDMA-based WiMAX system, and we propose to use a Network Utility Maximization (NUM) framework to formulate the DAMA strategy foreseen in the uplink of IEEE 802.16. Utility functions are chosen to achieve fair solutions attaining di ﬀ erent degrees of fairness and to further support the QoS requirements of the services in the system. Moreover, since the standard allocates resources in a terminal basis but each terminal may support several services, we develop a new decomposition technique, the coupled-decompositions method, that obtains the optimal service ﬂow allocation with a small number of iterations (the improvement is signiﬁcant when compared to other known solutions). Furthermore, since the PHY layer in mobile WiMAX has the means to adapt the transport capacities of the links between the Base Station (BS) and the Subscriber Stations (SSs), the proposed PHY-MAC cross-layer design uses this extra degree of freedom in order to enhance the network utility.


Introduction
The wireless community has recently directed much attention on a variety of topics related to Worldwide Interoperability for Microwave Access (WiMAX) technologies as a broadband solution.Two different standards are under this commercial nomenclature: the IEEE 802.16 [1], with its extension to mobile scenarios IEEE 802.16e [2], and the ETSI HiperMAN [3].Operating in the range of 2 GHz to 11 GHz, WiMAX enables a fast deployment of the network even in remote locations with low coverage of wired technologies, such as the Digital Subscriber Loop (DSL) family, and it can be used, among others, for wireless backhaul or last-mile applications.
The IEEE 802.16 standards family provides manufacturers with basically four different physical (PHY) layers [4].Two of them are based on single carrier transmissions and use Time Division Multiple Access (TDMA) whereas the other two are based on multicarrier modulations and use either TDMA or Orthogonal Frequency Division Multiple Access (OFDMA).Within the multicarrier subgroup, the WirelessMAN Orthogonal Frequency Division Multiplexing (OFDM) uses a 256-point Fast Fourier Transform-(FFT-) based OFDM modulation together with a TDMA scheme to deploy a Point-to-Multipoint (PMP) subnetwork in the frequency range from 2 GHz up to 11 GHz in Non-Line-of-Sight (NLOS) propagation conditions.This PHY layer has been accepted for fixed WiMAX applications, and it is often termed as fixed WiMAX.Finally, WirelessMAN OFDMA exploits the multicarrier principles to implement a more flexible OFDMA access scheme.As in WirelessMAN OFDM, it is intended for NLOS PMP applications in the 2 GHz-11 GHz range.However, it uses a variable-size FFT ranging from 128 up to 2048 subcarriers.This PHY layer has been accepted for mobile WiMAX applications, and it is usually termed mobile WiMAX.
Concerning network topology, the basic configuration is PMP with a Base Station (BS) serving many Subscriber Stations (SSs).Not with standing, there is also a mesh mode available where SSs can be linked directly to the BS or routed through other SSs.This last mode is out of the scope of this paper, where we consider the design of appropriate scheduling mechanisms in uplink using the WirelessMAN OFDMA PHY layer and a PMP network.The conceived scheduling mechanism is based on a Demand Assignment Multiple Access (DAMA) strategy that implements a Dynamic Bandwidth Allocation (DBA) solution (where bandwidth is understood as rate in a wide sense).Jointly with flow allocation, we consider the adjustment of the transmission parameters of the OFDM system, and hence, the joint approach proposes a cross-layer interaction between PHY and Medium Access Control (MAC) system layers.
Previous works related to Radio Resource Management (RRM) in WiMAX networks address a variety of scenarios, from PMP to mesh, from TDMA to OFDMA access types, and distinguishing single channel or multichannel networks, most of them from a physical (PHY) layer perspective, where the goal is to properly configure the transmission parameters.At the best of our knowledge, two main approaches are found in literature, namely: (i) formulate the problem in a mathematical optimization framework and (ii) develop heuristic algorithms.In the sequel, we briefly review some of the works.In [5], the author proposes an heuristic solution for the case of a single cell OFDMA WiMAX network that maximizes the network sum-rate under some fairness considerations by means of performing subcarrier and power allocation.The authors in [6] analyze how concurrent transmissions boost performance in mesh type networks by proposing an interference-aware routing and scheduling mechanism.In [7], the reader can find a discussion about the advantages of a multichannel network.Finally, [8] contributes with a mathematical optimization solution that falls into the Network Utility Maximization (NUM) framework, where a distributed optimal solution to the established NUM problem is obtained using a convex decomposition approach.The authors extend in [9] their original work to generic OFDMA mesh networks, and the contributions in [10][11][12] are within the same context.A common feature in the last three references is that they split the global rate control and resource allocation problem into independent and smaller subproblems in order to alleviate the complexity of the solution at the expenses of a certain loss in optimality.
Our work follows the NUM framework to define the underlying optimization problem as in [8] but modifies the formulation in order to exactly fit the DAMA process that is envisaged for the WiMAX uplink.The problem is then decomposed (without any loss in optimality) using the Mean Value Cross (MVC) decomposition method [13].It allows to separate the original joint problem into a flow optimization problem (given fixed link capacities) and a radio resource optimization problem (given fixed values of transmission rates).The latter results in a linear program that can be solved centrally at the BS, whereas a distributed solution that uses the novel proposed coupled-decompositions method is applied to the former.
The rest of the paper is organized as follows.Section 2 describes the system model.Section 3 reviews the MVC decomposition technique and introduces the novel coupleddecompositions method, whereas Section 4 solves the pro-posed joint problem in Section 2. Finally, Section 5 gives some numerical results, and Section 6 ends the paper with the conclusions.

System Model
Let us consider a PMP OFDMA WiMAX network as depicted in Figure 1, where a number of SSs share a subset of the subchannels in the system.A subchannel in WiMAX is made up of some of the system subcarriers and lasts for several OFDM symbols in time.There exist different ways to gather subcarriers into subchannels, which depend on the permutation types (see in [4] a good review on WiMAX aspects).In this work, we assume that the transmitting power per subchannel as well as the set of subcarriers that form it is given.Therefore, the different powers are not variables of our allocation problem.Furthermore, each terminal allocates the amount of power at each subchannel among the inner subcarriers in order to optimize the transmitting rate.This assumption can be found in [14], where the authors take into account intercell interference to constrain the subchannel transmitting powers.Note that one interesting extension is then the inclusion of subchannel power allocation but it is beyond the scope of this paper.In our framework, given a specific allocation of subchannels to terminals {ρ i } (top left part of the figure), each terminal is able to transmit at a rate c i (ρ i ), which is the sum of the rates that the SS attains in its active subchannel subset (the subset allocated to the terminal).
We further assume (as described in the IEEE 802.16 standard documents) that each terminal negotiates the resource allocation for all traffic flows that go through it, that is, it jointly requests transmission opportunities for the ongoing connections without doing it on a flow basis.The advantage of this procedure is that signaling is reduced, specially when a significant number of connections have to be managed.The disadvantage is that, depending on the particular mechanism used to find the solution of the problem, it may not be optimal.In that sense, solutions derived from distributed optimization do not sacrifice optimality.The price to pay is the time required to get the solution, and therefore, we are interested in techniques that converge fast.In Figure 1, the rate of the jth flow at the ith SS is labeled as r i j .The IEEE 802.16 standard defines five different scheduling services that will provide Quality of Service (QoS) differentiation among the multiple traffic types.These services are [4] (i) the Unsolicited Grant Service (UGS) (ii) the real-time Polling Service (rtPS) (iii) the non-real-time Polling Service (nrtPS) (iv) the Best-Effort (BE) service, and (v) the extended real-time Polling Service (ertPS).Let us model the DAMA solution implemented in the WiMAX uplink by means of a convex program [15] where the different scheduling services are mapped using three parameters: the minimum rate that has to be allocated to the connection (the jth flow at the ith terminal) or m i j , the rate requested or d j i , and the priority of the service or p i j .The desired QoS degree of each service depends then on both m i j and p i j .For example, the UGS that needs a constant rate can be requested just by plugging Terminals Subchannels that rate into m i j and fixing d i j = m i j regardless the value of p i j .Another example is the ertPS that can be requested with some amount of m i j for the fixed allocation part and some d i j > m i j for the variable rate part.The value p i j is then used to prioritize this flow against other competing connections.
In summary, the cross-layer system model used to characterize the DBA part of WiMAX, including PHY and MAC layer issues, responds to the following convex optimization problem in maximization form [15,Section 4.1.3]: where U i j (r i j ; p i j ) is the function that measures the utility perceived by the connection when the rate r i j is allocated.The function has p i j as a parameter.Furthermore, Γ = [ρ 1 , . . ., ρ N ] collects the subchannel allocation per user (ρ i ), and the symbols and stand for component-wise nonstrict inequalities.Finally, c i (ρ i ) = ρ T i c i , where c i contains the achievable rates of SS i at each possible subchannel, and C is the rate at which the BS can transmit.Note that in principle the allocation variables within each vector ρ i should take the integer values 0 and 1 so that a given subchannel is completely allocated to a certain SS, whereas the constraint Γ1 1 forces that no more than one terminal gets the subchannel.As it has been done in other works in literature [16], we relax the integer constraints to ρ k i ≥ 0, which allows us to represent the problem as a convex one (easy to solve).Once the solution of the relaxed problem is found, a suboptimal solution to the original problem (with integer constraints) is obtained by means of employing rounding algorithms.However, in the WiMAX scenario and taking into account that an allocation is kept during several timeslots, real-valued allocation variables have sense in practice (by time sharing of subchannels).Indeed, if we consider that the allocation lasts for T time slots, then it is possible to use values in Γ with a granularity of 1/T.
Not with standing, the problem in (1) itself does not guarantee a fair allocation of resources.Fortunately, such distribution can be attained by means of employing adequate utility functions, and a general formulation for fairness was introduced in [17] under the nomenclature of (p, α)proportional fairness.A feasible rate vector r † (i.e., it attains the generic network constraints Ar † c) is said to be (p, α)proportionally fair (where p = [p 1 , . . ., p N ] T and α are positive real numbers) if, given any other feasible rate vector r ‡ , it holds that Accordingly, the utility functions that accomplish this fairness criterion are [17] ( The reader can find in Figure 2 the plots of U i (r i ; p i , α) for α = 0.1, α = 1, and α = 3 (equal p i value).Let us fix p = [1, . . ., 1] T and move from α → ∞ to α = 0.With α → ∞, the solution is said to be max-min fair [18, Section 6.5], and it is not possible (given feasibility, i.e., Ar c) to increase any rate in the network, say r j , without decreasing another rate r p < r j .On the other hand, when α → 0, the flow allocation problem leads to a max sumrate approach, and therefore, it drastically favors the users  with better links (it is then unfair).Intermediate solutions allow a certain decrease in r p at the expenses of a greater increase in r j depending on α.Note that in Figure 2 the bigger the α value is, the higher the increase in r j will be in order to compensate a utility loss in r p .A common adopted solution in literature is α = 1, and it was termed by Kelly [19] as proportional fair.Moreover, this solution coincides with the Nash Bargaining one, and therefore, it accomplishes the recognized, axioms in game theory [20] of linearity, irrelevant alternatives and symmetry [21].We can conclude that there is no unique criterion to define fairness but a series of them are explicitly characterized with the utility functions in (3).Furthermore, some flows can be prioritized over the others within a specific fairness framework (fixed by α) by particular adjustment of the scale thanks to the parameters {p i }.In general, proportional fairness (α = 1) provides a reasonable trade-off between fairness and resource utilization (network throughput).

Decomposition in Convex Programming
Decomposition techniques are used to break down a given optimization problem into a number of smaller problems, usually termed the subproblems.The most used decomposition methods in communications literature and in relation to convex optimization are primal and dual decompositions [22,23].It is usual to employ these decomposition techniques as a tool to obtain distributed solutions to some problems, as it is the case in Network Utility Maximization (NUM) problems [24,25].The formulation in ( 1) is an adaptation of the classical NUM to match the DBA problem in OFDMA WiMAX.Recently, Palomar and Chiang provided an exhaustive review on primal and dual decompositions applied to the classical NUM and extensions of it [26].In particular, they proposed multilevel decomposition approaches to split the problem into different and coupled subsets of variables (e.g., link powers and transmission rates).However, the problem in primal and dual decompositions is that, in general, they converge slowly and that an adaptation step size has to be fixed by the user.So motivated, we base our work in two distinct decomposition techniques: the Mean Value Cross (MVC) decomposition [13] and the proposed novel coupled-decompositions method.In the following, we briefly review the former and describe the latter.

Mean Value Cross Decomposition.
Consider the following problem formulation from [13]: where c : and minimize it over the variable x, including the constraints that have not been taken into account in the Lagrangian definition, to obtain the function K(y, μ) as follows: which is convex in y and concave in μ [13].
From K(y, μ), the method defines the primal and the dual subproblem by fixing either the primal variable y or the dual variable μ.After some manipulations, the primal subproblem turns into and the dual subproblem into Finally, the method is completed by passing filtered versions of the primal and dual variables between the primal and dual subproblems, as it is summarized in the following algorithm.
Take starting points μ 0 0 and y 0 ∈ Y and let k = 1.(8).Get y k as the inner minimizer of d(μ k ).

Repeat
(2) Let and compute p(y k ) as in (7).Get μ k as the inner Lagrange multiplier of p(y k ). (3 Further details on the MVC decomposition method can be found in [13].

Coupled-Decompositions Method.
Let us consider now the following problem formulation: where 1 is a column vector with all J entries equal to one, and the subset Y is the cartesian product of J convex one-dimensional subspaces that include the minimum and maximum values of the variables {y j }, and thus, it is convex.We consider that μ is the dual variable associated to the coupling constraint 1 T y ≤ c.In the sequel, we briefly describe the algorithm that we propose in order to solve (9).However, the interested reader can find in [27,28] an extended and well-reasoned version of it.
The technique intertwines the primal and dual subproblems that are obtained when classical primal and dual decompositions [22,Section 6.4] are applied to (9).In primal decomposition, the J subproblems appear when y is fixed.Note that under this assumption the problem is fully decoupled.Similarly, in dual decomposition we can relax the coupling constraint 1 T y ≤ c (constructing a partial Lagrangian of the problem with dual variable μ), and J subproblems are defined (the problem fully decouples again) for a fixed value of μ.In both classical strategies, the successive updates of y and μ are driven by the primal and dual master problems.In the coupled-decompositions method, the result of the primal subproblems is transformed using a redefined dual master problem, the dual projection, and plugged to the dual subproblems.Similarly, the output of the dual subproblems is transformed using the primal projection and fed to the primal subproblems.A flow diagram of the  method is depicted in Figure 3.The algorithm starts with μ 0 = 0 and iterates as follows: dual subproblems → primal projection → primal subproblems → dual projection → dual subproblems.Since primal and dual subproblems are extensively analyzed in literature (its formulation appears in Figure 3), let us now detail the novel parts.Notwithstanding, a complete iteration is revisited during the proof of the method.On one hand, primal projection is pretty similar to the primal master problem in primal decomposition.Assuming that y 0 is constructed with the output of the J dual subproblems, the primal projection solves the following optimization problem: with the only particularity that the constraint 1 T y ≤ c must be attained with equality when the last update of the Lagrange multiplier is μ > 0. This is in accordance with the Karush-Kuhn-Tucker (KKT) conditions for convex problems [15, Section 5.5] (see more details in [27]).On the other hand, the dual projection takes the output values from the primal subproblems λ t 0 and selects the values within λ t 0 that have been obtained with primal variables y j not in the boundary of Y j .Let us collect this subset in λ t 0 .The motivation is that the nonselected values do not directly impact on the value of μ (it can be seen from the KKT conditions of the problem; see more details in [27]).Thereafter, the μ update is found as which updates μ with the value within λ t 0 that is closer to the previous μ value.
Proof of the method: See the appendix.

Proposed Solution
Our solution uses a combination of both decomposition techniques.First, an MVC decomposition is applied, making it possible to split the joint problem into one flow or bandwidth allocation subproblem and one subchannel allocation subproblem.The latter depends on variables that are available at the BS, and thus, it is not necessary to explore distributed computations in order to solve it.On the contrary, the former is distributed among the BS and the SSs in order to be standard-compliant (the BS allocates aggregate bandwidth to the SSs and these decide the final allocation to flows and services).In this case, we use a two-level coupleddecompositions strategy.
First, let us consider the problem in (1) and identify rates with x and subchannel allocation variables with y in the MVC decomposition formulation in (4).Rewriting the original joint problem as max where R = {r i j | m i j ≤ r i j ≤ d i j } and S = {ρ i | Γ1 1, ρ i 0}, we can define the primal subproblem of the MVC decomposition method as max for fixed values of {ρ i } and the dual subproblem as max for fixed values of the Lagrange multipliers {γ i } that are associated to the constraints that couple rates with subchannel allocation variables in (12).Note that the two subsets of variables are fully decoupled in (14), and thus, the maximization in {ρ i } can be done independently solving the following linear program: The joint problem is then solved as follows.

Until convergence.
Since ( 15) is solved at the BS, the remaining issue is to find the solution of (13).In order to avoid excessive DBArealted signaling in the subnetwork and to restrict ourselves to the standard, we propose to solve it using a two-level coupled-decompositions strategy.Note that we can rewrite (13) as max where Note also that the dual Lagrange variable γ i corresponds to the constraint y i ≤ ρ T i c i in (16).Therefore, we apply the coupled-decompositions method to solve (16) at the upper layer (BS), and we use it again at the lower layer (at each SS) to solve (17) when it is required by the upper layer.
The iterations of the resulting two-level flow allocation algorithm and the involved signaling are summarized in the following list as well as in Figure 4.
(1) The dual variable μ t (associated to N i=1 y i ≤ C) is spread through the network, reaching each connection.
(2) Each connection computes the allocation given μ t by means of solving the inner dual subproblems (the constraints in m i j and d i j can be obviated if desired without affecting convergence).The SSs and the BS get their own allocations by aggregation of the allocations below them. (1) (1) (1) (3) The BS corrects the previous allocations (primal projection) to attain N i=1 y i ≤ C and (4) The corrected allocations are used by the SSs to perform inner iterations (within each SS) of the coupled-decompositions method in order to obtain new candidates γ i .
(5) Finally, the BS updates the value of the dual variable to μ t+1 using the dual projection and the previous γ i values.
Intuitively, the multilayer coupled-decompositions strategy tries to find a consensus on the price μ that has to be paid for sharing the transport capacity C of the BS.Often, primal variables are interpreted from a resource-oriented perspective whereas dual variables take the role of prices to be paid to use the resources [15,Section 5.4.4].All CIDs participate in principle in finding such optimal value.However, the price of the connections within a particular SS may be distinct from the global price μ if, for example, its link capacity is small (hence forcing the price to locally increase).In these occasions, local prices γ i that differ from the optimal and global consensus price μ are found.
Other works in literature [10][11][12] study a similar problem within generic mesh OFDM networks.In general, they search for suboptimal but affordable solutions, which are based on decoupling the joint problem into independent optimization programs that manage only a subset of the variables without looking at the others.In this work, we suggest (for the particular PMP WiMAX case) the derivation of the joint optimal rate and subchannel allocation (under fairness considerations), and we propose a distributed scheme that achieves it.Moreover, the numerical results in the next section show the practical interest of the mechanism in terms of the number of iterations (i.e., directly related to the amount of signaling).As a matter of fact, the proposed method (possibly with extensions) can be used in other scenarios to speed up the computation of optimization problems or subproblems, either in optimal or suboptimal decoupling approaches.

Numerical Results
Let us consider the network setup depicted in Figure 5 with 4 SSs and 9 connections (CIDs) in total.We choose logarithmic utility functions (α = 1), Other policies balancing the solution towards the max-sumrate or the max-min-fair designs can be implemented by fixing other α values using the same algorithm (as discussed later).We fix all requests to 100 kbps (requests are emitted in WiMAX in terms of bytes of information but we transform them to rates taking into account the time basis) and all the minimum granted rates to 1 kbps.All connections have the same priority p i j = 1.The available number of subchannels is 7, all of them to be shared among the 4 SSs.We consider the following transport capacities (in kbps) per subchannel (10 kHz of bandwidth) and user (given one realization of flat-fading Rayleigh subchannels that have 10 dB of SNR in mean): Figure 5: Setup of the network under test.
Note that depending on the scheduling length (i.e., the number of contiguous time slots in time that are allocated in a single allocation phase, which fixes the granularity of the ρ i values) and on the channel characteristics (coherence time), it is reasonable to consider which values of c i may be really achieved within each allocation phase (mid-term values seem reasonable) so that one may resort to robust designs in order to compute them.The output rate capacity of the BS is 200 kbps, and the initial subchannel allocation is 49, 13.19, 13.37, 30.92].
Figure 6 shows the evolution of the subchannel allocation variables when we apply the proposed method, achieving new link capacities [c 1 , c 2 , c 3 , c 4 ] = [89.39, 86.83, 60.44, 49.23].In order to accelerate the convergence to the solution, we have used instantaneous values of {γ i } instead of the time-average that is proposed in the MVC decomposition method, averaging only the primal (allocation) variables.This solution has been derived by other authors [8] using a different approach (which validates it), and it is specially relevant in the first iterations where the {γ i } values show abrupt changes and very high values.Note that in the figure the final allocation is completely different from the initial one (only SS1 keeps using subchannel 1) but the solution still needs to be rounded to accommodate a practical scheduling implementation, which has its implications also in terms of convergence to the optimal solution because it may have sense to truncate the algorithm after some iterations and round that solution.
In Figure 7, we plot the resulting flow allocation per connection (that correspond to the CIDs ordered from left to right in Figure 5) and the final link capacity once the subchannel allocation has been obtained for the four scenarios specified in Table 1.The objective is to show how fairness considerations impact in the final allocation.The first Scenario is the same as in Figure 6, whereas Scenario 2 evaluates a different allocation scheme (with fairness parameter α = 0.1).In the next two scenarios, we study the effect of different priorities using again a proportional fairness approach (α = 1).The difference between Scenarios 3 and 4 is that Scenario 3 fixes the same requested rate for all the connections (100 kbps), whereas Scenario 4 has two possible requests (10 kbps and 100 kbps).We notice in the results of Scenario 1 that link capacities have been adjusted (with the subchannel allocation mechanism) in order to provide a similar allocation to all connections.In Scenario 2, the allocation scheme favors the best channels so that each subchannel is assigned to the SS that experiences the maximum achievable rate at that subchannel.Therefore, SS1 gets subchannels 1, 2, and 6; SS2 gets subchannels 3 and 4; SS3 gets subchannel 7; SS4 gets subchannel 5.The corresponding link capacities are [c 1 , c 2 , c 3 , c 4 ] = [105.02,88.54, 47. 15, 48.95].The final rate allocation is limited by the outcoming rate at the BS (200 kbps) so that SSs 3 and 4 limit their ongoing connections to a lower rate than the connections in SSs 1 and 2, which share the remaining transport capacity.When prioritized traffic flows appear, as in Scenario 3, granted rates are balanced toward services depending on their priority values.Accordingly, it can be seen that subchannel allocation provides more link capacity to SSs 3 and 4. In Scenario 4, we further modify the requested rates with respect to Scenario 3 and the highest priority services in Scenario 3, (the ongoing connections of SS4) reach their requests.As expected, remaining resources (remember that the BS can manage no more than 200 kbps) are redistributed in order to allocate more rate to services in SS3 (with priorities equal to 3) than to services within SS1 and SS2 (with priorities equal to 1), while subchannel allocation favors the link BS-SS3 as well.
Finally, our last result analyzes the efficiency of the novel coupled-decompositions method (used to solve the flow allocation subproblem) in terms of convergence speed.For that purpose, we extend Scenario 1 to 20 SSs with 5 ongoing connections on each.The mean received SNR is 15 dB, and each ongoing connection in SSs 1-15 requests 100 kbps, whereas each connection in SSs 16-20 requests 10 kbps.The transport capacity at the BS is now increased to 1200 kbps.We plot in Figure 8 the evolution of the dual variable μ, that is, negotiated between the BS and the SSs when we use both our novel proposed method and a classic dual decomposition approach using the same 2-layer architecture.Remember that classical decomposition methods need to adjust the value of the step size of the gradient-based update.
In this particular case, we have found that a setup with α(t) = 0.5/t at the highest level (i.e., between the BS and the SSs) and α(t) = 0.005/ √ t at the lowest (i.e., between SSs and connections) provides a satisfactory tradeoff between convergence and speed.However, the need of a good adjustment is in practice an obstacle of the method, and it is not easy to find a step providing that good tradeoff.On the contrary, one of the important advantages in the coupled-decompositions method is that any user-defined step is completely avoided.The other important advantage is in the number of iterations required.As shown in the figure, the novel technique converges in 5-6 iterations, contrary to the dual decomposition strategy (both obtain the same optimal solution), which needs more than 250 iterations.This drawback of dual decomposition appears in other works in literature, for example, in the numerical results of [10], where it is used to obtain a distributed solution that optimizes power and rate allocation within a mesh OFDM network.

Conclusions
In this work, we have proposed an algorithm that implements the DAMA mechanism foreseen in the IEEE 802.16 WiMAX standard.Initially, we have introduced our system model, which considers both flow and subchannel allocations in a cross-layer approach.Some PHY and MAC-layer aspects of WiMAX that are relevant to our work have been briefly reviewed as well as how to translate a series of fairness definitions into a convex optimization framework.All this has led us to formulate a network utility maximization problem.
Since the standard fixes that resources should be requested and granted in a terminal basis but we should consider several traffic flows within each SS (may be with different QoS requirements), we have proposed a distributed solution to the original convex optimization problem in order to fulfill these requirements while keeping the optimality in the allocation.Furthermore, we have explored the usage of our novel proposed coupled-decompositions algorithm and a recently proposed MVC decomposition method applied to distinct parts of the problem with the goal of achieving a more practical design than with classical primal and dual decompositions.
Results show that it is possible to find a solution to the flow allocation subproblem with very few iterations and without the manual setup of any parameter, as opposite to a classical dual decomposition.The last statement applies also to the subchannel allocation subproblem, which is able to give a good approximation to the solution within a reasonable number of iterations.Finally, we have shown with an example that our strategy is able to attain a fair distribution of resources and to support QoS by means of traffic prioritization.

A. Proof of Convergence of the Coupled-Decompositions Method
First of all, we assume that strong duality [15, Section 5.2.3] holds, which is usually verified in convex programs, so that the optimal primal variables attain the optimal dual variables when plugged into the subproblems and vice versa.In the following, the superscript t indicates iteration number although we omit it in some irrelevant occasions.Equivalently, the objective value of the problem is the same regardless it is solved directly (primal version) or by maximizing the dual function (dual version) [15, Section 5.2].We will prove that where the relation λ t 0 = 1μ is found by the application of the KKT conditions (see more details in [27]) and μ * is the optimum value of the dual Lagrange variable.In the following, we review a complete iteration of the method.
Let us consider that μ t < μ * (the proof is similar if μ t > μ * ) and recall the result in [28, Lemma 1], where it is shown that the primal variable y j at the jth subproblem (primal or dual) is a decreasing function of λ t 0j .This fact together with λ t 0 = 1μ t forces y j λ t 0 ≥ y * j , ∀ j, ( A .where equality is attained only when y * j ∈ bd Y j (boundary of the subset) and therefore 1 T y > c.
In the primal projection, it is verified that thanks to the lemma below.

Lemma 1.
Given the optimization problem in (10), its optimal solution can be expressed as y * = y 0 − k with k 0.
Proof.See Section B. Comparison between a classical dual decomposition strategy and the proposed coupled-decompositions method.Applying the relationship between the primal and dual variables of the subproblems to the previous y t value, it is fulfilled that Furthermore, given that y t is not the optimal value, it is verified that some of the λ t 0j values are λ t 0j ≤ λ * j whereas the remaining ones are λ t 0j ≤ λ * j , since it holds that 1 T y = c.In other words, some of the y j values attain y j ≥ y * j whereas the rest verify y j ≤ y * j .An example depicting the situation before dual projection can be found in Figure 9.
Consider now that λ t 0 contains a single element.Note that a null vector is not possible since we assume that the coupling constraint is active.Then we can prove the following lemma.Lemma 2. Let a primal point y attain 1 T y = c and y ∈ Y. Let also λ 0 be a vector containing the dual translation (computed by primal subproblems) of the values in y that verify y ∈ int Y (interior of the subset).Then, if the vector λ 0 is in fact a scalar, it is verified that where λ * is the optimum value of λ for the selected position in λ 0 (i.e., equal to μ * ).
Proof.Using Lemma 1, we can state that all the values within y except the kth element accomplish y i ∈ inf Y i (i / = k).Therefore, it holds that y k < y * k = y * 0k = y * k .Applying the relationship between subproblems (remember that both in primal and dual subproblems, primal variables are a decreasing function of dual variables and y(λ * 0 ) = y * ), we reach the desired result.
Finally, we update μ t+1 using (11).Collecting all the results obtained up to this point, we have that since every value in λ t 0 verifies λ t 0i > μ t .Furthermore, it is also true that since the value λ t 0i closer to μ t (dual projection) accomplishes λ t 0i < λ * i = μ * , which is derived from Lemma 2 and the discussion preceding it.Figure 9 provides a graphical explanation.We can finally conclude that μ t < μ t+1 < μ * . (A.8) The proof ends showing by contradiction that μ t cannot tend to a value smaller than μ * .Assume that there exists a value μ where successive iterations converge.Then μ is a stationary point of the method.In other words, a complete iteration of the method starting from μ returns exactly the same value.This enforces in the primal projection that y = y 0 (μ ), otherwise the values in λ 0 would increase and so the update in μ (dual projection).Given the relationship between primal and dual subproblems, we see that the previous equation is only attained if μ = μ * since a lower value μ < μ * would obtain a primal point y 0 (μ ) from dual subproblems such that 1 T y 0 (μ ) > c.
Before concluding this section, we want to note that it is possible to substitute the primal projection by the projection into 1 T y = c and the method still converges (it can be similarly proved).It is a more practical option since the projection can be analytically computed as [15, Section 8.1]

B. Proof of Lemma 1
First, note that a point y = y 0 − k with k 0 is feasible since it attains both 1 T y ≤ c and y ∈ Y (assuming that the intersection is not empty).Then, we have to proof that a point that does not accomplish the equation y = y 0 − k for positive values in k cannot be optimal for problem (10).
We proof this last result by induction.Assume a certain vector k, called k that attains 1 T (y 0 − k ) = c and k 0. Construct now a new vector k † from k by fixing its lth element k † l to −a with a > 0 and distributing the difference |k l − k † l | among the rest of elements in k † so as to attain the equality coupling constraint.In other words, Let us introduce some results from majorization theory [29] that we need to complete the proof.First, let the components of x ∈ R n be ordered in decreasing order and express it as and thus, any solution where one element within k is negative is not optimal (since the problem is convex and has a single solution).The proof ends by induction of this result to an arbitrary number of negative elements in k.

Notation
U i (r i ; p i , α): Utility achieved when entity i transmits at rate r i .The utility is parameterized by a priority p i (entity-dependant) and a shape factor α (common to all utilities) N: Number Maximum outgoing rate at the BS ρ i : Subchannel allocation vector at the ith SS c i : Achievable rates at the ith SS (includes all subchannels) c i (ρ i ): Maximum outgoing rate at the ith SS Γ: Subchannel allocation matrix: Γ = [ρ 1 , . . ., ρ N ] R: Feasible rates subset: R = {r i j |m i j ≤ r i j ≤ d i j }

Figure 2 :
Figure 2: Different degrees of fairness (α) in the definition of utility functions.

Figure 3 :
Figure 3: Flow diagram of the coupled-decompositions method.

Figure 6 :
Figure 6: Evolution of some subchannel allocation variables ρ m i .

Evolution of μ using 2 Figure 8 :
Figure 8: Evolution of μ value in the flow allocation subproblem.Comparison between a classical dual decomposition strategy and the proposed coupled-decompositions method.

Figure 9 :
Figure 9: Example of the situation before dual projection.
of SSs N i : Number of active connections at the ith SS r i j : Rate of the jth ongoing connection at the ith SS m i j : Minimum guaranteed rate to the jth ongoing connection at the ith SS EURASIP Journal on Wireless Communications and Networking