- Open Access
Sum rate maximization via joint scheduling and link adaptation for interference-coupled wireless systems
© Castañeda et al.; licensee Springer. 2013
- Received: 18 May 2013
- Accepted: 1 November 2013
- Published: 16 November 2013
The work presented in this paper addresses the sum rate maximization problem for the downlink of a wireless network where multiple transmitter-receiver links share the same medium and thus potentially interfere with each other. The solution of this problem requires the optimization of two aspects: the first one is the set of links that can be jointly scheduled, and the second is the set modulation and coding schemes (MCSs) that maximize the sum rate. A feasible link achieves a certain MCS if its signal-to-interference-plus-noise ratio (SINR) is above a threshold or target SINR associated with the MCS and the SINR of each link is coupled with the other links’ powers that are required to achieve their respective MCSs. Since the available MCSs form a finite set, the rate maximization problem has a combinatorial nature. We present iterative algorithms that find a suboptimal solution to the combinatorial problem by operating in two phases. Phase one verifies the feasibility of the MCS assignment by performing either eigenvalue analysis or power consumption analysis, and phase two uses the feasibility information delivered by phase one to modify either the set of links (user removal) or the MCS assignment if feasibility conditions are not fulfilled. Our approach extends the concept of user removal to the case of adaptive modulation, and this generalization allows us to schedule users more efficiently, improving outage probability figures. Numerical results show that the proposed algorithms achieved a good tradeoff between sum rate performance and complexity. Moreover, our algorithms are a low complex alternative to the state-of-the-art user-removal algorithms with minimum gap in outage performance.
- Power control
- Link adaptation
- Resource allocation
- User removal
- Sum rate maximization
The efficient resource allocation in wireless networks is fundamental to fulfill several practical quality of service (QoS) measures like data rate and outage probability. The number of wireless users and data services has increased dramatically over the last few years, and optimization of the resource allocation has become primal to guarantee both user and operator satisfaction without increasing system requirements, mainly bandwidth and power budget. Moreover, there are scenarios where the set of transmitter-receiver pairs (links) operate simultaneously in a shared medium, and interference mitigation techniques must be employed. The QoS is measured in practice by the signal-to-interference-plus-noise ratio (SINR) and recent works on resource allocation optimization for interference-coupled networks [1–8] show the relation between the SINR maximization and the efficient power and rate allocation. Since the achievable data rate depends on the SINR which is a global function of all transmit powers, the schemes of power control are fundamental to maximize either a global network utility [1, 6–8] or individual rates [2, 3, 9] in networks where interference is the main limitation and cannot be completely eliminated.
In interference-coupled networks, the successful scheduling of a set of interfering links is conditioned to the fulfillment of all individual QoS requirements and power constraints. An efficient scheduling policy must determine the proper set of links for which there exists a power and rate allocation that meets power constraints and SINR requirements. However, finding the set of links that maximizes a given network metric is a combinatorial problem which has been claimed to be NP-complete . A suboptimal solution to this problem is found in the so called user-removal techniques [10–12] where the users that violate the power constraints or QoS requirements are found iteratively and temporarily dropped.
The works related to resource allocation optimization and the ones of user removal have different objectives, and both fields remain isolated. The former assumes that for a given set of links, there exists an infinite number of solutions to the resource allocation problem, and the main objective is to find the allocation that maximizes a specific utility function such as sum rate and power consumption. The latter is concerned about the set of links that can be scheduled whose QoS requirements are fixed and the resource allocation can be achieved by conventional allocation schemes. There is a number of open issues that must be solved in interference-coupled networks, and the objective of this work is to provide a solution to the scheduling problem and simultaneously maximize the total sum rate by performing efficient resource allocation. Moreover, unlike the available literature, we consider that the values of the SINR are constrained to take values from a finite set of thresholds or targets associated with a given set of modulation and coding schemes (MCSs). By considering the different MCSs, we extend the philosophy of user removal for the case of non-fixed SINR targets, and at the same time, we provide a methodology to find the MCS allocation that maximizes the total sum rate.
1.1 Related works
Over the last 20 years, several theoretical [3, 5, 11, 13–16] and practical [2, 6, 12, 17] works have been developed to understand and solve the problems of power allocation and utility maximization for cellular, multihop, peer-to-peer, and digital subscriber line (DSL) networks. Early works on power control [13, 17–20] designed iterative algorithms under the standard interference function framework [3, 13] in order to guarantee the convergence of the algorithms to a unique and optimal power allocation solution. These works assume that the set of given links is always feasible, i.e., for such a set, there always exists a solution to the power allocation problem. More recent works [3, 6, 14] studied the relationship between the rate allocation and power control, specially for the high SINR regime. This is a common assumption because for interference-coupled systems, the mathematical modeling of the resource allocation problem in more tractable  and efficient iterative algorithms can be developed. For instance, in , the power control problem for rate maximization is formulated as a convex problem and its solution is found via geometric programming (GP) for wireless networks.
Related works for interference-coupled wired (DSL) networks [7, 8, 21] show that suboptimal yet efficient power allocation for rate maximization can be achieved when the non-convex rate maximization problem is approximated by alternative convex objective functions of the transmit powers. In , the authors show that relaxed forms of the objective function of the rate maximization problem lead to the convergence of the proposed algorithms, and the accuracy of such approximation depends on the convexity properties of the objective functions. In , the weighted sum rate maximization problem was extended to the multiuser multicarrier scenario in interference-coupled systems. The idea behind the algorithms presented in [7, 21] is to solve iteratively the original resource allocation problem by optimally solving in each iteration a relaxed version of the original problem. In each iteration, the local optimal solution bounds the solution of the original problem for a given resource allocation, and due to the properties of the relaxed objective function, the convergence to a local optimum is guaranteed. This is known as successive convex approximation (SCA) whose goal is to refine the solution found for the relaxed problem in order to close the gap between the approximated and the optimum resource allocation. A framework to solve general optimization problems under SCA and a comprehensive analysis of state-of-the-art dynamic spectrum management algorithms is presented in . The work in  solved the power spectrum management problem, implementing a SCA algorithm that exploits the characteristics of the feasible region of resource allocation solutions, and in each iteration, GP is used to solved the local power assignment problem. The works of Tan et al.  and Stanczak et al.  presented algorithms that solve optimally the joint power and rate allocation problem by exploiting the convex characteristics of the feasible rate region and generalized the resource allocation problem for both high and low SINR regimes.
The mathematical abstraction of an interference-coupled network has a strong connection to the theory of irreducible matrices [22, 23], and the Perron-Frobenius theorem [5, 22] is a fundamental part of several works [1–3, 5, 11, 14] that characterize the feasible rate region and solve optimally the resource allocation problem. The theoretical results derived from this theorem [1, 5, 11] allow to render the network optimization problem into an eigenvalue problem and to analyze and verify the feasibility of both, the set of scheduled links and its corresponding resource allocation. This mathematical tool has been used to solve the rate and power allocation problem for a fixed set of links (e.g., [1, 5, 6]), assuming that the rates and powers can take any real positive value, and their main objective is to find the optimum allocation that maximizes a given utility function under a set of power constraints and transmission schemes. The user-removal techniques  exploit the Perron-Frobenius theory to identify the infeasibility of a set of links in scenarios where, for a given set of power constraints and rate (SINR) requirements, the classical power control algorithms (e.g., ) do not converge.
The joint scheduling and resource allocation that maximize the sum rate is a complex combinatorial problem that grows exponentially with the number of links and depends strongly on the number of available MCSs. The optimal solution of such a combinatorial problem can be found via exhaustive search by selecting the set of links and MCSs that yield the maximum sum rate. Such an algorithm has prohibitive complexity; therefore, we propose suboptimal algorithms that solve the combinatorial problem efficiently and achieve a tradeoff between sum rate performance and complexity. Our algorithms merge the objectives of the resource allocation optimization and the user-removal techniques. This integration is achieved by operating over two dimensions of decision, the set of links and the set of available MCSs. Conceptually, the proposed algorithms work in two phases. The first phase establishes that for a given set of links, there exists a power and rate allocation that satisfies the SINR requirements imposed by some MCSs under a set of power constraints. In other words, this phase verifies if the set of links and their MCSs are feasible. The second phase modifies either the set of links or the MCSs, based on the feasibility measure provided by the first phase.
We show how this two iterative phases can be designed using the Perron-Frobenius theory by formulating the sum rate maximization problem as an eigenvalue optimization problem. Although, this approach achieves acceptable sum rate and outage figures, in each iteration it requires the computation of the eigenvalues that characterized the interference coupled network. For this reason, we design alternative solutions that only require either the evaluation of the power consumption or the estimation of the achievable SINR per iteration. Furthermore, we introduce a low complexity algorithm that performs a fast estimation of the set of links and MCSs that solve the sum rate maximization problem. Numerical results show that despite the fact that the proposed algorithms are suboptimal strategies, they are asymptotically optimal when the number of users in the network grows to infinity. Moreover, we show that our algorithms generalize the concept of user removal for the case of multiple SINR targets and that the proposed schemes are efficient low-complex alternatives to the state-of-the-art user-removal algorithms.
The remainder of the paper is organized as follows. Section 2 presents the system model. Section 3 describes the link selection and resource allocation problem. Section 4 addresses the algorithms that find a solution for the joint resource allocation and scheduling problem. Section 5 shows numerical examples for the assessment of the presented algorithms using different performance metrics. The main conclusions are drawn in Section 6.
Some notational conventions are as follows: matrices and column vectors are set in boldface. (·) T , |·|, ∥·∥ p denote transpose, set cardinality, and the p norm, respectively. An M×N matrix A is non-negative if am,n≥0, for all m and n, and write A≥0. The term ρ(A) denotes the Perron-Frobenius root (PF-root) which equals the largest modulus eigenvalue of matrix A[22, 23]. I is the identity matrix of compatible size; diag (x) denotes a diagonal matrix whose main diagonal is x. A[i] is the i th principal submatrix of A whose i th row and column are removed. The same notation is applied for a vector whose i th element is removed. Let y be a vector, then y i = [ y] i is the i th element. For two vectors x and y, x≥y is a componentwise inequality. is the set of strictly positive real numbers.
where G ki is the power attenuation from the transmitter on link i to the receiver on link k, taking into account propagation loss and fast and slow fading, and G kk is the power loss for the intended transmission at link k. The term represents the additive white Gaussian noise (AWGN) power for the k th receiver.
The SINR target of the k th link is constrained to take values from a finite set of targets where , γ(m-1)<γ(m), and M is the number of available MCSs. A larger value of γ k implies that link k attempts to maintain a more spectral efficient modulation scheme. Depending on the service, can be used to assign different priorities among users or to limit their achievable data rates. For instance, the rate demand in link i can be limited while link k can improve its performance, having a larger number of available modulations for some k≠i. For the sake of simplicity, we assume the same for all links, . The k th link is associated with a modulation index m k that defines the position of its SINR target in the set so that . The discrete set of targets is given by the available set of MCSs supported in the systems, which in practice is defined by the user equipment capabilities and the wireless network technology. The vector of SINR targets is defined as γ=(γ1,…,γ K ) T , and all SINR targets will be summarized in a diagonal matrix Γ=diag(γ).
The matrix B absorbs the total power constraints as an additional source of interference whose power is inversely proportional to the power P t . The properties of B provide us insight into the transmission reliability and how interference, powers, and SINR targets are coupled. For the case of IPC, there exists a set of K matrices B, and each one absorbs a different individual power constraint [5, 11].
Feasibility conditions. They are the sets of SINR requirements and power constraints that the resource allocation must fulfill.
Resource feasibility. We say that a power vector p is feasible if for such a vector, the feasibility conditions are met. A set of targets γ or Γ is called feasible if all targets in the set can be jointly achieved by power control. A set of links is called feasible if there exists at least one set of targets whose related powers meet the feasibility conditions.
where R(SINR(p)) is the achievable rate associated with a given SINR and can be given either by (5) or (6), depending on the specific network requirements. As the elements of γ can only take values from a finite set , (7) is a combinatorial problem whose complexity depends on the size of and the number of elements in .
Finding a solution to problem (7) requires the optimization over the set of feasible links that can transmit simultaneously (user selection) and their respective feasible modulations (and their associated powers). An exhaustive search algorithm attempting to solve problem (7) would require to test all the combinations of links and target vectors. The associated search space has a size of , where several configurations of links sets and target vectors are infeasible.
Since looking for the optimal solution in is extremely complex, we introduce iterative algorithms that find a suboptimal solution to problem (7) using two different approaches. On the one hand, we use the Perron-Frobenius theory to reformulate the sum rate maximization problem as an eigenvalue optimization problem. On the other hand, we design alternative algorithms that verify the resource feasibility using information provided by the power consumption or the achievable SINR so that the eigenvalue computations are avoided. The fundamental concept behind the presented algorithms is to find in each iteration, the link k∗ or its associated and that conditions the most the resource allocation feasibility. The algorithms make decisions in two phases. In the first phase, the fulfillment of the feasibility conditions is verified. If the feasibility conditions are violated, the second phase is in charge of finding k∗ and modifying the set of active links or the target vector γ based on the information that k∗ provides.
where is a power vector that distributes equally the total available power among all links, i.e., .
4.1 Perron-Frobenius root-based optimization
Figure 2 illustrates the feasible region which is the intersection of sublevel sets of the spectral radii of all non-negative irreducible matrices B taking into account TPC. The region can be completely characterized from its boundary since it is downward comprehensive , which means that any γ on the boundary of defines a set of target vectors that are within the feasible SINR target region. However, we are interested on the point in that maximizes the sum rate, which implies that the target vector that solves problem (7) must lie in the boundary of .
where the constraint in (11) absorbs both constraints in (7). A similar reformulation of (7) applies when IPC (5) is considered .
In order to solve problem (11), it is required to identify which link violates the most the feasible conditions. From the theory of irreducible matrices, it is known that if A is an irreducible square matrix and A[k] is a proper principal submatrix of A, then ρ(A[k])<ρ(A) [22, 23]. In the context of user removal , this property is fundamental to determine the link that must be dropped since it relates the most infeasible link to the minimum PF-root over all principal submatrices of B. In our context, the link k∗ that compromises the most the resource feasibility is the one whose is minimum. The fastest fulfillment of condition (9) is achieved whether k∗ is temporarily disconnected or its associated is relaxed so that the PF-root of the matrix B is minimized.
For the next iteration, the link k∗ reduces its target index by one unit , and its new SINR target is set to . In the case where link k∗ cannot reduce its minimum target, it is classified as infeasible or useless. Assigning any positive power to this link will create interference to the other links without achieving the minimum required SINR. At this point, the set of feasible links must be reduced so that , and without the useless link, the algorithms attempt to allocate the maximum target for the remaining users, . The steps performed to solve problem (11) are given in Algorithm 1.
Algorithm 1 PF-root optimization
4.2 Power consumption and target-to-SINR ratio-based optimization
In order to solve problem (7) using the tools provided by the Perron-Frobenius theory, it is required to optimize the PF-root of the matrix B. In each iteration of Algorithm 1, finding the worst link k∗ requires eigenvalue computations. If IPCs (5) are imposed, finding k∗ would require PF-root evaluations per iteration . Moreover, the criterion used to evaluate the feasibility of the resource allocation, i.e. condition (9), is also based on the PF-root computation.
We design an alternative algorithm where the infeasibility of γ is determined through the characteristics of p. In , it was shown that if the targets are feasible, the link that consumes more power is the one that maximizes the PF-root in condition (9). The link with maximum power requirements compromises the most the resource feasibility. In order to identify the link k∗ that has the maximum demand of power, we define an algorithm that exploits the characteristics of the power vector computed by the distributed power control (DPC) algorithm . DPC has been used as part of algorithms that find the optimum power allocation in scenarios where is a feasible target vector (e.g., [1, 15]). It has been proved that DPC is a fixed point algorithm  that converges to the optimum unique p if and only if γ is feasible.
where x is the right eigenvector associated with ρ(Γ V) and C is a constant depending on the initial vector p(t=1) and the coupling matrix V. The power vector goes to infinity as the number of iterations (t) increases due to the fact that ρ(Γ V)>1. As our objective is to allocate the maximum SINR target for all links (γ k =γ(M),∀k), this initial conditions may not be feasible, leading to (13). This implies that we cannot use directly the power vector found by DPC to select the link that consumes more power. Nevertheless, if the powers are normalized in each iteration so that ∥p(t)∥1=1, then it can be found within a finite number of iterations τ, a link k∗ whose associated power is maximum, i.e., . If feasibility conditions are violated, we modified either the set of active links or the target so that the power consumption of k∗ is reduced in the next iteration. The initial conditions t=1 of the vector of normalized powers of the DPC algorithm are given by the constrained power vector in (8), i.e., . The DPC algorithm with normalized powers is described in Subalgorithm 1:
Subalgorithm 1 DPC algorithm with normalized powers
where for IPC, k∗ indicates which link consumes more power regarding its individual power constraint. Finding the infeasible link k∗ using the DPC implies a two-time scale algorithm, and there are two processes of optimization over the vector of powers: the first one is finding a p whose components are in , and the second is adjusting p according the power constraints.
where is given by (8). The steps performed to solve the problem (7) using both the minimum power consumption approach (14) and (15) or the target-to-SINR ratio approach (16) are described in Algorithm 2.
Algorithm 2 Power consumption and target-to-SINR ratio-based optimization
4.3 SINR target increment-based optimization
The objective of Algorithms 1 and 2 is to reach a point γ in the region of targets (10), ideally on its boundary. Both approaches start with initial conditions that may or may not be out of the feasible region of targets. The number of iterations required for the algorithms to converge depends on the channel instance. When the channel conditions are not favorable, many links will achieve low SINR, which implies that the number of iterations required for the algorithms to converge would be large. In order to find a faster way to compute the solution of problem (7), we introduce an algorithm that operates in two stages: in the first stage, it finds a target vector γ inside the feasible target region; in the second stage, it attempts to reach the closest point to the boundary of the feasible target region from inside out by tightening up the SINR requirements of specific links.
where the target is related to the highest spectral efficiency, i.e., the rate of the selected target (MCS) is the closest but below the achievable capacity of the SINR [4, 25]. The values of the achievable SINR targets from (17) are adjusted by (18), and the links whose adjusted are not in the set of available targets are dropped. Notice that the computation of uses the maximum available power vector , and this lack of power control may discard feasible links that could be scheduled.
The steps performed to solve problem (7) using this approach are described in Algorithm 3. Notice that more than one link can be discarded in steps 3 and 4, and still, is not guaranteed to be feasible. In such a case and in order to reduce outage, steps 21 and 22 are used to drop only one extra link whose associated adjusted target is minimum.
Algorithm 3 Target increment-based optimization
5.1 Examples of the MCS selection
5.2 Distributed antenna scenario
In order to assess the algorithms, we consider a distributed antenna system (DAS) scenario as the one depicted in Figure 1b. In DAS, the remote antenna units (RAUs) are geographically separated and connected by a dedicated link to a central unit where processing is jointly performed. These type of access interfaces are based on the concept of space diversity and cell splitting in order to improve coverage and spectral efficiency. The deployment of the distributed antennas consists of N RAUs: one at the center of the cell and N-1 distributed RAUs uniformly deployed at a distance of the cell radius from the cell center. We consider that the RAUs are coordinated only to control their transmit powers, and no signal processing is used (e.g., beamforming). The channels are modeled as Rayleigh fading and are affected by a path loss component and a shadowing fading component modeled as a log-normal distributed variable with parameter s σ .
5.2.1 Performance evaluation: sum rate and outage probability
Let us analyze two particular cases of U. For the first case, consider U=N=7, where in Figure 5, the achieved sum rate of the three algorithms is similar when TPC (6) is imposed. The sum rate maximization via PF-root optimization in Algorithm 1 identifies with more accuracy which link must relax its target, which results in an extra gain of 0.73 bps/Hz compared to the other approaches. Algorithm 3 achieves the same performance of Algorithm 2 since it allocates higher MCSs to its links. However, its performance in terms of outage probability is worst than the other approaches since it may discard feasible links in its first stage. In Figure 6, IPC (5) is considered, and the performance of Algorithms 3 and 2 via (16) are outperformed by Algorithm 2 via (14) and (15), which indicates that when individual power constraints are imposed, target relaxation based on power consumption achieves an extra gain of 0.95 bps/Hz compared to the optimization based on the target-to-SINR ratio.
The second case of analysis, U=70≫N, provides a rich MUD, and the performance of all algorithms reaches similar values of the sum rate for both sets of power constraints. This means that under favorable channel conditions, finding an acceptable solution to problem (7) can be achieved with the low complex Algorithms 2 and 3, avoiding the PF-root optimization.
5.2.2 Optimal joint link selection and resource allocation
5.2.3 Performance evaluation for the low-high SINR regimes
5.2.4 Application for user removal
In this paper, we present different algorithms to solve the sum rate maximization problem in interference-coupled wireless networks. This problem has a combinatorial nature since the SINRs are constrained to take values from a finite set. The algorithms find a set of links for which exists a feasible resource allocation that maximizes the sum rate. The presented algorithms are based on the criteria derived from the Perron-Frobenius theory or derived from the implicit information contained in the power consumption or achievable SINR. In addition, we present a low-complexity fast algorithm for link selection and resource allocation which converges in few iterations, and its robustness allows us to achieve acceptable performance under favorable channel conditions.
Numerical results show how our algorithms achieve a good tradeoff between complexity and accuracy compared to the optimal solution. Moreover, our results show that the low-complexity algorithms that avoid the PF-root computations are suitable for scenarios with favorable channel conditions (e.g., rich MUD). In such scenarios, their sum rate performance is similar to the one achieved by approaches depending on the eigenvalue computation.
Furthermore, we show that our proposed algorithms can be used for user removal under different sets of power constraints whose performance is close to the one achieved by state-of-the-art algorithms but with significant reduction on complexity. A secondary application of our algorithms is to group users for time-sharing scheduling where transmission is provided only to useful users that can be jointly supported whilst useless users can be served in a later time slot or handed to another channel or base station.
The authors wish to thank the anonymous reviewers for the useful comments and suggestions which improved the quality of the paper. The work presented in this paper was supported by the Portuguese FCT CADWIN (PTDC/EEA TEL/099241/2008) and CROWN (PTDC/EEA-TEL/115828/2009) projects and the FCT grant (SFRH/BD/70130/2010) for Eduardo D Castañeda.
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