Generalizing and optimizing fractional frequency reuse in broadband cellular radio access networks
 Lei Chen^{1}Email author and
 Di Yuan^{1}
https://doi.org/10.1186/168714992012230
© Chen and Yuan; licensee Springer. 2012
Received: 18 November 2011
Accepted: 30 June 2012
Published: 24 July 2012
Abstract
For broadband cellular access based on orthogonal frequency division multiple access (OFDMA), fractional frequency reuse (FFR) is one of the key concepts for mitigating intercell interference and optimizing celledge performance. In standard FFR, the number of OFDMA subbands and the reuse factor are fixed. Whereas this works well for an idealized cell pattern, it is neither directly applicable nor adequate for reallife networks with irregular cell layouts. In this article, we consider a generalized FFR (GFFR) scheme to allow for flexibility in the total number of subbands as well as the number of subbands in each celledge zone, to enable networkadaptive FFR. In addition, the GFFR scheme takes power assignment in consideration. We formalize the complexity of the optimization problem, and develop an optimization algorithm based on local search to maximize the celledge throughput. Numerical results using networks with realistic radio propagation conditions demonstrate the applicability of the GFFR scheme in performance engineering of OFDMA networks.
Introduction
Orthogonal frequency division multiple access (OFDMA) is a current technology for broadband radio access. In OFDMA, the radio spectrum is split into a large number of channels, referred to as subcarriers. Data transmission is performed simultaneously over multiple subcarriers, each carrying a lowrate bit stream. OFDMA is very flexible in exploring multiuser diversity with high spectrum efficiency and scalability. The technique is part of the downlink air interface in the fourth generation cellular systems based on 3GPP long term evolution (LTE) standards (see [1]), and IEEE 802.16 WiMAX (see [2]).
Orthogonal frequency division multiple access subcarriers are orthogonal to each other. As a result, intracell interference is not present. Intercell interference, on the other hand, becomes a performancelimiting factor. For this reason, interference mitigation has become an important topic in performance engineering of OFDMA networks.
Fractional frequency reuse
One subcarrier allocation scheme in multicell OFDMA networks is frequency reuse with factor one (reuse1), in which the entire spectrum is made available in all cells. Here and throughout the article, a reuse factor of X means that the spectrum block in question is used in every group of X cells. Users that benefit from the reuse1 scheme are those located close to a base station antenna, i.e., users in the cell center. Because the channel condition is good and the interference from other cells is relatively low, the throughput in cellcenter zones grows by bandwidth. For users located at celledge areas, however, the performance typically suffers severely in the reuse1 scheme, because of the high interference from the surrounding cells in relation to the signal of the home cell. In other words, celledge zones are much more sensitive to interference than bandwidth. Previous studies (e.g., [3]) indicate that, if the overall system throughput is the only performance target, then reuse1 is the best choice. At the same time, the resulting performance in the celledge zones tends to be unacceptably low.
For the cell layout in Figure 1, the standard subband allocation pattern with reuse3 is very intuitive. The allocation pattern ensures that the subband of a celledge zone is not reused in any of the neighboring cells. In reallife cellular networks, however, the amount of interference is very irregular over the service area. The cells differ greatly in the number of significant interferers as well as the respective amounts of interference, causing difficulties in applying standard FFR. As the number of surrounding cells varies from one cell to another, allocating the subbands optimally is not straightforward. For the same reason, a single reuse factor, if applicable at all, is no longer optimal. In addition, because the sensitivity to interference varies by cell, the allocation of one subband per edge zone may not be adequate. Finally, scalability becomes an issue, because it is not optimal to replicate the allocation pattern of one part of the network to another.
A generalized FFR scheme
In this article we present and evaluate a generalized FFR (GFFR) scheme, in order to overcome the limitations of standard FFR in dealing with reallife networks with irregular cell layout. As a result, the allocation pattern is adapted to the characteristics of each individual network.
The GFFR scheme that we consider extends the standard one in three aspects. First, the frequency band used for celledge zones can be partitioned into any number of subbands. Doing so is potentially useful for interference avoidance in cells with many surrounding interfering neighbors. Second, the number of subbands allocated to the edge zone is cellspecific. Having this flexibility is important when many subbands are created, since the celledge zones in real networks differ in their levels of sensitivity to interference respective spectrum bandwidth. Some edge zones may benefit from using few subbands which are not reused at all in the surrounding areas, whereas other edge zones can tolerate higher interference and consequently should be allocated more subbands with a higher reuse level. As the third extension of FFR, power assignment is part of the optimization framework of GFFR. In standard FFR, power is typically not a variable. The GFFR scheme allows the power to vary by cell, in order to achieve further gain in addition to that enabled by subband reuse optimization.
We focus on celledge user performance in this article. The performance metric that we use to assess subband allocation and power assignment targets throughput guarantee in the celledge zones. In the system model (see Section “The system model”), the service area is represented by a large number of pixels. For example, in one of the test network scenarios for performance evaluation (Section “Experimental results”), a pixel is a square area of size 20×20 m. For each celledge pixel, the performance metric is defined by the data throughput that can be guaranteed when all the interfering cells are active. In effect, this metric corresponds to the average downlink throughput with uniformly distributed celledge users and roundrobin scheduling, and provides a throughput map over the celledge areas. Note that, by associating pixels with nonuniform utility parameters, the metric is easily adapted to any given user distribution with uneven traffic demand.
Assuming that power is evenly distributed over the subbands in every cell[7–10], we consider the power assignment scheme where each cell can select, among a given set of power levels, the power to be used per subband. The total power used for the celledge zone of a cell is thus the product between the chosen power level and the number of allocated subbands. Thus GFFR deals with optimization in the dimension of power, in addition to spectrum.
We formulate the task of subband allocation and power assignment as an optimization problem, and prove its complexity. A local search algorithm is developed for problem solution. The search in the algorithm is based on a neighborhood structure that finds the exact optimum of subband allocation and power assignment of one cell, provided that the solutions of the other cells are tentatively kept fixed. This singlecell optimization process is not straightforward, we show however that it can be implemented to run in polynomial time.
We use networks with realistic radio propagation conditions and very irregular cell layout for performance evaluation. The optimized allocation significantly goes beyond the performance of FFR with three subbands, which, in its turn, delivers better throughput than reuse1 in celledge zones. The improvement that we obtain enables better tradeoff between cellcenter and celledge performance. In addition, the information required by the algorithm’s key operation can be restricted to be local to each cell. This observation forms a promising basis for developing distributed allocation algorithms.
Paper outline
The remainder of the article is organized as follows. In Section “Related works” we review related works and summarize the contributions of the current article. The system model and notation are given in Section “The system model”. In Section “The optimization problem and its complexity”, we formalize the optimization problem and prove its complexity. The local search algorithm is detailed in Section “Solution algorithm”. We present and analyze experimental results in Section “Experimental results”. Finally, Section “Conclusions” is devoted to conclusions and an outline of further research.
Related works
There is a growing amount of research on FFR and its performance evaluation. Simulations for assessing the performance of the standard FFR scheme have been conducted in [11–14]. The main conclusion is that FFR is able to improve the performance at celledge zones. The impact of celledge size on throughput has been analyzed in [15]. In [16], an analytic model for estimating the throughput of FFR is presented. The idea of using analytic models instead of simulation to assess the signaltointerferenceandnoise ratio (SINR) for FFR is adopted also in [17]. In [18], it is shown that, if FFR is performed in a distributed fashion, and each cell selfishly optimizes the assignment of resource to its own users, the system will converge to a Nash equilibrium. Chang et al. [19] take a graph theoretical approach and formulate FFR subband allocation as a graph coloring problem. In [20, 21], FFR subband allocation has been performed based on an interference graph connecting the base stations. The studies consider an ideal cell layout pattern in performance evaluation. Comparative studies of FFR and other interferencemitigating schemes, in particular the soft frequency reuse (SFR) scheme, are provided in [22, 23]. Simulation tools and architecture support for FFR are presented in [24, 25]. The application of FFR and its performance evaluation in femtocell environments are examined in [26–29].
For a given set of users, the problem of allocating resource at the subcarrier level in FFR is formulated using mathematical optimization models in [30, 31]. The constraint in resource allocation is the minimum required user throughput. The solution approach is based on a relaxation of the original problem to obtain convex optimization formulations. A similar problem setting for OFDMA resource allocation is considered in [32], and solved using Lagrangian relaxation.
Extensions and variations of the basic FFR scheme have been examined in a number of recent articles. In [33], the authors propose an incremental frequency reuse (IFR) scheme to adapt resource allocation to cell load. Similar schemes that adapt subband allocation to traffic and user distributions are proposed in [34, 35]. Luo et al. [36] propose algorithms to improve the throughput of besteffort traffic in OFDMA systems through automated formation of the FFR pattern. The work in [37] validates a clusteringbased FFR scheme. In [38], FFR is combined with interference suppression for improving spectral efficiency and power expenditure. Rahman and Yanikomeroglu [39] proposes to perform FFRbased interference avoidance at two levels by base station and a central controller, respectively. The latter conducts intercell coordination. An FFR scheme based on generating reports of the interference levels among neighboring cells is presented in [40]. The work in [41] augments FFR by enabling adaptive spectral sharing, and proposes a graphtheoretical solution approach. In [8], Ali and Leung presents a twolevel FFR scheme. At the RNC level, groups of subcarriers are allocated to cells. Resource allocation to individual users is then performed opportunistically by base stations. In [42], Tara and Hakima proposes an FFRbased frequency planning scheme utilizing zone switching diversity for multicell mobile WiMAX networks.
The current article presents significant extensions to our preliminary studies of optimal FFR in largescale networks with irregular cell layout [23, 43]. The works reported in the two references have several limitations. First, no theoretical result in problem complexity is provided. Second, resource allocation of every edge zone is restricted to one single subband. Third, uniform power is assumed in all cells. Fourth, the impact of the SINR threshold used to define cell edge is not considered. In the current article, we provide a formal proof of problem complexity and present a new optimization algorithm that overcomes the performancelimiting assumptions in [23, 43]. In addition, we study how the celledge SINR threshold influences the performance.
A problem domain related to optimal FFR is frequency assignment in second generation cellular networks. In frequency assignment, the number of frequencies to be allocated to each cell is given. Typically, the objective is to minimize the total number of frequencies or to minimize the interference for a fixed set of available frequencies. For frequency assignment, we refer to the extensive surveys in [44, 45] and the references therein, and [46, 47] for generalizations of frequency assignment to frequencyhopping networks. In Section “The optimization problem and its complexity” we outline the structural differences between frequency assignment and GFFR.
Another related topic consists in the practical implementation of FFR in OFDMAbased systems, such as LTE networks. Static and semistatic FFR implementations have been discussed in [4, 6, 25], and dynamic FFR implementations have been investigated in [8, 26, 41]. By the flexibility of GFFR, it admits to be implemented both statically or dynamically. For example, GFFR can be implemented at the top level (e.g., the RNC level in the architecture proposed in [8]) in a hierarchical resource management framework, with opportunistic resource allocation in the lower level.
Depending on the way of implementation (centralized, distributed, cluster based, etc.), information gathering and exchange are necessary at one or more levels in a network. The information required by GFFR does not differ from that in the previously proposed FFR implementations. For example, in a centralized scheduling scheme, such as the one at the top level in [8], the scheduler needs user channel state information to coordinate the resource allocation among cells. The same type of information would be gathered for GFFR. When implemented in a distributed manner, GFFR is able to, as was mentioned earlier, base its decisions on local information, provided that the neighboring cells coordinate the sequence of decision making. In such a case, the resource allocation status needs to be exchanged between eNodeBs. In LTE, this can be done through the standard X2 interface. To summarize, GFFR is a more generalized and enhanced version of FFR, and can be implemented as an integral part of the overall resource allocation scheme.
The system model
In this section, we first present the basic elements of the system model, along with introducing notation. Next, the power assignment scheme is discussed. We then present celledge throughput calculation for any given subband allocation and power assignment.
Preliminaries
We use $\mathcal{C}=\{1,\dots ,C\}$ to denote the set of cells of an OFDMA network. The service area is represented by a regular grid of a large number of pixels $\mathcal{J}=\{1,\dots ,J\}$. Each pixel $j\in \mathcal{J}$ is a small square area within which radio propagation is considered uniform. We use g_{ ij } to denote the total gain between the cell antenna of i and pixel j. The gain value is typically obtained by measurements and/or prediction models of signal propagation. The service area of a cell is divided into a center zone and an edge zone. The latter represents locations that are prone to interference from the surrounding cells. Denote by ${\mathcal{J}}_{i}^{e}$ the set of pixels forming the edge area of cell i. Note that, for reallife networks, the shape of cell edge is in general irregular, just like the cell layout. Moreover, it may happen that a cell does not have an edge zone. This occurs if the signal of the cell antenna is strong over the entire cell area. Merely for simplifying the notation, we assume that all cells have edge zones in our system modeling.
The total downlink transmit power of any cell antenna is denoted by P^{ Tot }. We assume that P^{ Tot } does not vary by cell, again for the sake of simplifying the notation. The spectrum is partitioned into two parts to be used by the center and edge zones, respectively. We denote the total bandwidth and the bandwidths allocated for center and edge zones by B, B_{ c }, and B_{ e }, respectively, where B=B_{ c } + B_{ e }.
The cellcenter band with bandwidth B_{ c }is allocated with reuse1 in all cellcenter zones, which we do not discuss further in the system model. The celledge band is split into equalsized subbands. Let K denote the number of edge subbands, and $\mathcal{K}=\{1,\dots ,K\}$. The bandwidth of each subband is ${B}_{\mathrm{sub}}=\frac{{B}_{e}}{K}$. Moreover, we use σ_{ j }to denote the thermal noise effect in pixel j.
In (1), ${\mathcal{C}}_{k}$ contains the subset of cells being allocated subband k. Clearly, each cell will appear in at least one of the set elements in c. For subband allocation pattern c, we use N_{ i }(c)to denote the number of subbands allocated to the edge zone of cell i; this equals the number of set elements in ccontaining cell i.
Power assignment
The number of subbands of an edge zone can range between one and K in GFFR. As a result, the maximum value that p_{ i }can take depends on the number of subbands used in cell i. Without loss of generality, we assume that P^{ L } equals the total power available to any cell for its edge zone (i.e., this maximum can be reached on a subband, provided this is the only subband allocated in a cell), and set its value to be equal to the total cell power scaled by the edge bandwidth, i.e., ${P}^{L}=\frac{{P}^{\mathrm{Tot}}{B}_{e}}{B}$. Power assignment has to respect the constraint that the product between p_{ i }and the number of subbands allocated in cell i does not exceed P^{ L }. Thus for subband allocation pattern c, the candidate subband power level of cell i becomes _{ i }(c)={1,2,…,L^{ ′ }}, where L^{ ′ } is the largest integer such that ${P}^{{L}^{\prime}}{N}_{i}\left(\mathbf{c}\right)\le {P}^{L}$.
Celledge throughput
The optimization problem and its complexity
The objective function is monotonously nondecreasing, and concave when the Shannon formula (4) is used for throughput calculation. System (6)–(8) defines a combinatorial optimization problem. In its general form, the problem is NPhard. We formalize this result below.
Theorem 1
The subband allocation and power assignment problem defined by (6)–(8) is NPhard.
Proof
The proof of the NPhardness of problems, or, NPcompleteness of its decision version, involves a polynomialtime reduction from one of the wellknown NPcomplete problems. For our problem, we give a polynomialtime reduction from the vertex coloring problem. Consider any instance of vertex coloring defined on graph G=(V,E). Let n=V. The decision problem is to determine, for a positive integer K>1, whether or not there is a feasible coloring with at most K colors. We assume that n≥3, K≥3, and K<n, as otherwise determining the feasibility is trivial.
The gain values ${g}_{i{j}_{{i}^{\prime}}}$ and ${g}_{{i}^{\prime}{j}_{{i}^{\prime}}}$, and noise effect σ_{ j } are set such that the throughput over one subband at pixel ${j}_{{i}^{\prime}}$ is 1.0, if the subband is not allocated to cell i and the power level in cell i^{ ′ } is P^{1}. In this case, the signaltonoise ratio in ${j}_{{i}^{\prime}}$ is 3.0. If the subband is reused in i, the throughput of ${j}_{{i}^{\prime}}$ is $\frac{1}{2{n}^{2}K}$, if the power values are P^{ L }and P^{1} in cells i^{ ′ } and i, respectively. For cell i, the gain values of the potentially interfering cells (corresponding to some of the nodes in graph G) are set such that the throughput of j_{ i } over one subband is $\frac{1}{\mathrm{nK}}$, if none of the potentially interfering cells reuses the subband and cell i uses power level P^{1}. If reuse takes place, the throughput of j_{ i } is at most $\frac{1}{2\mathrm{nK}}$, even if cell i uses power P^{ L } and there is only one interfering cell with power P^{1}. It can be realized that gain and noise values satisfying the above conditions can be constructed easily.
After setting the gain and noise values, we will show in this paragraph that, at optimum, any cell i∈{1,…,n} will use one single subband, whereas any cell i^{ ′ }∈{1^{ ′ },…,n^{ ′ }}will use all the K subbands. Suppose it is optimal that cell i uses multiple subbands (with power P^{1} on each). We consider two scenarios. Assume first cell i^{ ′ } uses more than one subband (and thus also with power P^{1}on each). Keeping the power fixed at P^{1}and supposing that cell i gives up any of the allocated subbands, the only cell experiencing lower throughput is i, and the loss is at most $\frac{1}{\mathrm{nK}}$. The throughput increase in cell i^{ ′ } on the subband is $1\frac{1}{2{n}^{2}K}$ or 1, depending on whether or not the subband is currently in use in i^{ ′ }. Since $1\frac{1}{2{n}^{2}K}=\frac{\mathrm{nK}\frac{1}{2n}}{\mathrm{nK}}>\frac{1}{\mathrm{nK}}$, the subband removal operation leads to better overall performance, contradicting the assumption of optimality. Assume now that cell i^{ ′ } uses a single subband with power P^{ L } (using P^{1} is not optimal, because cell i^{ ′ }does not generate interference to other cells, see Figure 3). Consider removing the subbands of i except one, and let i^{ ′ } use all the K−1 interferencefree subbands with power P^{1}. Before the modification, the throughput in i and i^{ ′ }are at most $(K1)\frac{1}{\mathrm{nK}}$ and $\frac{1}{2}\underset{2}{log}(1+3K)$ (because for i^{ ′ } the SNR with power P^{1}is 3, and P^{ L }=K P^{1}), respectively. Disregarding the throughput in cell i or potential performance improvement in any other cell, the new throughput of i^{ ′ } alone due to the modification is at least K−1. Comparing K−1 to $(K1)\frac{1}{\mathrm{nK}}+\frac{1}{2}\underset{2}{log}(1+3K)$, the former is greater for any K≥3, contradicting the optimality assumption. Therefore cell i uses exactly one subband at optimum.
In this and the next paragraph, we will conclude the proof through the connection between the constructed vertex coloring instance and our problem. Observe that the optimum choice of any cell i^{ ′ }∈{1,…,n^{ ′ }} is one of the two options: To use power P^{1} on all the K subbands, among which K−1 are interferencefree, or power P^{ L }on one single subband. The former gives a throughput of at least K−1, whereas the latter gives no more than $\frac{1}{2}\underset{2}{log}(1+3K)$, which is smaller than K−1 for any K≥3. In conclusion, using all the K subbands is optimal in cell i^{ ′ }. Consequently, the throughput of i^{ ′ }at optimum is at least K−1, and strictly below $K1+\frac{1}{2{n}^{2}K}$ (because the additional term can be reached only with power P^{ L }). The total throughput of {1^{ ′ },…,n^{ ′ }}is in the interval $\left[n(K1),n(K1)+\frac{1}{2\mathrm{nK}}\right)$.
For cells 1,…,n, the total throughput reaches $\frac{1}{K}$, if there is a feasible coloring in G with no more than K colors. In this case, the overall throughput in the entire network is at least $n(K1)+\frac{1}{K}$. If conflict in coloring has to take place between some nodes in G, the overall throughput of our problem instance is strictly less than $n(K1)+\frac{1}{2\mathrm{nK}}+\frac{n1}{\mathrm{nK}}+\frac{1}{2\mathrm{nK}}=n(K1)+\frac{1}{K}$. Thus the feasibility of the vertex coloring instance is equivalent to whether or not the overall throughput of our problem instance reaches $n(K1)+\frac{1}{K}$. In addition, the reduction is clearly polynomial. Hence the decision problem of (6)–(8) is NPcomplete, and the optimization problem itself is NPhard. □
To some extent, optimizing subband allocation in GFFR is similar to the frequency assignment problem (FAP) in second generation cellular networks. There are several versions of FAP, see [45]. The version that resembles most subband allocation in GFFR is minimuminterference (MI) FAP. Although both problems amount to allocating frequencies (or frequency bands) to cells, there are several structural aspects in which they differ significantly. First, the number of frequencies to be allocated to each cell is part of the input of FAP, whereas for GFFR it is a decision variable. Second, MIFAP addresses interference between pairs of cells. In our case, in contrast, celledge throughput is in focus, and thus the pixels forming the service area are modeled explicitly. The third difference lies in the nonlinearity in the objective function (6). Due to these differences, solution algorithms developed for FAP do not apply to subband allocation in GFFR.
Solution algorithm
In view of the problem complexity of (6)–(8), and the objective of applying GFFR to largescale networks, we consider a local search algorithm aimed at finding highquality solutions timeefficiently. A local search algorithm iteratively seeks solution improvement by repeatedly introducing modifications to the current solution and evaluating the outcome. The search strategy is defined by the operation used in making trial modifications. Solutions generated by the modification operation are considered neighbors to the current one, and the definition of the modification operation is also known as the neighborhood structure. The algorithm stops when no improvement can be obtained by solution modification, that is, the current solution is locally optimal in respect of its neighborhood. In this section, we present the design of the local search algorithm, in particular the neighborhood structure and how to timeefficiently evaluate the solutions in the neighborhood.
Initial solution
To obtain a starting solution, we apply a type of greedy algorithm that allocates one single subband to each cell. The algorithm goes through all cells 1,…,Cone by one. For each cell, the total throughput over the celledge pixels is computed for each of the subbands in {1,…,K} with the same power as reuse1, and the subband leading to the highest throughput value is chosen. Once allocated a subband, the allocation for the cell in the initial solution is fixed when considering the remaining cells. In effect, the algorithm tends to select the subband for which the cell under consideration and the cells having the subband allocated have least interference to each other in the edge areas.
The amount of calculations in generating the initial solution is polynomial in C and K. Note that, with an ideal hexagonal cell layout, the starting solution will coincide with standard FFR if K=3.
Search strategy
For (6)–(8), we will show that computing the exact optimum of subband allocation and power assignment of one cell can be performed in polynomial time, provided that the solutions of the remaining cells are tentatively kept fixed. Based on this result, we define the neighborhood structure as the set of alternative solutions obtained by optimizing the subband allocation and power assignment of one cell at a time. Among the neighboring solutions, the one having largest overall improvement is selected, and replaces the current solution. The search is repeated until the current allocation is locally optimal in every cell, i.e., no improvement can be reached by changing the allocation of any single cell.
Let the current solution be $\mathbf{c}=({\mathcal{C}}_{1},\dots ,{\mathcal{C}}_{k},\dots {\mathcal{C}}_{K})$ and p=(p_{1},…,p_{ i },…,p_{ C }). The task is to find the new, optimal subband allocation and power assignment for each cell $i\in \mathcal{C}$, assuming that in the computation for cell i, the solutions of the other cells remain those specified by c and p. Suppose, for a moment, that power assignment is independent from subband allocation. Then changing the allocation of one subband does not have any impact on the throughput over the other subbands. If a subband is added to cell i with a positive power level, the throughput of cell i grows, and that of the cells currently reusing the subband will go down. The net effect can be calculated without considering the allocation of the remaining subbands in cell i. The same holds when a currently allocated subband changes power or is deleted completely from cell i.
Consider now the connection between subband allocation and power assignment. Changing the number of subbands in a cell may enlarge or shrink the set of candidate levels in power assignment. Hence the computation does not decompose in the same way as above. As there are $\sum _{m=1}^{K}\left(\genfrac{}{}{0.0pt}{}{K}{m}\right){L}_{m}$ potential allocation patterns, where L_{ m }denotes the number of candidate power levels for m subbands, the amount of computation is seemingly exponential in K. We will show, however, that determining the global optimum can be implemented to run in polynomial time.
The idea is to, for each cell i, go through the candidate numbers of subbands, and, for each m between 1 and K, power levels 1,…,L_{ m }. The overall number of combinations is polynomial in K and L. For each of these combinations, the computational task boils down to finding, for the given power level, the m best celledge subbands for cell i. Toward this end, all subbands presently used in cell i are removed first, so the cell’s subband allocation becomes empty. The throughput in cell i becomes zero, and the throughput of the remaining cells that use these subbands are updated. Next, for each subband k∈{1,…,K}, the throughput of cell i on subband k and the throughput loss in the other cells using this subband are computed. The values are put together as the performance value of subband k. Note that this computation is not dependent on the allocation nor the throughput values of the other subbands. Once the computation is complete for all subbands in {1,…,K}, the m subbands giving the highest performance values form the optimum, provided that the number of subbands allocated to cell i is restricted to be m. Repeating the procedure for m=1,…,K results in the global optimum of cell i, assuming that the allocations in the other cells are unchanged.
The algorithm’s key steps can be summarized as follows.

Step 1: use current solution as the starting solution and, for each cell, find the best number of subbands, together with the power allocation, provided that the solutions for the remaining cells are tentatively kept fixed.

Step 2: calculate the throughput change for the current cell.

Step 3: repeat Steps 1–2 for all cells.

Step 4: find the largest throughput improvement among all cells.

Step 5: if the throughput improvement in Step 4 is positive, use the corresponding solution to replace the starting solution. Go back to Step 1.
A formal description of the algorithm is provided in Algorithm 1. The initial solution consists in subband allocation c and power assignment p. These are updated by local search and returned by the algorithm as the output. For each cell considered in the main loop, the locally optimal solution (p^{∗},c^{∗}) is initialized using pand c, in lines 4–5. Thus the computations for the cells all start with the same solution. Then, subband removal in a cell is done in lines 6–11, resulting in a new temporary initial solution $\stackrel{\xc2\xaf}{\mathbf{c}}$. The corresponding throughput of each subband in all cells using the subband is recalculated and updated in line 10. The forloop spanning lines 12–36 considers the number of subbands to be allocated in the cell under consideration, and forms the bulk of the computation. For each number m, another loop takes place over the corresponding candidate power levels (lines 13–35). For the power level under consideration, lines 15–19 go through all subbands and calculate the throughput change over each subband if the subband is allocated to cell i. This results in a throughput change vector $\overrightarrow{\delta}$. Line 20 sorts the vector in descending order, resulting in a sorted vector $\overrightarrow{{\delta}^{\prime}}$ and the corresponding index vector I. Line 21 calculates the sum of the m highest values in $\overrightarrow{{\delta}^{\prime}}$. If the sum improves over the currently best value ξ^{∗}(initialized to zero), the optimal solution of the cell is updated (lines 22–34), and the cell is allocated the first m subbands in the sorted sequence with the power level selected (lines 25–33). Once all cells have been considered after a major iteration, c and p are replaced by the best singlecelloptimized subband allocation c^{∗}and power assignment p^{∗}, respectively. The algorithm stops when an iteration does not produce any improvement (i.e., ξ^{∗}=0).
Algorithm 1 Local search
1: repeat
2: ξ^{∗}⇐0
3: for all$i\in \mathcal{C}$do
⊳initialize
4: c^{∗}⇐c
5: p^{∗}⇐p
6: for all$k\in \mathcal{K}$do
⊳remove the current subband allocation and update
7: if$i\in {\mathcal{C}}_{k}$then
8: ${\stackrel{\u0304}{\mathcal{C}}}_{k}\Leftarrow {\mathcal{C}}_{k}\setminus \left\{i\right\}$
9: end if
10: ${\stackrel{\u0304}{f}}_{k}\Leftarrow \sum _{h\in \stackrel{\u0304}{{\mathcal{C}}_{k}}}\sum _{j\in {\mathcal{J}}_{h}^{e}}f\left(\mathrm{SIN}{R}_{\mathrm{hj}}\right(\stackrel{\u0304}{\mathbf{c}},\mathbf{p}),j)$
11: end for
12: for m=1,..,K do
⊳loop over all possible numbers of subbands
13: for$\ell \in \{1,2,\mathrm{..},L\}:{p}_{i}^{\ell}m\le {P}^{L}$do
⊳loop over candidate power levels
14: ${p}_{i}^{\prime}\Leftarrow {p}^{\ell}$
15: for all$k\in \mathcal{K}$do ⊳loop over subband with the current power level
16: ${\mathcal{C}}_{k}^{\prime}\Leftarrow \stackrel{\u0304}{{\mathcal{C}}_{k}}\cup \left\{i\right\}$
17: ${f}_{k}^{\prime}\Leftarrow \sum _{h\in {\mathcal{C}}_{k}^{\prime}}\sum _{j\in {\mathcal{J}}_{h}^{e}}f\left(\mathrm{SIN}{R}_{\mathrm{hj}}\right({\mathbf{c}}^{\prime},{\mathbf{p}}^{\prime}),j)$
18: ${\delta}_{k}\Leftarrow {f}_{k}^{\prime}\stackrel{\u0304}{{f}_{k}}$
19: end for
20: $[\mathbf{I},{\overrightarrow{\delta}}^{\prime}]\Leftarrow \mathrm{sort}\text{\_}\mathrm{decent}({\delta}_{k},(k=1,\mathrm{..},K\left)\right)$
21: ${\xi}^{\prime}\Leftarrow \sum _{k=1}^{m}{\delta}_{k}^{\prime}$
⊳accumulate the highest m throughput changes
22: if ξ^{ ′ }>ξ^{∗}then ⊳update the currently best solution
23: ξ^{∗}⇐ξ^{ ′ }
24: ${p}_{i}^{\ast}\Leftarrow {p}_{i}^{\prime}$
25: for h=1,..,K do
26: if h<m then
27: l=I_{ h }
28: ${\mathcal{C}}_{l}^{\ast}\Leftarrow {\mathcal{C}}_{l}^{\prime}$
29: else
30: l=I_{ h }
31: ${\mathcal{C}}_{l}^{\ast}\Leftarrow \stackrel{\u0304}{{\mathcal{C}}_{l}}$
32: end if
33: end for
34: end if
35: end for
36: end for
37: end for
38: c⇐c^{∗}
⊳update the best solution for one major iteration
39: p⇐p^{∗}
40: until ξ^{∗}=0return(c,p)
Experimental results
Test networks
The downlink bandwidth equals 4.5 MHz (i.e., the basic block of LTE). In the experiments, we use B_{ c }=1.8 MHz and B_{ e }=2.7 MHz. Thus the cellcenter areas lose 60% of the bandwidth in comparison to the reuse1 scheme. The number of subbands K varies from 2 to 15.
Network statistics and parameter setting
Berlin  Lisbon  

Area size (m^{2})  7500×7500  4200×5000 
Number of cells  148  164 
Area size (pixels)  22,500  52,500 
Pixel size (m^{2})  50*50  20*20 
Total DL Tx power (dBm)  46  46 
Thermal noise (dBm)  −107  −107 
Celledge pilot SINR threshold (dB)  ≤−6.5/−5.8/−5.3/−1.1  ≤−5.0/−4.4/−4.0/−0.8 
Number of cells with cell edge  64/83/109/124  128/147/158/161 
Reuse1 celledgethroughput (Mbps)  1.10/1.23/1.33/1.47  1.50/1.66/1.78/1.93 
In Table 1, the cell total transmit power P^{ Tot }is 46 dBm, which corresponds to approximately 40 W. Since the celledge band takes 60% of the entire bandwidth, the total power over the edge subbands is P^{ L }=24 W. We require that all cells allocate at least 0.1 W to cell edge (if any). The rest of the candidate power levels are created with a step size of 0.1 W.
The optimization algorithm is implemented in C++ and runs on a Dell E6410 notebook with an Intel Core i7 CPU (2.8 GHz) and 8 GB RAM. For each of the scenarios, we conduct 200 replications by changing the sequence of the cell during the generation of the initial solution. The results we present represent the average of the replications.
Celledge performance
As can be seen from the figures, the optimization algorithm brings substantial throughput gain to cell edge in comparison to reuse1. Taking the Berlin network and 5% cell edge as an example, the edge throughput grows from 1.2 to 4.5 Mbps for small K, and further to 4.9 Mbps for some of the larger K values. Considering the Lisbon network and the same percentage of edge area, the throughput is 1.7 Mbps with reuse1, and lies between 3.4 and 4.1 Mbps for GFFR. Summarizing the results, the ranges of the improvement factor are [2.9, 4.7] and [2.0, 2.9] for the two networks, respectively, for K=15. Here, the improvement factor refers to the ratio between the optimized GFFR throughput and that of reuse1 in the celledge zones.
From the figures, it is apparent that the improvement decreases by the threshold value used to define the cell edge. Thus considering larger area as cell edge brings down the average throughput. This is because a high threshold means that larger areas are considered cell edge, although parts of these areas are bandwidthsensitive rather than interferencesensitive. The throughput loss of these bandwidthsensitive users contributes significantly to the decrease of the average celledge throughput.
From the figure, the improvement of GFFR over both standard FFR and [23, 43] is apparent. Taking the Lisbon networks as an example, standard FFR with K=3gives an average throughput of 2.44 Mbps. Applying the optimization algorithm in [23, 43], which follows a search strategy being different from that of the current article, the average throughput result is 2.71 Mbps, giving an improvement of approximately 11% over standard FFR. GFFR has its strengths in providing the additional flexibility of power optimization as well as allowing multiple subbands in a celledge zone. For K=3, GFFR yields an average throughput of 3.53 Mbps, outperforming standard FFR by 45% and the results of our previous works by 30%. For K>3, additional improvement is achieved. With K=15, the corresponding GFFR throughput value is 4.10 Mbps, for which the relative improvement over standard FFR is 68%. Similar observations can be made for the Berlin network scenario. In conclusion, the merit of GFFR in comparison to standard FFR and [23, 43] is significant.
Performance tradeoff
Revisiting the results, it is clear that the improvement due to GFFR is strongly related to the proportion of the area that is considered to be cell edge. Low SINR threshold leads to small celledge areas with very high interference sensitivity, and therefore more improvement is achieved by GFFR in comparison to reuse1 for the celledge areas. Another factor having a strong influence on performance is the relation between B_{ c }and B_{ e }. In practice, setting the bandwidth values is networkspecific, and, to a large extent, the optimal choice depends on the user distribution as well as the performance target. For example, if the celledge zones have many users with high qualityofservice requirement, more bandwidth has to be allocated to B_{ e }. For the results reported in Section “Celledge performance”, B_{ e }corresponds to 60% of the total bandwidth. The value is quite typical in previous studies of FFR. It results in a throughput loss of 60% for the cellcenter zones. In general, the performance gain of FFR in the celledge zones means a degradation in the cellcenter throughput, as well as lower overall throughput. Similar observations were made in earlier works (e.g., [7]).
Conclusions
To overcome the limitations of standard FFR and to address the performance of FFR in largescale networks with irregular cell structure, we have presented a GFFR scheme that offers higher flexibility in resource allocation. By optimizing subband allocation and power assignment, the scheme adapts the utilization of the spectrum and power resource to the level of interference sensitivity of each celledge zone. For highly interferencesensitive celledge zones, interference is minimized by subband isolation or power reduction, whereas for the other celledge zones more bandwidth is allocated if this leads to better performance.
We have studied the complexity of the GFFR optimization problem. A local search algorithm has been developed for problem solution for largescale networks. Computational experiments show that optimized GFFR delivers substantially higher throughput than reuse1 at cell edge. In addition, the potential of GFFR, as indicated by the results, is highly useful in dealing with the tradeoff between cellcenter and celledge performance.
By construction, the optimization process in the local search algorithm decomposes by cell. For each cell, the information used in the allocation decision is local (the performance gain and loss of this cell and the neighboring cells). Therefore a promising line of further research is to design distributed implementations of the GFFR scheme. A second topic is to adopt nonuniform power allocation over the subbands of each cell. This potentially brings additional performance improvement because it exploits further the diversity over the subbands. As the problem complexity grows with this additional dimension of power allocation, it is of particular interest to optimize nonuniform power jointly with the development of distributed and lowcomplexity algorithms.
Declarations
Acknowledgements
The work has been supported by the ELLIIT Excellence Center and CENIIT, Linköping University, Sweden, and the EU FP7People200731IAPP218309 Marie Curie project.
Authors’ Affiliations
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