Autonomous algorithms for centralized and distributed interference coordination: a virtual layerbased approach
 Martin Kasparick^{1}Email author and
 Gerhard Wunder^{2}
https://doi.org/10.1186/168714992014120
© Kasparick and Wunder; licensee Springer. 2014
Received: 27 January 2014
Accepted: 28 June 2014
Published: 16 July 2014
Abstract
Interference mitigation techniques are essential for improving the performance of interference limited wireless networks. In this paper, we introduce novel interference mitigation schemes for wireless cellular networks with space division multiple access (SDMA). The schemes are based on a virtual layer that captures and simplifies the complicated interference situation in the network and that is used for power control. We show how optimization in this virtual layer generates gradually adapting power control settings that lead to autonomous interference minimization. Thereby, the granularity of control ranges from controlling frequency subband power via controlling the power on a perbeam basis, to a granularity of only enforcing average power constraints per beam. In conjunction with suitable shortterm scheduling, our algorithms gradually steer the network towards a higher utility. We use extensive systemlevel simulations to compare three distributed algorithms and evaluate their applicability for different user mobility assumptions. In particular, it turns out that larger gains can be achieved by imposing average power constraints and allowing opportunistic scheduling instantaneously, rather than controlling the power in a strict way. Furthermore, we introduce a centralized algorithm, which directly solves the underlying optimization and shows fast convergence, as a performance benchmark for the distributed solutions. Moreover, we investigate the deviation from global optimality by comparing to a branchandboundbased solution.
Keywords
1 Introduction
Increasing bandwidth requirements, not least due to the fast growing popularity of handheld devices with high data rate consumption, bring cellular networks to the brink of their capacity. Until recently, an end to the growth of this demand is not yet in sight. In order to optimally exploit the available bandwidth, current cellular networks experienced a paradigm shift towards frequency reuse1. Consequently, this leads to an increased susceptibility to interference such that current and future cellular networks are usually interference limited. This situation is aggravated by a trend towards ever smaller cell sizes. Especially users at the celledge are affected by high intercell interference (ICI). In theory, fully coordinated networks, where neighboring base stations act as a large distributed antenna array, promise a vast boost in performance [1, 2]. However, this makes great demands on synchronization and backhaul bandwidth. In fact, the promised gains from such schemes turn out to be hard to implement in practice [3]. As a consequence, distributed schemes for interference mitigation, incorporating joint scheduling and adaptive power allocation, are of utmost interest. However, due to mobile users and varying channel conditions, such algorithms have to be dynamic and able to operate autonomously^{a}. Moreover, future cellular systems, including pico and femto cells, must be selforganizing to maintain flexibility and scalability. Therefore, selfoptimizing interference coordination schemes are needed.
 (i)
Network utility maximization (NUM). By formulating a network utility maximization problem, we steer the system to a desired operating point. This includes fairness goals (like proportional fair and maxmin).
 (ii)
Virtual model. On top of regular scheduling and resource allocation, we maintain a virtual model of the network that captures and simplifies the complicated interference situation in the real world. It comprises a static ‘cooleddown’ version of the network based on longterm gains, thus suppressing the influence of fastfading. We use this model to obtain granular power control decisions. Thus, it can be seen as a new layer for longterm resource allocation decisions in a cooperative way (including corresponding message exchange).
 (iii)
Suitable shortterm scheduling. Instantaneously, we employ a popular gradient scheduler which is known to asymptotically converge to the solution of the underlying utility maximization problem. The scheduler has to take the power constraints into account (which can be strict or average constraints) that are obtained in the virtual layer to manage interference.
As mentioned before, we focus on fixed codebookbased schemes in SDMA networks. In particular, we assume that each base station maintains a fixed codebook of a certain size comprising precoding vectors, called ‘beams’. These beams can be used to support multiple users on the same timefrequency resource. Using fixed codebooks is a practical assumption and allows us to compare our algorithms with practical schemes used in current cellular systems.
The optimization within the virtual layer can be organized either in a centralized or in a distributed manner. Clearly, a distributed implementation is favorable but in order to accurately quantify the tradeoffs involved, we also investigate a centralized solution. This does not only provide a valuable benchmark for the distributed algorithms, but may also be a feasible option for small networks, where a central controller is indeed possible. Our centralized baseline algorithm is based on an alternating optimization approach, solving scheduling and power optimization in the virtual control plane separately. Since user rates are strongly coupled via transmit powers, we employ a successive convex approximation technique to tackle the inherent nonconvexity. Due to the nonconvex nature of the underlying optimization problem, global optimality cannot be guaranteed. Therefore, we additionally assess the deviation from global optimality by comparing our approach to an optimal solution based on branchandbound (BNB) in a simplified setting.
1.1 Related work
There is a significant amount of research on interference mitigation in cellular networks [5], which is often treated as a specific aspect of selforganization in cellular networks. A comprehensive survey on this is provided in [6]. A straightforward approach to avoid interference is to use a frequency reuse factor greater than one or some fractional frequency reuse scheme [7]. Another line of research targets reuse1 networks and interference mitigation by power control and resource allocation. There have been a variety of suggestions for joint multicell power control and scheduling in cellular networks such as [8, 9] (see also [10] for an overview). Multicell coordination via joint scheduling, beamforming, and power adaptation is considered in [11]. Thereby, fairness requirements (leading to concave utility functions) are fundamental for current and future cellular standards. The work [12] considers joint power allocation and user assignment to cells in the NUM context, taking into account a mixture of concave and nonconcave utilities. In [13], a gradient algorithmbased scheme for selforganizing resource allocation in LTE systems is proposed. However, a multitude of information has to be exchanged between coordinating sectors.
Although many of the aforementioned references consider distributed schemes, none treats resource allocation and interference management in multiuser MIMO systems. By contrast, our framework explicitly aims to exploit the freedom in terms of resource and power allocation offered by SDMA. Thereby, our framework builds upon and extends the framework introduced in [14, 15] for singleantenna (SISO) networks.
It is commonly accepted that the underlying optimization problems, which are nonconvex in general, can be solved optimally only for a limited set of problems and utilities in reasonable time. Existing solutions in cellular networks often rely on uplink downlink duality [16]. Since they attempt to solve such nonconvex problems directly, successive approximation techniques become an increasingly popular tool to treat this nonconvexity, used for example for power control in DSL [17] and multihop networks [18]. Global optimality is often achieved using branchandboundbased approaches [19, 20] or monotonic optimization [21], however at a high computational complexity. Other directions include interference pricing [22, 23] or game theoretic approaches [24]. Only recently, distributed coordination schemes in cellular networks have gained increasing popularity [9, 14, 25] due to the high complexity of centralized approaches. For example, [26] considers the derivation of transmit beamformers, also based on interference prices, for dense small cell networks.
1.2 Organization
The paper is organized as follows. In Section 2, we introduce the considered system model, introduce the notation, and describe the optimization problem that we address. In Section 3, we present a virtual control layer for solving this problem and introduce three distributed algorithms that are based on different realizations of this virtual control plane. In Section 4, we propose an alternative centralized scheme, while in Section 5, we present an optimal solution based on branchandbound. In Section 6, we present systemlevel simulation results that evaluate the performance of the distributed algorithms and moreover investigate simpler scenarios to compare these to the centralized and the optimal baselines. Eventually, in Section 7, we state the most important conclusions.
2 System model and notation
We consider the downlink of a cellular OFDMA network, where each cell is subdivided into three sectors. In total, we have M sectors m∈{1,…,M}. Each sector is served by a base station having n_{T} transmit antennas with a corresponding sector controller responsible for user selection and resource allocation. There are I users randomly distributed in the system, each equipped with n_{R} receive antennas. In the following, we assume n_{R}=1. We assume that each user has pending data at all times. Let I_{ m } be the number of users associated to sector m.
Important notation
Notation  Definition 

${P}_{\mathit{\text{jb}}}^{m}$  Power assigned to beam b on PRB j insector m 
${\stackrel{\u0304}{P}}_{\mathit{\text{jb}}}^{m}$  Average power constraint (‘target’ power) of beam b on PRB j in sector m 
${\stackrel{\u0304}{P}}^{m}\left(t\right)$  Current total allocated power in sector m 
P _{max}  Maximum total base station power 
${c}_{\mathit{\text{jb}}}(k,{\stackrel{\u0304}{P}}_{j}^{m})$  Power cost/consumption of beam b on PRB j in sector m 
${C}_{\mathit{\text{jb}}}^{m}\left(k\right)$  Virtual power cost/consumption of sector m on beam b and PRB j 
π(l)  Probability of fading state l 
$\mathcal{K}\left(l\right)$  Set of possible scheduling decisions given fading state l 
${\mathit{h}}_{\mathit{\text{ij}}}^{m}\left(t\right)$  Channel vector from base station m to user i on PRB j at time t 
${\stackrel{\u0304}{\mathit{h}}}_{\mathit{\text{ij}}}^{m}$  Average channel from base station m to user i on PRB j 
${\mathit{\mu}}_{\mathit{\text{jb}}}^{l}\left(k\right)$  Vector of user rates on PRB j and beam b given decision k and fading state l 
${\stackrel{\u0304}{\mathit{X}}}^{m}$  Vector of average total rates of users insector m 
${\varphi}_{\mathit{\text{jk}}}^{\mathit{\text{lm}}}$  Fraction of time that scheduling decision k is chosen on PRB j given fading state l 
${\varphi}_{\mathit{\text{ijb}}}^{m}$  Fraction of time that user i in sector m is scheduled using beam b on PRB j 
${G}_{\mathit{\text{ij}}}^{m}$  Longterm gain of user i to base station m for his best beam on PRB j 
${G}_{\mathit{\text{ijb}}}^{m}$  Longterm gain of user i to base station m for beam b on PRB j 
${R}_{\mathit{\text{ij}}}^{m}\left(k\right)$  Virtual rate of user i on PRB j given decision k 
R _{ j }  Vector of virtual user rates in PRB j 
${R}_{\mathit{\text{ijb}}}^{m}$  Virtual rate of user i on PRB j and beam b 
${X}_{i}^{m}$  Virtual average rate of user i in sector m 
X(t/n_{ v })  Vector of virtual average rates at t th virtual scheduling run 
${F}_{\mathit{\text{ijb}}}^{m}$  SINR of user i on beam b and PRB j 
${D}_{\mathit{\text{jb}}}^{(\widehat{m},m)}$  Sensitivity of sector m to a power change of beam b on PRB j in sector $\widehat{m}$ 
${D}_{\mathit{\text{jb}}}^{m}$  Sum sensitivity to a power change of beam b on PRB j in sector m 
n_{ j }(k)  Number of beams that are activated on PRB j if decision k is chosen 
λ_{ j b }(t)  Dual parameter: deviation of power on PRB j and beam b from the target value 
${\alpha}_{\mathit{\text{jb}}}^{m}\left(k\right)$  Scales target beam powers to instantaneous powers ‘costs’ 
${\alpha}_{\mathit{\text{ijb}}}^{m},{\beta}_{\mathit{\text{ijb}}}^{m}$  Approximation constants in concave lower bound 
2.1 Problem statement
Thereby, ${\mathit{\mu}}_{\mathit{\text{jb}}}^{l}\left(k\right)={\left({\mu}_{\mathit{\text{ijb}}}^{l}\left(k\right)\right)}_{i\in \{1,\dots ,{I}_{m}\}}$ is a vector comprising elements ${\mu}_{\mathit{\text{ijb}}}^{l}\left(k\right)$, which represent the rate that user i is assigned on PRB j and beam b when the system is in fading state l and scheduling decision k is chosen. They can be zero if the particular resource is not assigned to user i by decision k. ${\varphi}_{\mathit{\text{jk}}}^{\mathit{\text{lm}}}$ denotes the fraction of time that scheduling decision k is chosen on PRB j, provided the system is in fading state l.
In a nutshell, we are not interested in a specific ‘snapshot’ of the system, but only in ergodic rates. Therefore, a possible control algorithm should not adapt to a specific system state but should be able to optimize the system performance over time.
where the fixed parameter β>0 determines the size of the averaging window. The gradient scheduler is known to asymptotically solve the problem (for β→0) without knowing the fading distribution. In case of logarithmic utilities, the gradient scheduler becomes the wellknown proportional fair scheduler^{c}.
3 Autonomous distributed power control algorithms for interference mitigation
We now turn to the distributed power control schemes for autonomous interference management in cellular networks, which are designed to enable network entities to locally pursue optimization of the global network utility. We introduce three basic approaches. It is important to note that all algorithms are special cases of the wellknown gradient algorithm [28, 29], whose convergence behavior has been thoroughly analyzed. Therefore, we refrain from reproducing this theoretical analysis. However, in Section 6, for illustration purposes, we present numerical results indicating a fast convergence behavior. The main difference between the algorithms that we propose is the granularity of power control.
The first algorithm uses an opportunistic scheduler which only adapts the power per frequency subband, which is then distributed equally among activated beams. We call this opportunistic algorithm (OA). It leaves full choice to the actual scheduler as to which beam to activate at what time. The scheduler can therefore decide opportunistically (⇒ opportunistic algorithm). However, the power budget per PRB which is distributed (equally) among activated beams is determined by an associated control scheme.
The second algorithm is the virtual subband algorithm (VSA), which enforces strict power constraints on each beam by requiring all beams to be switched on all the time (with power values given by the associated control). This has the advantage of making the interference predictable (assuming known power values). However, it leaves only limited freedom for the actual scheduler, whose task is reduced to user selection for each beam. Since a beam is always turned on, it can be treated as an independent resource for scheduling, just like a ‘virtual’ subband (⇒ virtual subband algorithm).
The third is a hybrid approach, which permits opportunistic scheduling at each time instance but, in addition, enforces average power constraints per beam. We call it costbased algorithm (CBA). It leaves more freedom for opportunistic scheduling than the virtual subband algorithm. In contrast to requiring all beams to be used at all times with strict power values, we only require the target beam power values to be kept on average. Thus, instantaneously, the scheduler is free to make opportunistic decisions based on the current system state. In order to assure that the average power constraints are kept, we introduce an additional cost term into the utility maximization and the gradient scheduler (⇒ costbased algorithm).
We focus on the applicability in different fading environments, comparing the overall performance with respect to a networkwide utility function as well as the performance of celledge users. Thereby, we show that although the algorithms behave differently in different user mobility scenarios, in general, it is more beneficial to impose average rather than strict power constraints.
As we demonstrate later, the three algorithms perform differently with different mobility assumptions on the users. A problem that arises with increased mobility is that the (virtual) model lacks behind the actual network state. Especially when the controllers are restricted to a gradual power adaption process on a perbeam granularity, they might not always be able to fully exploit multiuser diversity. Since the proposed algorithms put different emphasis on opportunistic scheduling in power adaption and resource allocation decisions, they perform differently when facing user mobility and fast fading.
Let us now turn to the control plane, the virtual layer, of the considered algorithms. They all have the following general procedure in common. The goal of the control plane is to obtain estimates of the partial derivatives of the network utility with respect to the power allocation of particular resources and to adapt the power control policy accordingly. These estimates can be seen as estimates of the sensitivity of the network utility to changes of the allocation strategies. Thereby, the allocation can be, as in the OA of Section 3.1, the power allocation of a PRB which is then divided equally among activated beams. Or it can even be, as in the VSA of Section 3.2, the power allocation of an individual beam. Or it can also be, as in the CBA of Section 3.3, simply an average power constraint of an individual beam, which does not have to be kept at every single time instance.
A further similarity between all algorithms is that in addition to ordinary shortterm CSI, the mobiles (not necessarily often) report longterm feedback to their base stations, which is then used to calculate (virtual) user rates and, accordingly, (virtual) average rates. These average rates are a good representation of the interference coupling throughout the network and are used to calculate the ‘sensitivities’ to power changes on particular resources in other sectors (and in the own sector). This sensitivity information is compiled in messages which are exchanged between the sectors^{d}. Upon reception of the message vectors, the sector controllers are now able to calculate the desired estimate of the system utility’s sensitivity to power changes on particular resources and adjust powers (or power constraints) accordingly. It is important to always distinguish between actual rates and virtual rates. The actual average rates are the ones tracked by the ordinary (proportional fair) scheduler and determine the sector’s utility. The virtual average rates are based on longterm feedback of averaged channel gains and do not have an immediate physical meaning in the ‘real world’. They are created by a ‘virtual’ scheduler based on ‘virtual’ scheduling decisions. The set of all these rates forms a ‘virtual’ model of the system which is used to derive power adaption decisions.
The question remains how the virtual average rates and accordingly the sensitivities to power changes are calculated. Besides the granularity of power control, this virtual model, which can be also seen as a virtual layer for interference mitigation above the actual shortterm scheduling, is the main difference between the investigated algorithms. In the following, we will discuss this in detail.
3.1 Opportunistic algorithm
where ${\stackrel{\u0304}{h}}_{\mathit{\text{ij}}}^{m}$ is the channel from user i to its sector controller on PRB j, averaged to eliminate the influence of fastfading.
In this paper, we use ρ(x)= log(1+x). Thereby, ${\stackrel{\u0304}{P}}_{j}^{m}\left(k\right)$ is the current power value of PRB j divided by the number of users scheduled, thus depending on decision k. Moreover, ${\stackrel{\u0304}{P}}_{j}^{{m}^{\prime}}{\stackrel{~}{G}}_{\mathit{\text{ij}}}^{{m}^{\prime}}$ represents a longterm estimate of the interference of sector m^{′} on PRB j (which can be measured by the mobiles).
Having defined the virtual user rates R, virtual average user rates X and sensitivities are derived by virtual scheduling (based on a similar procedure in [14]) as follows. We run the following steps n_{ v } times per TTI in each sector m and for any PRB j. Thereby, the parameter n_{ v } determines how long the virtual scheduler runs before accepting the sensitivities. Consequently, a larger value means more overhead by the virtual layer but better results.

We determine the virtual scheduling decision k^{∗} using a gradient scheduler according to${k}^{\ast}\in arg\underset{k}{max}\nabla {U}^{m}\left({\mathit{X}}^{m}\left(\frac{t}{{n}_{v}}\right)\right){\mathit{R}}_{j}\left(k,\frac{t}{{n}_{v}}\right).$

We update virtual average user rates according to$\mathit{X}\left(\frac{t+1}{{n}_{v}}\right)=\left(1{\beta}_{1}\right)\mathit{X}\left(\frac{t}{{n}_{v}}\right)+{\beta}_{1}J{\mathit{R}}_{j}\left({k}^{\ast},\frac{t}{{n}_{v}}\right).$

We update sensitivities according to$\begin{array}{ll}{D}_{j}^{\left(\widehat{m},m\right)}\left(\frac{t+1}{{n}_{v}}\right)& =\left(1{\beta}_{2}\right){D}_{j}^{\left(\widehat{m},m\right)}\left(\frac{t}{{n}_{v}}\right)\\ \phantom{\rule{1em}{0ex}}+{\beta}_{2}\sum _{i=1}^{{I}_{m}}\frac{\partial {U}^{m}\left({X}^{m}\right)}{\partial {X}_{i}^{m}}\frac{\partial {R}_{\mathit{\text{ij}}}^{m}\left({k}^{\ast},\frac{t}{{n}_{v}}\right)}{\partial {\stackrel{\u0304}{P}}_{j}^{\widehat{m}}\left(t\right)}.\end{array}$(9)
Starting with equal power, the adaption of the PRB powers can be summarized as follows. From time to time, the sensitivities are exchanged and summed up by each sector controller for each beam and PRB. Since each ${D}_{j}^{\left(\widehat{m},m\right)}$ is an estimation of the sensitivity of sector m’s utility to a power change in sector $\widehat{m}$, the summation gives an estimate of the network utility’s sensitivity. Then, the power is increased on the PRB with the largest positive sum and decreased on the PRB with the largest negative sum.
3.2 Virtual subband algorithm
summing up the sensitivities of all sectors (including itself) to a power change of beam b on PRB j in sector m and which can be either positive or negative. Note that sector indices m and $\widehat{m}$ in the RHS of (15) are interchanged compared with (14), since in (14), we are interested in how the beam in sector $\widehat{m}$ interferes with sector m, while in (15), it is of interest how the beam in sector m interferes with (all) sector(s) $\widehat{m}$.
Since ${D}_{\mathit{\text{jb}}}^{m,\widehat{m}}$ represent estimates of the sector utilities to a power change on jb in sector m, ${D}_{\mathit{\text{jb}}}^{m}$ clearly is an estimate of the sensitivity of the system’s utility to a power change on the respective beam. Depending on the ${D}_{\mathit{\text{jb}}}^{m}$, we can now make a power adjustment which steers the system operating point towards a greater utility in the virtual model.
 (1)Pick a virtual resource (j b)_{∗} (if there is one) such that ${D}_{{\left(\mathit{\text{jb}}\right)}_{\ast}}^{\left(m\right)}\left(t\right)$ is the smallest among all virtual resources jb with ${D}_{\mathit{\text{jb}}}^{\left(m\right)}\left(t\right)<0$ and ${P}_{\mathit{\text{jb}}}^{m}\left(t\right)>0$. Now, set${P}_{{\left(\mathit{\text{jb}}\right)}_{\ast}}^{m}\left(t+1\right)=max\left\{\underset{{\left(\mathit{\text{jb}}\right)}_{\ast}}{\overset{m}{P}}\left(t\right)\Delta ,0\right\}.$
 (2)If ${\stackrel{\u0304}{P}}^{m}\left(t\right)<{P}_{\text{max}}$, pick (j b)^{∗} (if there is one) such that ${D}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{\left(m\right)}\left(t\right)$ is the largest among those jb with ${D}_{\mathit{\text{jb}}}^{\left(m\right)}\left(t\right)>0$. Set${P}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{m}\left(t+1\right)={P}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{m}\left(t\right)+min\left\{\Delta ,{P}_{\text{max}}{\stackrel{\u0304}{P}}^{m}\left(t\right)\right\}.$
 (3)If ${\stackrel{\u0304}{P}}^{m}\left(t\right)={P}_{\text{max}}$ and $\underset{\mathit{\text{jb}}}{max}{D}_{\mathit{\text{jb}}}^{\left(m\right)}\left(t\right)>0$, pick a pair ((j b)_{∗},(j b)^{∗}) (if there is one) such that ${D}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{\left(m\right)}\left(t\right)$ is the largest, and ${D}_{{\left(\mathit{\text{jb}}\right)}_{\ast}}^{\left(m\right)}\left(t\right)$ is the smallest among those virtual resources jb with ${P}_{\mathit{\text{jb}}}^{m}\left(t\right)>0$ and ${D}_{{\left(\mathit{\text{jb}}\right)}_{\ast}}^{\left(m\right)}\left(t\right)<{D}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{\left(m\right)}\left(t\right)$. Set$\begin{array}{ll}{P}_{{\left(\mathit{\text{jb}}\right)}_{\ast}}^{m}\left(t+1\right)& =max\left\{\underset{{\left(\mathit{\text{jb}}\right)}_{\ast}}{\overset{m}{P}}\left(t\right)\Delta ,0\right\},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\\ {P}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{m}\left(t+1\right)& ={P}_{{\left(\mathit{\text{jb}}\right)}^{\ast}}^{m}\left(t\right)+min\left\{\Delta ,\underset{{\left(\mathit{\text{jb}}\right)}_{\ast}}{\overset{m}{P}}\left(t\right)\right\}.\phantom{\rule{2em}{0ex}}\end{array}$
By intuition, the algorithm reallocates power to the beams with large positive utilitysensitivity. Note that for numerical reasons, it may be necessary to specify a certain minimum power per beam ${P}_{b}^{\text{min}}$ instead of allowing beam powers to be reduced to zero. In this case, the changes to the algorithmic notation above are straight forward, so we do not explicitly state them here.
3.3 Costbased algorithm
with ${\stackrel{\u0304}{P}}_{j}^{m}=\sum _{b}{\stackrel{\u0304}{P}}_{\mathit{\text{jb}}}^{m}$.
Average user rates ${\stackrel{\u0304}{X}}^{m}\left(t\right)$ are maintained and updated as in (6). The above algorithm can be seen as an application of the greedy primal dual (GPD) algorithm presented in [29].
The virtual control plane differs from VSA in the following. In contrast to the VSA, where every beam is switched on all the time, the situation is different here. To enable the calculation of derivatives of the rates with respect to beam powers (needed in (22)), we introduce scaling factors ${\alpha}_{\mathit{\text{jb}}}^{m}\left(k\right)$, which scale target beam powers ${\stackrel{\u0304}{P}}_{\mathit{\text{jb}}}^{m}$ to powers ‘costs’^{e}${C}_{\mathit{\text{jb}}}^{m}\left(k\right)$ that are instantaneously used by the virtual scheduler. Thus, ${\alpha}_{\mathit{\text{jb}}}^{m}\left(k\right){\stackrel{\u0304}{P}}_{\mathit{\text{jb}}}^{m}={C}_{\mathit{\text{jb}}}^{m}\left(k\right)$.
${I}_{\text{intra}}=\sum _{{b}^{\prime}\ne b}{\alpha}_{j{b}^{\prime}}^{m}\left(k\right){G}_{\mathit{\text{ij}}{b}^{\prime}}^{m}{\stackrel{\u0304}{P}}_{j{b}^{\prime}}^{m}$, and ${I}_{\text{inter}}=\sum _{\widehat{m}\ne m}\sum _{\widehat{b}}{G}_{\mathit{\text{ij}}\widehat{b}}^{\widehat{m}}{\stackrel{\u0304}{P}}_{j\widehat{b}}^{\widehat{m}}$.
Again, ${\stackrel{~}{\varphi}}_{\mathit{\text{jk}}}^{m}$ are optimal time fractions of resource usage for sector m; however, in contrast to VSA where those time fractions were calculated explicitly, CBA uses the approach of OA to determine the virtual average rates (and implicitly the time fractions) through virtual scheduling. As before, to distinguish real and virtual scheduler, we use capital letters for virtual scheduler quantities whenever possible. In each TTI, the virtual scheduler performs n_{ v } scheduling runs. In each run, the following steps are carried out on each PRB j:

We determine the virtual scheduling decision k^{∗} similar to (17).

We update virtual average rates similar to (6).

We update the virtual average power costs for each beam b similar to (18).

We update sensitivities for each beam b and sector $\widehat{m}$ (β_{2}>0 small) according to$\begin{array}{ll}{D}_{\mathit{\text{jb}}}^{\left(\widehat{m},m\right)}\left(\frac{t+1}{{n}_{v}}\right)& =\left(1{\beta}_{2}\right){D}_{\mathit{\text{jb}}}^{\left(\widehat{m},m\right)}\left(\frac{t}{{n}_{v}}\right)\\ \phantom{\rule{1em}{0ex}}+{\beta}_{2}\sum _{i=1}^{{I}_{m}}\frac{\partial {U}^{m}\left({\mathit{X}}^{m}\right)}{\partial {X}_{i}^{m}}\sum _{{b}^{\prime}}\frac{\partial {R}_{\mathit{\text{ij}}{b}^{\prime}}^{m}\left({k}^{\ast},\frac{t}{{n}_{v}}\right)}{\partial {\stackrel{\u0304}{P}}_{\mathit{\text{jb}}}^{\widehat{m}}\left(t\right)}.\end{array}$(22)
The power adaption is then carried out similar to the other algorithms. For each resource, each sector sums up values ${D}_{\mathit{\text{jb}}}^{\left(\widehat{m},m\right)}$ from all sectors and increases the power level on the beam with largest positive sum while decreasing the power level on the beam with largest negative sum. However, CBA adapts only power constraints per beam, not actually used beam powers.
Apart from the intuitive benefits of instantaneously allowing opportunistic scheduling, we observe that from a practical point of view, average power constraints are further justified since hybrid automatic repeat request (HARQ) coding is essentially performed over multiple successive transmissions.
One can show the following:
Theorem 1
Let 0<ε<1 be finite and U^{ m } be an increasing concave utility function, defined in (0,∞). Then, the family of functions f_{ ε }(R) (defined by (23)) with R_{ i j b }≥c>0 (∀i,j,b) is differentiable everywhere and converges for any sequence ε_{ n }→0 to f in (13) (which is continuous) in the uniform sense.
Proof
The proof can be found in Appendix 1.
By Theorem 1, we can replace our utility function with a smooth, uniformly convergent approximation, which can be locally maximized in the power control loop. Note that this replacement is already incorporated in the calculation of sensitivities (Equation 14).
4 An alternating optimizationbased approach
Given the distributed nature of the above presented algorithms, the question arises: Can a centralized controller do better? Therefore, in the following, we want to compare the algorithms of Section 3 with a centralized solution.
with ${R}_{\mathit{\text{ijb}}}^{m}\left(\mathit{P}\right)=log\left(1+\underset{\mathit{\text{ijb}}}{\overset{m}{F}}\left(\mathit{P}\right)\right)$ and ${F}_{\mathit{\text{ijb}}}^{m}$ defined in (11). As described in Section 3, the power allocation problem is so far solved using a distributed gradient ascent procedure. However, when we allow a centralized controller for the network, we can instead solve (24 to 26) directly each time a power update is desired and use the resulting power allocation directly for actual resource allocation.
Since the scheduling subproblem is convex (cf. Lemma 2), we only have to care about the power allocation subproblem. We try to tackle this problem by a successive convex approximation (SCA) approach similar to [17]. The subproblem in P is still highly nonconvex. However, using Lemma 3, we obtain a convexified version of the power allocation subproblem.
Lemma 2
With constant P, optimization problem (2426) is a convex optimization problem in Φ.
Proof.
The proof follows since nonnegative weighted addition and scalar composition preserve concavity [31].
Lemma 3
is a convex optimization problem.
Proof.
The proof can be found in Appendix 2.
The algorithm is initialized as described in the first step of Algorithm 2, which is equivalent to a high signaltointerferenceplusnoise ratio (SINR) approximation (which can be seen when applying this initialization in (27)). The highSINR approximation assumes that for large SINR, log(1+SINR)≈ log(SINR). This is a common initialization for this kind of algorithm [17, 32]. Each iteration of the algorithm comprises two steps, a maximize step and a tightenstep. In the maximize step, we find a solution to the current convexified version (28 to 29) of the power control problem. This solution is then used in the tighten step to update the convex approximation parameters α and β for each link according to (30). The algorithm converges when the tighten step (30) does not produce any (significant) changes. Being based on an inner approximation framework by Marks and Wright [33], it can be shown that Algorithm 2 converges at least to a KKT point of the PA subproblem.
5 Approaching global optimality: comparison with branchandbound
It is known that the underlying optimization problem is nonconvex; thus, the gradient ascentbased algorithms presented in Section 3 as well as the alternating optimizationbased algorithm of Section 4 will most likely converge to a local maximum. Thus, although simulation results (cf. Section 6) show already high gains in utility, the question remains how good the solution found actually is, that is how much of the achievable performance gains is actually realized?
with ${R}_{\mathit{\text{ij}}}^{m}=log\left(1+\frac{\underset{\mathit{\text{ij}}}{\overset{m}{G}}{P}_{j}^{m}}{{\sigma}^{2}+\sum _{{m}^{\prime}\ne m}{G}_{\mathit{\text{ij}}}^{{m}^{\prime}}{P}_{j}^{{m}^{\prime}}}\right)$, is highly nonconvex (even in the twosector, twouser case). Moreover, since this is essentially a joint optimization in time fractions Φ and power values P, the complexity is significantly higher than with power control only. To gain insight into the deviation from the optimal solution of (31 to 33), we compare our algorithm’s performance in the simplified setting with a (near) optimal solution based on BNB. Although computationally very expensive, it gives us a good impression on how much of the achievable performance is actually reached. We assume, without loss of generality, that the maximum sum power available in each cell in (33) is normalized to one.
Branchandbound is a standard algorithm for global optimization, which creates a search tree where at each node, an upper and a lower bound to the problem are evaluated. Details can be found for example in [34]. It is based on constantly subdividing the feasible parameter region and for each node of the resulting tree calculating the upper and lower bounds to the objective function. For ease of notation, we will combine all parameters to our objective function in one vector $\widehat{P}=\left({p}_{1},\dots ,{p}_{n}\right)$. Let ${\mathcal{P}}^{\left(0\right)}$ denote our initial parameter region. This region forms a convex ndimensional polytope with V extreme points (or vertices) ${\widehat{P}}_{v}$. We collect the corner points of polytope ${\mathcal{P}}^{\left(0\right)}$ in set ${\mathcal{V}}^{\left(0\right)}$. Consequently, the parameter region associated with node l is denoted as ${\mathcal{P}}^{\left(l\right)}$ with associated vertices ${\mathcal{V}}^{\left(l\right)}$.
Equation 35 is still not concave since it includes a sum of a concave function (${\stackrel{~}{f}}_{\mathit{\text{ij}}}^{m}\left(\widehat{P}\right)$) and a convex function (${\stackrel{~}{g}}_{\mathit{\text{ij}}}^{m}\left(\widehat{P}\right)$). To get a concave objective function, we replace ${\stackrel{~}{g}}_{\mathit{\text{ij}}}^{m}\left(\widehat{P}\right)$ by it’s convex envelope (cf. [34]), defined as follows:
with λ=(λ_{1},…,λ_{ V }), $\widehat{P}\in {\mathcal{P}}^{\left(l\right)}$, ${\widehat{P}}_{v}\in {\mathcal{V}}^{\left(l\right)}$, and V being the number of vertices collected in ${\mathcal{V}}^{\left(l\right)}$.We use three different approaches to obtain a lower bound at node l. The first lower bound is obtained by taking the maximum of the objective function values at each of the corner points of the respective parameter region. Second, we evaluate the optimal parameter vector from calculating the upper bound. Third, we use a standard solver to compute a (local) optimum of the original nonconvex problem. If one of the three methods leads to a higher value, the current global lower bound is replaced.
6 Numerical evaluation
In this section, we present numerical results to evaluate the distributed algorithms of Section 3 as well as the centralized and BNBbased solutions of Section 4 and Section 5, respectively.
6.1 Systemlevel simulations
In order to compare the performance of the three algorithms in a setting close to practice, we conduct systemlevel simulations based on LTE.
6.1.1 Simulation setup
General simulation setup
Parameter  Value 

Number of sectors (M)  21 
Total number of terminals (I)  105 to 315 
Mobile terminal velocity  0 km/h, 3 km/h 
Number of PRBs (J)  8 
Number of beams (B) (coordination)  4 
Number of beams (B) (baseline)  8 
Base station antennas (n_{T})  4 
Number of terminal antennas (n_{R})  1 
Simulation duration  10,000 TTI 
Power adaption step size (Δ)  0.5% of the initial power 
Traffic model  Full buffer 
6.1.2 Baseline algorithm
As baseline, we use a noncoordinative scheduling algorithm called greedy beam distance (GBD) algorithm, with a codebook size of 3 bits (eight beams). GBD requires feedback from each user, comprising a CDI and a channel quality information (CQI) value. Note that the CQI is only an estimate of the users’ SINR values in case the user alone is scheduled on the PRB (with full power), since the scheduling decisions cannot be known in advance. Thus, it does not contain intrasector interference. Moreover, it contains only an estimate of the intersector interference. On each PRB, the users are greedily scheduled on their best beams (using their proportionally fair weighted CQI feedback as utility). Thereby, a minimum beam distance has to be kept, in order to minimize the interference between users scheduled on the same PRB. This distance is based on a geometrical interpretation of beamforming. It means that given a user is scheduled on a certain beam, adjacent beams (up to a certain ‘distance’) are blocked and users that reported one of those beams are excluded from the list of scheduling candidates for the respective PRB. We use a minimum distance of 3, that is, with the used eightbeam codebook at most three users can be scheduled on the same PRB. Of course, no adaptive power allocation is performed; the power is distributed equally among the PRBs (and further among the thereon scheduled users).
6.1.3 Simulation results
6.2 Performance of centralized solution based on alternating optimization
In summary, the centralized solution has some clear advantages. First, it is invariant to the initialization of the system and is thus able to adapt very fast to changing environmental conditions. Second, especially when the channels are varying fast and the SNR is high, the centralized scheme offers a higher performance in comparison to the distributed scheme. However, one should note that at realistic SNRs, the distributed solution still achieves almost the same performance and at the same time scales smoothly with the network size, while a centralized solution can only be implemented in very restricted scenarios.
6.3 Global optimality
In the following, we compare the performance of the distributed algorithm to an (nearly) optimal solution based on BNB. To simplify the analysis in this section and due to the high complexity of the BNB algorithm, we restrict ourselves to the simple case of two sectors with two users (no mobility) and two PRBs and a single antenna.
Given a certain tolerance ε, BNB converges to an optimal solution of (31). However, it has exponential complexity and is therefore only feasible in very small settings. We compare our distributed coordination algorithm with an equal power noncooperative scheduler and the BNB algorithm described in Section 5.
In the ‘equal’ interference case, the gains to the interfering base stations are roughly the same as the gains to the own base stations; thus, ${G}_{\mathit{\text{ij}}}^{m}\approx {G}_{\mathit{\text{ij}}}^{{m}^{\prime}}(m\ne {m}^{\prime})\phantom{\rule{2.77626pt}{0ex}}\text{for all}i,j,m,{m}^{\prime}$. It can be observed that the power control algorithm leaves a larger portion of the available performance gains unused, compared to the weak interference case, although it shows significant improvements over noncooperative scheduling.
The strong interference case is characterized by ${G}_{\mathit{\text{ij}}}^{m}<{G}_{\mathit{\text{ij}}}^{{m}^{\prime}}(m\ne {m}^{\prime})$, for all i,j,m,m^{′}; thus, for all users, the link gains to the interfering base stations are higher than the gains to the ‘own’ base stations. Here, the gap in the noncooperative algorithm’s performance, compared to the BNB result, is significantly worse than in the other cases. By contrast, the power control algorithm shows roughly the same performance gap than in the case of equal gains.Again, to illustrate the convergence, Figure 13 gives an example of the performance over time (here, for the weak interference case). Obviously, the power control algorithm converges to a local maximum soon.
7 Conclusions
We proposed and compared three distributed algorithms for autonomous interference coordination in cellular SDMA networks. The algorithms are based on a virtual layer that models the interference interdependencies in the network and gradually adapts power control levels. The proposed algorithms differ in granularity of power control, required feedback, signaling overhead, and the virtual model itself. Systemlevel simulations indicate high gains both in overall utility and in celledge user throughput for all three algorithms in static environments without user mobility. While VSA offers a very fine granularity of power control on a perbeam level, it suffers from the lack of freedom to instantaneously perform power allocation in an opportunistic manner. In fact, in an environment with significant user mobility, only CBA achieves significant gains in both metrics. This demonstrates the superiority of CBA’s approach to enforce average power constraints but instantaneously allowing opportunistic scheduling. Comparisons to a centralized benchmark scheme reveal that although the convergence of the centralized scheme is much faster than in the distributed case, the performance in overall utility is comparable.
Endnotes
^{a} Note, that in this paper, we use the term autonomous not in the sense of an autonomous operation of the different network entities, such as base stations, but to the ability of the network to find a suitable operating point without the needed of prior planning or human interaction.
^{b} Each PRB consists of a fixed number of OFDM symbols in time and has a total length of 1 TTI.
^{c} We use $\sum _{i}log\left(\underset{i}{\overset{m}{\stackrel{\u0304}{X}}}\right)$ as sector utility in this paper, leading to a proportional fair operating point.
^{d} To reduce messaging overhead, sector controllers could limit the message exchange to strongest interferers, e.g., consider only neighboring base stations.
^{e} To avoid confusion, we denote variables belonging to the real scheduler with lower case symbols, and variables from the virtual scheduler with corresponding upper case symbols.
Appendices
Appendix 1: proof of Theorem 1
First, we show that f_{ ε }(R) is everywhere differentiable. For this purpose, we use the following Lemma 4 and set H(x,y):=U(R,Φ).
Lemma 4
is continuously differentiable in [a_{1},a_{2}]. (This can be generalized to the multidimensional case. Namely, the domains [a_{1},a_{2}] and [a_{3},a_{4}] can be replaced by arbitrary convex compact sets in finitedimensional vector spaces, and derivatives replaced by gradients).
with $(n\in \mathbb{N})$. We can use Dini’s Theorem to show that ${\left\{{f}_{n}\right(R\left)\right\}}_{n\in \mathbb{N}}$ converges uniformly to f. This theorem states that if ${\left\{{f}_{n}\right\}}_{n\in \mathbb{N}}$ (${f}_{n}:K\to \mathbb{R},n\in \mathbb{N}$, being a sequence of continuous functions and K being a compact metric space) converges pointwise to f ($f:K\to \mathbb{R}$ being a continuous function) and if f_{ n }(x)≥f_{n+1}(x) (∀x∈K and $\forall n\in \mathbb{N}$), then ${\left\{{f}_{n}\right\}}_{n\in \mathbb{N}}$ converges uniformly to f.
The rate space K:=[c,B]^{ I J N } is clearly compact, since R_{ i j b }∈[c,B],B<∞,c>0. Moreover, for each x^{∗}∈K, f_{ n }(x^{∗}) converges to f(x^{∗}) when n→∞ (since ε_{ n }→0), and since ϕ_{ i j b }∈[0,1], we have ϕ^{1−ε}_{ n }≥ϕ^{1−ε}_{ m } (∀i,j,b) if m≥n. Thus, f_{ n }(x)≥f_{n+1}(x).
Appendix 2: proof of Lemma 3
is convex, and thus, ${\stackrel{~}{R}}_{\mathit{\text{ijb}}}^{m}$ is concave. Noting that nonnegative weighted addition and scalar composition preserve concavity concludes the proof.
Declarations
Authors’ Affiliations
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