Wireless Communications and Mobile Computing

Volume 2018, Article ID 9163783, 15 pages

https://doi.org/10.1155/2018/9163783

## Impact of Resource Blocks Allocation Strategies on Downlink Interference and SIR Distributions in LTE Networks: A Stochastic Geometry Approach

^{1}INRIA CNRS UMR 5668 LIP, University Lyon 1, France^{2}CNRS UMR 8623 LRI, University Paris Saclay, France

Correspondence should be addressed to Anthony Busson; rf.airni@nossub.ynohtna

Received 26 January 2018; Revised 5 May 2018; Accepted 28 May 2018; Published 26 June 2018

Academic Editor: Natalia Y. Ermolova

Copyright © 2018 Anthony Busson and Iyad Lahsen-Cherif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a model based on stochastic geometry to assess downlink interference and signal over interference ratio (SIR) in LTE networks. The originality of this work lies in the proposition and combination of resource blocks assignment strategies, transmission power control, and realistic traffic patterns into a stochastic geometry model. For this model, we compute the first two moments of interference. They are used to parameterize its distribution from which we deduce the SIR distribution. Outage and transmission rates (modulation and coding rate) are then derived to evaluate the system performance. Simulations that cover a large set of scenarios show the accuracy of our proposal and allow us to compare these strategies with more complex ones that aim to minimize global interference. Numerical evaluations highlight the behavior of the LTE network for different traffic patterns/load, eNodeB density, and amount of resource blocks and offer insights about possible parameterization of LTE networks.

#### 1. Introduction

The amount of mobile data that cellular networks must carry is continuously increasing. The capacity of the wireless systems must continuously increase in order to satisfy the growing demand of traffic from users and applications. Long-Term Evolution and Long-Term Evolution-Advanced [1] (LTE-A) have been recently standardized to improve the network capacity and support this traffic growth. One of the solutions brought by LTE is the enhancement of the radio spectrum reuse. The smallest radio resource that can be allocated to a user is a* resource block* (RB). An RB is a channel (an OFDMA channel composed of a set of OFDM subcarriers) for the duration of one time slot. Considering the number of RB is finite, they are reused in different cells generating potential intercell interference. The algorithms that assign RB to users located in different cells have thus an important role in the system performance. A static RB assignment where disjoint resources are distributed to each cell may lead to an inefficient resource usage as the unused RB in a cell cannot be reused in another one. Instead, algorithms that assign RB can be centralized in a scheduler/controller that controls a certain number of neighborhood cells and adapts the RB assignments to the cells load. Also, it may improve the spatial reuse while ensuring a low level of interference.

Several studies have proposed assignment strategies performed at the scheduler to minimize global interference [2, 3]. These strategies aim to minimize interference for a given configuration and are evaluated exclusively through simulations. However, the assignment strategies have to be evaluated for more general scenarios and at larger scale.

Stochastic geometry offers a powerful tool to analyze large scale networks through a few parameters and to understand the role of these key parameters on the whole system. The other benefit of stochastic geometry is to consider realistic Base stations (BS) or eNodeB (*evolved Node B*) locations. It uses random point processes rather than deterministic (grid or hexagonal patterns for instance) or predetermined locations of BS/eNodeB. For instance, the Poisson Point Process (PPP) has been shown to be suitable to model the spatial location of BS [4–6]. Nevertheless, interference as experienced by a user is not generated by all BS but only by the ones using the same radio resources. The resource allocation strategies have thus to be mapped to the point process modeling BS/eNodeB to determine which points/BS are interfering with a given communication. Consequently, the traffic demand must also be taken into account as it sets the number of resources used at a given time.

In this work, we propound a combination of several assignment strategies, realistic traffic demands, and transmission power control mechanisms into a stochastic geometry model. We begin by reviewing related works and our contributions.

##### 1.1. Related Work

In a downlink LTE system, a resource block (RB) is the smallest radio resource unit that can be allocated to a user. The LTE system has to schedule and assign RB to users as a function of the link qualities, traffic demands, and potential quality of service requirements. In this paper, we focus on a system where a controller assigns RB for a set of eNodeB. We do not overview RB assignment techniques in LTE network as they aim to optimize RB assignments and modulation/coding rates for a given topology and a traffic demand. Instead, this paper deals with the macroscopic design of the network: the impact of eNodeB density, allocation scheme, and power allocation on the global performance of a downlink LTE system. Nevertheless, the reader can refer to [7, 8] for recent surveys. Also, interesting contributions on the optimization of the downlink system for a given configuration are described in [3, 9–12].

Stochastic geometry has emerged as an efficient tool to analyze the performance of cellular networks. It offers, through simple models, a way to study wireless architectures at a large scale. Recent surveys [13, 14] summarize the numerous wireless architectures and models for which stochastic geometry has been applied. One of the main difficulties in the analysis of large wireless systems is to characterize interference. This quantity does not depend only on BS location and radio environment (path loss, shadowing/fading, etc.), but also on the way that radio resources (time, frequency, and power) are allocated. The point process modeling interfering nodes is thus of crucial importance. The PPP offers an accurate model to describe BS location [4–6]. This process is tractable, and it is possible to derive closed formulas for some key performance metrics of the system: interference, coverage, outage, Signal over Interference plus Noise Ratio (SINR), etc. But the PPP models all BS/eNodeB and not the subset of interfering eNodeB for a given communication. The process has thus to be thinned to take into account interference coordination (IC) techniques and radio resources assignment, for example, leading to processes that are no more Poisson. In the next paragraph, we focus on recent contributions, and on studies where resources allocation and more generally IC techniques are taken into account.

IC refers to techniques that aim to mitigate interference at the receivers. Surveys on such techniques can be found in [2, 15]. A common IC approach consists in controlling the allocated radio resources (frequency/time/power) in order to alleviate the interference impact on communications. In [16], the authors consider a random resources allocation strategy where the BS are distributed as a PPP. This simple and tractable strategy allows model interfering BS as an independent thinning of a PPP and deriving closed formulas for the coverage probability. They also deduce the minimal reuse factor achieving a given coverage probability. The performance of strict FFR (Fractional Frequency Reuse) and SFR (Soft FFR) allocation strategies is evaluated using stochastic geometry in [17]. With these two techniques, different radio resources are allocated to users that are at the edge of a cell (Voronoï cells here) with regard to the ones close to the BS. The criteria distinguishing core and edge users is based on the SINR at each user computed from the underlying PPP modeling all BS. For strict FFR, the radio resources used at the edge and in the core are disjoint. Instead, the radio resources may be reused between the two regions for SFR. For these two strategies, the authors derive closed formulas for the coverage probability and discuss pros and cons of these approaches. A superior interference reduction is observed for FFR but SFR benefits from a greater resource efficiency. This work is generalized in [18, 19] to the context of K-tier and heterogeneous networks considering different point processes for each tier or network technology. It is also extended and studied in [20] with the dynamic strict FFR (DSFFR) where the edges of the cells are dynamically divided into sectors with the help of directional antennas. In [21], a coordinated beamforming is employed to ensure that a set of closed BS, “a cluster”, will use different resources. A user associated with a BS is then not subject to interference from BS belonging to the same cluster. The authors derive analytical expressions for the Signal over Interference Ratio (SIR) for this strategy and discuss the impact of the clusters cardinality. A similar approach is used in [22], where the set of coordinated BS corresponds to the most interfering ones. Interference level takes into account path loss and long-term shadowing. The interfering BS are outside this set. They are selected randomly and independently leading to a thinned PPP. For this model, the authors study the coverage probability for different scenarios. In [23], a user is served by its 1 or 2-closest BS according to the position of these BS with regard to the user. When the two BS are coordinated, the transmission power is split into the two transmissions. The total transmission power is thus the same with one or two coordinated BS. Interference is generated by the other BS without restriction which is assumed to be distributed through a PPP. The authors derive a closed-form expression for the SIR distribution and the network coverage probability and discuss the benefit of this approach. In [24], an IC technique is evaluated for a user at the edge of its cell. When the resource of this communication is used by neighboring cells, they may not transmit any signal for a certain period to mitigate interference at this user. This coordination technique is analytically evaluated assuming that interfering nodes are still distributed as a PPP.

Besides the modeling of IC, [22, 25–27] propose spatial and tractable models that take into account the traffic demand in the interference computation, but they do not consider concrete RB assignment algorithms. In particular, the authors in [26] study SIR coverage for a cellular network based on PPP. A queue is associated with each BS that determines the BS transmission activity as a function of the traffic. Considering the traffic at each BS is independent, interferers at a given time are then an independent thinning of the initial PPP and are still Poisson. This model differs with this paper as we do not take into account eNodeB activity as a function of the traffic but instead the resource allocation as a function of the number of associated users to each eNodeB. Also, stochastic geometry models can be specific to certain power control scheme [28] or radio technologies as in [29] where the authors consider a K-tier heterogeneous network with transmissions operating on the millimeter wave band.

##### 1.2. Contributions

The primary contribution of this work is to offer an analytical model based on stochastic geometry to evaluate the performance of a downlink LTE system taking into account RB allocation strategies, power control, and traffic demands. All these mechanisms have never been combined into a single stochastic geometry model. The number of allocated resources for an eNodeB is assumed to follow the distribution of the number of clients in an M/M/C/C queue. It models the number of communications in progress when both the interarrival of the communications and their duration follow an exponential distribution. Such assumptions are pertinent in cellular networks as it has been recently shown in [30]. We associate to these traffic demands several resource allocation strategies. All these algorithms are combined with a power control mechanism that depends on the channels quality. Allocation strategies lead to non-PPP as correlation appears between the locations of the interfering nodes. It prevents the use of the convenient properties of the PPP to compute interference distribution. Nevertheless, we propose approximations that allow us to deal with these correlations and to obtain an analytical method that is shown very accurate with regard to simulations.

We compare our model to classical optimization approaches where, for a given configuration/sample, the allocation is optimized with regard to an objective function. To our knowledge, such comparison has never been done before. It shows that geometry stochastic based model may be relevant to offer tight approximations on wireless system performance.

Models are evaluated through a large set of simulations that highlights benefits of our approach to design some key parameters of the wireless system. Results show that the obtained values for SIR, coding, and modulation rates correspond to the reference values of the standards and technical LTE documents, empirically proving that our model is able to approximate performance of real systems. This work has been partially presented in [31].

##### 1.3. Paper Organization

The remainder of this work is organized as follows. In the next section we present the system model. We expose the assignment strategies in Section 3. In Section 4, we derive the first and second moments of interference for each allocation strategy. SIR distribution is assessed in Section 5. Numerical results and simulations are presented and discussed in Section 6. We conclude the paper in Section 7.

#### 2. System Model

##### 2.1. eNodeB Location and Interference

eNodeB location is modeled by a point process distributed in with intensity . Its distribution is detailed in Section 2.2. The eNodeB are numbered with regard to their distance to the origin, eNodeB at being the closest one. We consider a downlink system between a typical user and its attached eNodeB. Without loss of generality, we assume that this user is located at the origin. The users are assumed to be associated with their closest eNodeB with regard to the Euclidean distance. The channels between eNodeB and the typical user are modeled through a sequence of i.i.d. random variables . The transmission power between an eNodeB and one associated user is given by the function . This function models the power control algorithm implemented by the eNodeB and depends on the distance between a user and its attached eNodeB. The scheduler/controller manages a set of resource blocks that are common to all eNodeB. They are thus shared between eNodeB. The RB are numbered from to . The scheduler assigns one RB for each user, but the model can be easily extended with a random number of RB for each demand. Traffic demands exceeding at an eNodeB are not served. The RB with index is allocated to the typical user. We show, in the appendix, that this choice does not impact the computations, and any other index could be chosen instead. An eNodeB interferes with the typical user if and only if it reuses this RB. Interference at the typical user can be expressed aswhere is the path loss function. The argument of interference is the distance between the typical user and its eNodeB (eNodeB ). expresses interference when this distance is random and depends on the r.v. . expresses interference when this distance is given and equal to . This notation is motivated by the fact that the r.v. and are correlated to . Moreover, the mean and the variance of interference will be computed for both a given value of and with regard to its distribution. The r.v. indicates whether eNodeB interferes with the typical user ( or ). is the random variable modeling the location of a user attached to the eNodeB (at ). We assume that is uniformly distributed in the Voronoï cell formed by the process and with nucleus . The main notations used throughout this paper are given in Table 1.