Analytical evaluation of heterogeneous cellular networks under flexible user association and frequency reuse

doi:10.1016/j.comcom.2017.11.014

Computer Communications

Volume 116, January 2018, Pages 147-158

https://doi.org/10.1016/j.comcom.2017.11.014 Get rights and content

Abstract

Offloading mobile users from highly loaded macro base stations (BSs) to lightly-loaded small cell BSs is critical for utilizing the full potential of heterogeneous cellular networks (HCNs). However, to alleviate the signal-to-interference-plus-noise ratio (SINR) degradation of so called biased users, offloading needs to be activated in conjunction with an efficient interference management mechanism. Fractional frequency reuse (FFR) is an attractive interference management technique due to its bandwidth efficiency and its suitability to orthogonal frequency division multiple access based cellular networks. This paper introduces a general mathematical model to study the potential benefit of load balancing in conjunction with two main types of FFR interference coordination: Strict-FFR and soft frequency reuse (SFR)- in the downlink transmissions of HCNs. For some special but realistic cases we were able to reduce the rather complex general mathematical expressions to much simpler closed-forms that reveal the basic properties of BS density on the overall coverage probability. We show that although Strict-FFR outperforms the SFR mechanism in terms of SINR and rate coverage probability, it fails to provide the same spectral efficiency. Finally, we present a novel resource allocation mechanism based on the BSs bias values and FFR thresholds that achieves an even higher minimum user throughput and rate coverage probability.

Introduction

Cellular network operators are forced to increase their network capacity in order to cope with the rapidly rising demand on data rate. In the recent years the mobile data usage has grown up to 200% per annum [1]. Introducing new tiers that comprise of base stations (BSs) with smaller transmission ranges (called small cell BSs), is a potential and cost effective approach to increase the capacity of cellular networks [2].

The design of cellular networks with optimal parameter settings requires to have efficient methods and models to analyze the performance of heterogeneous cellular networks (HCNs). Because of the uncontrolled nature of small cell BS distributions in HCNs, conventional models such as Wyner [3] and hexagonal models are considered to be obsolete for modeling. A recent approach to describe the random nature of BS locations is to apply point process theories, leveraging techniques from stochastic geometry [4]. Its accuracy in abstracting realistic BS deployments has been validated in numerous contributions [5], [6], [7]. In [5], the authors showed that even in a single tier network Poisson point processes (PPPs) provide an accuracy at least as high as grid models. However, the grid model does not exhibit the same level of analytical tractability as PPPs. Owing to their advantages in tractability and accuracy, PPPs have been extensively employed to model and analyze HCNs [8] in recent years.

The authors in [5] derived important performance metrics such as coverage probability and average rate for a given system model in a single-tier cellular network. Their analysis is generalized in [9] for a K-tier cellular network. They proved that in open access and interference-limited networks with identical target signal-to-interference-plus-noise ratio (SINR) for all tiers the overall coverage probability is independent of the number of tiers and the density of BSs. The model is further developed in [10] to incorporate a user-based load definition into the analysis. In [11], the authors presented a framework to evaluate the coverage probability of indoor users in urban two-tier cellular networks. In this model, environments are partitioned by walls with a certain penetration loss to distinguish between the outdoor BSs in line of sight (LOS) and non-line of sight (NLOS). They showed that an increasing number of small cell BSs can reduce the impact of the building safeguard against the aggregate interference.

Most of the users in HCNs are assigned to macro BSs due to the lower transmission power of small cell BSs [12]. Hence, the network encounters an imbalanced load distribution among its tiers. A common approach to tackle this issue is biasing the users, as proposed by 3rd Generation Partnership Project (3GPP) in Release 10 [13]. Biasing follows the idea of pushing the user assignment towards low-power BSs by artificially increasing their coverage region.

In [14], [15] the authors considered the problem of user association and power allocation in the downlink of HCNs with the goal of maximizing the network energy efficiency. The authors in [16] evaluated the performance of cellular networks under an interference coordination mechanism when the users are associated to the closest BS. In [17], the authors developed a model to compute the coverage probability and the average throughput in a K-tier cellular network using a flexible user association. They showed that biasing the users without an efficient interference coordination mechanism always reduces the overall coverage probability of HCNs. It is a direct result of forcing the users with a lower SINR to a BS. In [18] the authors proposed a model to evaluate the performance of a two-tier cellular network under a simple resource partitioning mechanism. They assumed that a ξ fraction of resources is allocated to the macro cell users and the unbiased small cell users. The remaining $(1 - ξ)$ fraction of the resources, in which the macro cell shuts down the transmission, is assigned to the biased small cell users. Since the interference level in the fraction of resources for the biased users is lower than the shared part, those users experience a better SINR distribution.

The interference coordination mechanism under consideration in [18] is not bandwidth efficient due to reserving a fixed fraction of resources solely for the biased users. Achieving a high spectral efficiency, the use of the total bandwidth in all cells, is one of the key objectives of long-term evolution (LTE) systems [19]. Fractional frequency reuse (FFR) is a popular interference coordination strategy due to its good spectral efficiency and its suitability to orthogonal frequency division multiple access (OFDMA) based cellular networks [20], [21], [22]. It is included in the 3GPP-LTE standard since Release 8 [23]. The FFR mechanism partitions the cell into two regions (interior and edge regions) and applies different frequency reuse factors to each region [24], [25]. In this paper we consider two main types of FFR mechanism: Strict-FFR and soft frequency reuse (SFR).

Strict-FFR mechanism: In a Strict-FFR system the entire frequency band W is partitioned into a common part $W_{In}^{FFR} = β W$ and a reuse part $W_{e}^{FFR} = (1 - β) W$ where 0 ≤ β ≤ 1. The common part of resources is shared by the cell interior users of each tier. The reuse part is divided among the network tiers and $W_{e, k}^{FFR}$ is utilized by the kth tier, where $W_{e}^{FFR} = \sum_{k = 1}^{K} W_{e, k}^{FFR}$ . Hence, the cell edge users of a different tier use a disjoint set of resources. Furthermore, the reuse part of the kth tier is further partitioned into Δ_k sub-bands and each BS randomly chooses one sub-band to transmit to the cell edge users.

SFR mechanism: In an SFR system the entire frequency band in the kth tier is divided into Δ_k sub-bands. One of the Δ_k sub-bands is randomly allocated to the cell edge users of each tier, and the cell interior users share the rest of the sub-bands with the edge users of other cells. Since the BS employs the sub-bands used for the edge users of other cells to connect to its own cell interior users, the downlink data of cell interior users is typically transmitted with a lower transmit power to decrease the interference level of the other cells. Each tier has two possible power levels, i.e., a high power level P_k and a low power level m_kP_k where 0 < m_k < 1. The BS assigns a high power P_k to transmit to its cell edge users and a low power m_kP_k for the cell interior users. Since the BS uses two different power levels to transmit to cell edge and cell interior users in an SFR mechanism, we divide the users into two groups based on the BS power level. a) Category I user: the user which is located close to a BS and is associated to that BS even though the BS uses a low transmit power level m_kP_k. b) Category II user: the user that is associated to a BS applying a high power level P_k. In such case if the BS apply a low power level m_kP_k, the user of this tier receives the maximum long-term biased received power from BSs of other tiers. A rigorous mathematical description is presented in Section 2.

The author in [26] proposed a frequency resource allocation mechanism to decrease the interference of a two-tier cellular network. In this model the macro cell BSs use the FFR mechanism for the frequency resource allocation. The small cell BSs located in the cell edge region utilize the whole frequency band while the small cell BSs located in the interior region use different sub-bands from the macro cell interior to mitigate the interference level. In [27] the overall coverage probability and the average rate of HCNs with an FFR interference management technique is derived when the mobile users are associated to the nearest BS of each tier.

Contributions: The main contribution of this work is the development of a general analytical model to evaluate the performance of HCNs downlink transmissions under a flexible user association and two main types of FFR interference coordination: Strict-FFR and SFR. We extend the work in [27] by presenting the per-tier coverage probability of cell edge and cell interior users. In the design of cellular networks it is important to find the minimum achievable user rate and the number of cell edge and cell interior users for the optimal setting of network parameters such as the frequency reuse threshold. But [20], [27] just provide the average ergodic rate of networks. In this paper, we derive the per-tier average rate of the cell edge and the cell interior users as well as average number of cell edge and cell interior users of each tier, rate coverage probability, and the minimum achievable user rate. Although the mathematical results for very general scenarios are not in closed form, we provide closed form expressions for the coverage probability of users under several special cases. They in turn allow a deeper insight in the performance dependency on various parameters. Last but not least a contribution of this paper is proposing a novel resource allocation mechanism based on the cell edge and cell interior load of a BS and the user SINR distribution that achieves higher minimum user throughput and rate coverage probability.

In this paper, we present many novel observations and provide design insights. Particularly, our analysis demonstrates that even in an interference limited network when all tiers experience the same path loss exponent and we employ an unbiased user association, the coverage probability of the network is not independent of the BSs density. This result contradicts recent observations which show the coverage probability is independent of the BS density and the number of tiers [5], [9], [17]. Also, we observe that in an SFR mechanism with a low power control and a high frequency reuse factor, the users are always in coverage with almost sure probability. We show by our simulation examples that in an interference limited network, when all tiers experience the same path loss exponent, the average ergodic rate is only loosely correlated to the BS density. Furthermore, we show that the overall coverage probability of Strict-FFR outperforms the SFR mechanism, while the SFR mechanism exhibits better performance in terms of average ergodic rate and minimum average user rate.

The rest of the paper is organized as follows: Section 2 describes the system model employed along with deriving some important network metrics such as the user association probability, and distance distribution. The coverage probability of the cellular network is computed in Section 3. We then continue with other important performance metrics, such as the average ergodic rate of HCNs, average user throughput and the rate distribution in Section 4. Section 5 presents closed form expressions and proposes a novel resource allocation mechanism. Numerical results are presented in Section 6 before the paper is concluded in Section 7.

Section snippets

System model

In this paper, we consider downlink transmissions in OFDMA based K-tier cellular networks. Fig. 1 shows the network architecture in a sample two-tier cellular network. Let us denote k (where $k \in K,$ and $K = {1, 2, 3, \dots, K})$ the index of tier k, and assume that the BSs in tier k are spatially distributed as a PPP, $ϕ_{k} \in R^{2},$ of density λ_k. The locations of the users in the network are modeled by another independent homogeneous PPP, $ϕ_{u} \in R^{2},$ with a non-zero density λ_u. Without loss of generality, we perform all

Coverage probability

The coverage probability is the probability that the SINR of a user located at the origin is greater than a predefined target SINR value. In the following we derive the coverage probability for two different interference management mechanisms.

Average ergodic rate

Finding the average ergodic rate (average cell throughput) is important for the planning and design of the cellular network. The average ergodic rate of network under an SFR interference management $R^{SFR}$ is given as $R^{SFR} = \sum_{i = 1}^{K} (R_{e}^{SFR} (i) + R_{In}^{SFR} (i)) .$ The average ergodic rate of tier i cell edge users $R_{e}^{SFR} (i)$ and tier i cell interior users $R_{In}^{SFR} (i)$ are computed by $R_{e}^{SFR} (i) = A_{1, i} R_{1, e}^{SFR} (i) + A_{2, i} R_{2, e}^{SFR} (i),$ $R_{In}^{SFR} (i) = A_{1, i} R_{In}^{SFR} (i),$ where $R_{1, e}^{SFR} (i),$ $R_{2, e}^{SFR},$ and $R_{In}^{SFR} (i)$ are the rate of cell edge users in

Special cases of interest

As the general coverage probability has not been derived in a closed form in the previous section, this section provides more insight by some special cases where $α = 4$ for all the tiers and $σ^{2} = 0$ . Because of the high BS density in HCNs, in general the noise power is negligible compared to the interference power (wireless networks are interference limited systems). Besides, the choice of the path loss exponent $α = 4$ is commonly accepted in practice as long as users are not too close to the BS. Due to

Numerical results

For the verification of our results we carried out Monte Carlo simulations under 3GPP compliant scenarios. In particular, we consider a two-tier cellular network (macro cell BSs and small cell BSs). In our simulation of two-tier cellular networks, we assume $W = 10$ MHz and for both tiers we set the SINR threshold to $- 3$ dB, the frequency reuse threshold to 3 dB, and the rate threshold to 1 Mbit/s. The macro cell BSs are distributed with density $λ_{1} = \frac{1}{π 500^{2}} m^{- 2}$ over a field of 10 000 × 10 000 m² with a

Conclusion

This paper proposes a general framework for the performance analysis of HCNs with flexible user association and FFR mechanisms. We evaluate the per-tier and overall coverage probability, as well as the rate coverage probability of such networks, and the average user and cell throughput for the cell edge and the cell interior users. Supported by extensive numerical results, it is shown that SFR outperforms other mechanisms in terms of spectral efficiency and average minimum rate of users.

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^☆: This work has been supported by the INWITE project.

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Analytical evaluation of heterogeneous cellular networks under flexible user association and frequency reuse☆

Abstract

Introduction

Section snippets

System model

Coverage probability

Average ergodic rate

Special cases of interest

Numerical results

Conclusion

IEEE Wirel. Commun.

Coexistence of Wi-Fi and heterogeneous small cell networks sharing unlicensed spectrum

IEEE Commun. Mag.

Self-organization in disaster-resilient heterogeneous small cell networks

IEEE Netw.

Shannon-theoretic approach to a Gaussian cellular multiple-access channel

IEEE Trans. Inf. Theory

Stochastic geometry and random graphs for the analysis and design of wireless networks

IEEE J. Sel. Areas Commun.

A tractable approach to coverage and rate in cellular networks

IEEE Trans. Commun.

Spatial stochastic models and metrics for the structure of base stations in cellular networks

IEEE Trans. Wirel. Commun.

Using Poisson processes to model lattice cellular networks

Proceedings of IEEE INFOCOM Conference

Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: a survey

IEEE Commun. Surv. Tutor.

Modeling and analysis of K-tier downlink heterogeneous cellular networks

IEEE J. Sel. Areas Commun.

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International Workshop on Link-and System Level Simulations (IWSLS)

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IEEE J. Sel. Areas Commun.