Wireless Communications and Mobile Computing

Volume 2017 (2017), Article ID 7189090, 16 pages

https://doi.org/10.1155/2017/7189090

## On Carrier Sensing Accuracy and Range Scaling Laws in Nakagami Fading Channels

^{1}School of Information, Central University of Finance and Economics, Beijing, China^{2}College of Information Engineering, Shenzhen University, Shenzhen, China^{3}School of Software, Beijing University of Technology, Beijing, China

Correspondence should be addressed to Liang Chen; nc.ude.uzs@nehcl

Received 6 April 2017; Revised 8 August 2017; Accepted 24 August 2017; Published 26 November 2017

Academic Editor: Xianfu Lei

Copyright © 2017 Yue Wang et al. 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 make a detailed study on carrier sensing of 802.11 in Nakagami fading channels. We prove that to maximize sensing accuracy, the optimal channel accessing probability is solely determined by the path-loss SIR (Signal to Interference Ratio). We define -interference range and -carrier sensing range for fading channels and prove that their scaling laws in Nakagami fading channels are similar to those in the static channel. The newly derived theoretical results show a unified property between the static and fading channels. By extensive simulations, we reveal that fading depresses the probability of a dominating transmission state, and therefore it can mitigate severe hidden and exposed terminal problems, but fading harms the average sensing accuracy for an optimally adjusted carrier sensing threshold.

#### 1. Introduction

IEEE 802.11 has gained broad and far-reaching applications in our life, due to its agile deployment and cheap maintenance. A lot of previous works in wireless network models assume that signal strength is invariant over time in their study. It may be enough to consider the simple static channel in indoor WiFi. However, multipath fading comes into the picture when we deal with the outdoor environment where signal strength varies with time because stations are moving or surroundings are changing.

Although some network models have begun to incorporate multipath fading, and measurements and high-fidelity simulations show it is necessary to do so, some basic parts are missing, which are of concern to networking people, such as understanding the interplay between interference and carrier sensing, and the extension of interference/carrier sensing ranges to fading channels and their properties.

In this work, we study a two-link topology so as to examine basic principles as done in [1, 2]. Compared with related works (Section 2), we present some new insights on how and why the randomness of fading may affect carrier sensing, rather than incorporating fading into models and simply making calculations. We deal with the Nakagami fading model, which represents a general family of fading channels, and carry out simulations for the main bit rates of 802.11 a/b/g/n. The broad settings of parameters make our results confident. Note that our work differs from existing communication papers on fading, and the paper mainly talks about the interplay between interference and carrier sensing, which is less addressed in the literature.

Specifically, we make the following contributions.

##### 1.1. Interplay between Carrier Sensing and Interference: Optimal, Case Study, and Average Carrier Sensing Accuracies (Section 4)

We prove that, in order to maximize carrier sensing accuracy in Nakagami fading channels, whether the transmitter accesses the channel or not is solely determined by whether the path-loss SIR (Signal to Interference Ratio) is larger than or less than the SNR threshold, independent of channel time-variations (Theorem 3).

Simulations show that the four transmission states of (idle, success), (idle, failure), (busy, success), and (busy, failure) are mixed together in fading channels. Since fading caused randomness depresses the probability of a dominating state, it can mitigate severe hidden and exposed terminal problems.

We further study the* average* sensing accuracy averaging on different interferer locations within a disk plane. Intensive simulations show that fading degrades the average sensing accuracy for an optimally adjusted carrier sensing threshold.

##### 1.2. Definitions of Interference/Carrier Sensing Range and Their Scaling Laws in Nakagami Fading Channels (Section 5)

We define -interference range and -carrier sensing range, as interference and carrier sensing are mentioned in a probability manner in fading channels. We prove the range scaling laws of fading channels, which are similar to the ones of the static channel (Theorems 6 and 8). Besides the application to visualize channel conditions, the scaling laws make it enough to experiment on one topology only rather than trying all different topologically relevant parameters one by one.

The paper is organized as follows. Section 2 discusses the related work and highlights our difference. Section 3 presents the system model, symbols, conventions, and assumptions. Section 4 studies the fading effects on carrier sensing considering the interplay with interference. Section 5 defines interference range and carrier sensing range for fading channels and proves their scaling laws and applications. Finally, Section 6 concludes the paper.

#### 2. Related Works

Multipath fading needs to be considered in order to model 802.11 accurately. We make a comprehensive review of related works. Here we first summarize some representative ones of them and then highlight the difference of ours.

##### 2.1. Literature Review

Since the last century, there have been some early works on the capture effect of multipath fading. Hansen and Meno derived the distribution function of reception power considering both lognormal shadowing and Rayleigh fading [3]. Sowerby and Williamson calculated the outage probability (i.e., packet reception failure probability) of a wanted communication in face of multiple interferers in the Rayleigh fading channel based on the capture threshold model or the SNRT (Signal to Noise Ratio Threshold) model with the independence assumption [4]. Yao and Sheikh extended the result in [4] to the Rician fading [5]. These works did not consider carrier sensing since they talk about a cellular cell.

In this century, multipath fading is incorporated into analysis models of CSMA, evolving from simplified models to complex models, from one WLAN to an ad hoc network scenario, from Rayleigh fading to a general fading model.

The first class of 802.11 modelling works assumed* perfect* carrier sensing and only studied fading effect on capture. Perfect carrier sensing means that all ongoing transmissions on the channel can be sensed by a ready-to-transmit wireless node and carrier sensing accuracy is not influenced by fading. Kim and Lee analyzed the CSMA/CA throughput under multistation interference in composite Rayleigh and shadowing channels [6], where they used the SNRT reception model and the carrier sensing mechanism of exchanging of RTS and CTS. Song et al. incorporated fading errors and extended Bianchi’s CSMA model for saturated traffic [7]. Daneshgaran et al. further analyzed throughput performance for unsaturated traffic in the Rayleigh fading channel [8]. Leonardo and Yacoub also analyzed throughput performance of unsaturated traffic, but they provided an extensive calculation for the Hoyt, Rice, and Nakagami- fading channels [9]. In contrast to the above works dealing with a symmetric WLAN cell, Sheng and Vastola proposed a model for an ad hoc network in fading channels [10], by extending Chang et al.’s model in the static channel [11].

The second class of 802.11 modelling works considered* imperfect* carrier sensing, but they assumed a naive capture mechanism: two transmissions will collide if and only if they overlap in time. Early researchers studied the performance of nonpersistent [12], 1-persistent [13], and -persistent CSMA [14] under imperfect carrier sensing. Chong et al. analyzed the throughput and delay performance of CSMA/CA as a function of sensing error probability [15]. Sheng and Vastola incorporated the outage probability of carrier sensing into Bianchi’s model for the Rayleigh channel [16], where sensing error was determined by fading, instead of a predefined constant [12–14] or due to improperly set carrier sensing thresholds [15] assumed in previous works. The above works dealt with a symmetric WLAN cell and verified their analysis by simulation. Recently, Kai and Liew noted the partial carrier sensing relationship in their real experiment [17], and they said, “the carrier sensing relationship between the links are often probabilistic and can vary dynamically over time. This is the case even if the distance between the links is fixed and there is no drastic change in the environment,” suggesting that imperfect carrier sensing needs to be considered carefully.

The third class of 802.11 modelling works studied the* combined* effects of fading on capture and carrier sensing. Based on Bianchi’s model, Sheng and Vastola [18] modelled the throughput of 802.11 DCF in a symmetric circular topology and jointly considered the effects of Rayleigh fading on capture and carrier sensing. They noted that a fixed carrier sense threshold might not achieve the best performance at different network radii. Dai and Yamao proposed a probability analysis method that can predict packet delivery ratio for a multihop linear topology under fading environment, and they analyzed the impact of hidden terminal caused intraflow interference and compared two carrier sensing thresholds (−81 dBm and −85 dBm) [19]. Mittag and Hartenstein simulated V2V networks and they concluded that fading had only a slight impact on the effectiveness of CSMA [20]. Schumacher and Tchouankem showed that the severity of fading substantially influenced packet delivery based on their collected empirical data of highway traffic [21]. However, they used a fixed carrier sensing threshold and did not study its impact.

Recently, stochastic geometry [22] is noted as a powerful tool for analyzing and planning of a large dense CSMA network. We name some of them in the following. In [23], Baccelli et al. used the Matern hard-core process [22] to model the CSMA protocol to study maximum network throughput. The Matern hard-core process is a thinning of the Poisson point process and it guarantees that each transmitter in the center of a disc contains no other transmitters than itself, which is suitable to model carrier sensing. A fixed carrier sensing range that avoids collisions was used in the paper. In [24], also based on the Matern process, Nguyen et al. derived the closed-form formulas for the probability of coverage of the network and for the average throughput per user for a dense IEEE 802.11 network. A constant carrier sensing range of 300 m was used in the simulation. In [25], Haenggi proposed a unified framework that incorporates the fading process into the point process model (the point Poisson process was used) and he derived the formulas for node connectivity, broadcast reliability, broadcast transport capacity, and so forth in Nakagami- fading channels. No carrier sensing was considered in the paper since it studied the broadcast issues from a node. In [26], Haenggi and Ganti derived the interference statistics considering a constant carrier sensing range for the network topologies from lattices to homogeneous and clustered Poisson models to general motion-invariant ones. In [27], Kaynia et al. considered the performance of the ALOHA and CSMA MAC protocols in wireless ad hoc networks in the presence of fading. In their network model, packets belonging to specific transmitters arrive randomly in space and time according to a 3D Poisson point process. They found that the introduction of fading is added to the hidden and exposed node problems of CSMA, resulting in an up to 75% increase in the outage probability. In [28], Yang et al. devised a carrier sensing range in the presence of fading such that two nodes cannot sense each other larger than a certain probability. And they derived the upper bound of outage probability which has a functional relation to the carrier sensing range (see Theorem , [28]). In [29], Elsawy and Hossain proposed a modified hard-core point process to mitigate the node intensity underestimation flaw for any fading environment. As shown by their analysis, decreasing carrier sensing threshold decreases the intensity of simultaneously active transmitters as well as the hidden node problem, and there exists an optimal carrier sensing threshold which depends on the operating conditions of the network. In [30], Alfano et al. extended previous stochastic geometry models of CSMA networks and obtained throughput distributions, in addition to spatial averages. They observed that the carrier sensing threshold has a dramatic impact on the spatial fairness among the nodes. A large sensing threshold, while increasing transmission probability, can also cause strong interference and make a significant fraction of APs (those in unfavorable topological conditions) experience throughput starvation. Besides, they found that higher diversity in the fading distribution increases the spatial fairness and alleviates the starvation.

##### 2.2. The Difference of Our Work

Although there are some works addressing carrier sensing in CSMA under fading environment, our work made one more step to discuss in detail the carrier sensing issue in fading channels.

First, many previous works focused on incorporating fading into CSMA models and getting accurate model prediction, but the interplay mechanism between interference and carrier sensing in fading channels was not thoroughly discussed, and the results might not be consistent sometimes, depending on the used network topologies or channel conditions. We want to focus on and give a clear understanding on carrier sensing solely.

Second, we study this issue from the micro-view perspective and focused on carrier sensing accuracy instead of the whole CSMA. Although the two-link topology in our work is much simpler than that in the stochastic geometry works, we revealed some nontrivial results. For instance, we found by case study that fading can depress the probability of a dominating transmission state, and in some scenarios it can mitigate severe hidden or exposed terminal problems when they are significant (see Figure 3). The case of the average sensing accuracy also shows that fading tends to moderate the sensing accuracy. For a carrier sensing threshold set inadequately, fading can upgrade or degrade the sensing accuracy (see Figure 5), while for the optimally adjusted carrier sensing threshold, we found by extensive simulations (see Appendix A.2) that the average sensing accuracy in the static channel is always higher than that in fading channels, and their difference is more pronounced when the channel variation becomes larger.

Third, we worked out some new theoretical results. We proved the condition of optimal channel accessing in fading channels. And we extended the definition of interference/carrier sensing ranges to fading channels and proved their scaling laws.

#### 3. System Model

##### 3.1. Experimental Topologies

We consider a two-link topology in Figure 1. The target link is - (a) where is the transmitter and is the receiver, and the interfering link is - where is the interfering transmitter, called the interferer. , , and are defined as the link distance, interference distance, and carrier sensing distance, respectively. is determined by , , and their angle . Figure 1(b) is a simplified scenario of (a) where , , and are on a line, and their coordinates are 0, , and , respectively. Here, we fix to be a positive value and can be positive or negative. The two simple topologies are usually used as the first step of discussion (e.g., in [1, 2]) as we did here.