Random access transport capacity of multihop AF relaying: a throughput-reliability tradeoff
© Lee et al.; licensee Springer. 2013
Received: 4 July 2012
Accepted: 7 March 2013
Published: 17 April 2013
To determine the capacity of distributed wireless networks (i.e., ad hoc networks), the random access transport capacity was proposed as the average maximum rate of successful end-to-end transmission in the distance. In this article, we consider the random access transport capacity for multihop relaying to find the end-to-end throughput of a wireless ad hoc network, where each node relays the signal using an amplify-and-forward (AF) strategy. In particular, we analyze the exact outage probability for multihop AF relaying in the presence of both co-channel interference and thermal noise, where interferers are spatially distributed following a Poisson distribution. In our numerical results, it is observed that the maximum random access transport capacity is achieved at a specific spatial density of transmitting nodes due to the throughput-reliability tradeoff as the number of transmitting nodes (=interferers) increases. We compute the optimal spatial density of transmitting nodes that maximize their random access transport capacity. As a result, we can obtain the actual random access transport capacity of multihop AF relaying and predict the maximum number of transmitting nodes per unit area to maximize their performance.
Cooperative communication is a promising and emerging technique for enhancing the coverage and reliability in wireless networks [1, 2]. In particular, dual-hop transmission systems employing amplify-and-forward (AF) relaying, where a relay simply retransmits a scaled version of the received signal to the destination, are being spotlighted, due to their low complexity and delay benefits. In addition, the performance analysis of dual-hop AF systems has been an important area of research in recent years [3–5]. However, since dual-hop transmission over long distances requires a very high transmission power, multihop transmission in which a source communicates with a destination via a number of relays has been proposed as an effective method of establishing connectivity between the nodes of a network [6, 7]. More recently, the multihop transmission with AF strategy has drawn considerable attention in the literature [8–10]. In [8, 9] examined the ergodic capacity and outage probability of multihop transmission with AF strategy using Jensen’ inequality and the inequality between the harmonic and geometric means.  computed the optimal number of hops for linear multihop AF relaying with equal resource allocation in terms of the random coding error exponent. All of these previous works on multihop AF relaying focused on noise-limited fading environments for ideal configurations without interference. Since network interference is inevitable in practical wireless networks, due to spectral reuse, AF relaying in the presence of co-channel interference has been studied [11–14]. However, [11, 12] considered only dual-hop transmission and [11–13] neglected either noise or interference at each node for analytical tractability. Furthermore, most of the prior works on AF relay networks, including , analyzed the outage probability using some approximation methods, such as the harmonic mean, and assumed that the locations of the network nodes are deterministic without spatial randomness.
Meanwhile, under the assumption that the interferer locations are random, there have been some state-of-the-art works on the ad hoc network capacity [15–23]. In , the transmission capacity was proposed as the maximum allowable density of transmitting nodes to satisfy the data rate and outage probability constraints. The transmission capacity framework has been successfully used to characterize the physical layer on the ad hoc networks [15–20], such as through the use of multiple antennas , interference cancellation  and spectrum sharing in tiered cellular networks .  further studied the tradeoffs of the transmission capacity between the throughput, delay, and reliability in wireless ad hoc networks. However, since most of the prior works on computing the transmission capacity were limited to single hop transmission without noise,  derived the transmission capacity of dual-hop relaying while considering the thermal noise. Recently, to account for multiple hops and retransmissions,  developed a new metric for quantifying the end-to-end throughput termed random access transport capacity. The random access transport capacity was defined as the average maximum rate of successful end-to-end transmissions over some distance.
In this article, we consider a realistic communication environment in wireless ad hoc networks where all of the transmitting nodes are randomly scattered and uncoordinated following a Poisson law over a plane. For decentralized wireless networks, we consider the random access transport capacity to find the overall end-to-end throughput for multihop AF relaying. To compute the random access transport capacity, we analyze the exact outage probability of multihop transmission with AF strategy in a Poisson field of interferers without neglecting the noise at all of the nodes. From our numerical results, we observe that even when the number of transmitting nodes increases, the overall throughput of multihop relaying does not increase consistently due to interference. In other words, the random access transport capacity has a maximum value at a certain spatial density of transmitting nodes, due to the throughput-reliability tradeoff, and shows that the increase in the throughput is inversely proportional to the reliability as the number of transmitting nodes(=interferers) increases. Thus, in this article, we compute the optimal spatial density of transmitting nodes that maximize their random access transport capacity, and this helps us to predict and manage the maximum available number of transmitting nodes per unit area to maximize their performance. Moreover, since each relay node amplifies the interference, as well as the thermal noise, our results show that the performance of multihop AF relaying can be degraded due to the accumulated interference caused by the increasing number of transmission hops.
The remainder of this article is organized as follows. Section 2 presents the system model with channel model and interference model. We analyze the random access transport capacity of multi-hop AF relaying with both noise and Poisson interference in Section 3. Section 4 presents the optimal density of transmitting nodes to maximize their random access transport capacity. Section 5 compares the simulation results with the analytical results. Finally, Section 6 concludes this article.
2 System model
2.1 Channel model
where F12 captures the small scale fading which obeys a Rayleigh fading model and l(∥x1−x2∥)=∥x1−x2∥−α characterizes the effect of large scale path loss with path loss exponent α. In this article, we consider 2<α<5 for the main physical range of interest for path loss exponent in wireless networks. We assume that the channel responses for all of the nodes are independent and quasi-static.
2.2 Interference model
We consider a slotted ALOHA transmission without centralized scheduling [19, 22, 23], where all of the transmitters which cause interference are randomly scattered and uncoordinated. Since all of nodes are drawn from a homogeneous 2-D PPP of intensity λ, all nodes generating interference at the n th hop transmission are also modeled as a homogeneous PPP with λ. Then, the sets of interferers in each hop are denoted as . In particular, we define the interferer set as where I n is the location of interferers. In the case of the transmission power, we assume that all of the interferers in the n th hop use the same power . Moreover, as the only α>2 for which the aggregate interference has a distribution expressible in closed-form is for α=4 [19, 25], we assume a large wireless ad hoc network where each multihop branch is located relatively far away with each other.
3 Random access transport capacity of multihop AF relaying
where p s is the probability that the packet is successfully decoded by the destination D, λ is the density of transmitting nodes (=interferers), ζ is the target SINR, and d is end-to-end distance. To compute the success probability of the random access transport capacity, the outage probability of multi-hop AF relaying should be analyzed first.
Proof 1. See Appendix Appendix 1.
where x0 = 1.
Proof 2. See Appendix Appendix 2.
where is the PDF of n th hop. In this case, the complexity of computing the exact outage probability increases as the number of hops increases, because setting the interval of integration for each variable from (7) is very difficult. However, from our result (13), we can obtain the exact outage probability of multihop AF relaying with low complexity by using only the .
Proof 3. See .
where 1−Pou t(R) is the success probability, λ is the density of interferers, ζ is the target SINR with 2 K R −1, and d S D is the distance between the source and destination. From the above random access transport capacity, we can find the spatial density of successful transmissions at rate log(1+ζ) that spans a distance d S D when each node relays its data using the AF strategy.
4 Optimal spatial density of transmitting nodes for multihop AF relaying
On the contrary, in the region of , the random access transport capacity is a monotonic decreasing function due to . Thus, the optimal spatial density of transmitting nodes exists in the region of .
Using this bound (27), we can predict and manage the maximum number of nodes per unit area that maximizes their random access transport capacity of multihop AF relaying.
5 Numerical results
In this section, we present some numerical results concerning the outage probability and random access transport capacity for multihop AF relaying with both interference and noise in a Poisson network. To illustrate our analytic and simulated results, we consider a Rayleigh fading channel having equal resource allocation with equal transmission power P n =P T , equidistance d n =d S D /K, identical interference power , and fixed noise variance . Moreover, we consider 30 randomly distributed nodes following a Poisson distribution for each hop with the radius of the interference region , where λ is the density of transmitting nodes (=interferers).
5.1 Outage probability
5.2 Random access transport capacity
From a different viewpoint, Figure 4. presents the random access transport capacity of multihop AF relaying with K=2 and 4 as a function of the spatial density of transmitting nodes λ for two different interference powers P I N F =2 dB and 5 dB with the fixed transmission power P T =15 dB. This figure also shows that multihop AF relaying with K=4 can achieve higher end-to-end throughput than dual-hop relaying. Meanwhile, the general overall throughput of ad hoc networks increases when many nodes transmit data simultaneously, but the random access transport capacity does not increase consistently owing to the increasing interference as the spatial density of transmitting nodes increases. Especially, in Figure 4, the random access transport capacity has the maximum value at the certain spatial density, because the effect of increasing throughput is larger than that of decreasing reliability in the relatively low-spatial density regime. In both figures, we observe that there is a tradeoff between the throughput and reliability and a need to compute the optimal density of transmitting nodes for the purpose of maximizing the random access transport capacity.
5.3 Optimal spatial density of transmitting nodes
This article considered a multihop transmission with AF strategy in the simultaneous presence of both noise and interference, allowing the use of a Poisson interference model. In particular, as all of the nodes are randomly distributed in a wireless ad hoc network, we considered a metric termed random access transport capacity of decentralized wireless networks, to measure the maximum rate of successful end-to-end transmission over some distance in wireless ad hoc networks. Moreover, to compute the random access transport capacity, we derived the exact outage probability of multihop AF relaying, because the performance gap between the conventional bounds and the exact value increases as the number of hops increases. The analytic and simulated results showed that multihop transmission with the AF strategy amplifies the interference as well as the thermal noise and causes an error floor phenomenon in terms of the outage probability. In addition, since there is a tradeoff relationship between the throughput and reliability in terms of the random access transport capacity, we computed the optimal spatial density of transmitting nodes in wireless ad hoc networks. From this article, we can obtain the actual random access transport capacity and predict the maximum number of transmitting nodes per unit area to maximize their performance.
7 Appendix 1
7.1 Proof of end-to-end SINR
Expanding the above results, we can obtain the end-to-end SINR of multihop AF relaying (7).
8 Appendix 2
8.1 Proof of outage probability of multihop AF relaying
This research was supported by “The Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) support program supervised by the NIPA” (NIPA-2012-H0301-12-3002). The work of S. Kim was supported by the Sogang University Research Grant of 2011.
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