ICTC 2014-Invited 1570021873
A Huge Challenge With Directional Antennas: Elastic Routing Jangho Yoon1 , Won-Yong Shin2 , and Sang-Woon Jeon3 Electrical Engineering, KAIST, Daejeon 305-701, Republic of Korea Computer Science and Engineering, Dankook University, Yongin 448-701, Republic of Korea 3 Information and Communication Engineering, Andong National University, Andong 760-749, Republic of Korea Email:
[email protected];
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a sector and whose sidelobe forms a circle (backlobes are ignored in this model). The antenna beam pattern has a gain value Gm for the mainlobe of beamwidth θ ∈ [0, 2π), and also has a sidelobe of gain Gs of beamwidth 2π−θ. The parameters θ Gm and Gs are then related according to 2π Gm + 2π−θ 2π Gs = 1, where 0 ≤ Gs ≤ 1 ≤ Gm . In our work, we assume that Gm = Θ(1/θ) and Gs = Θ(1), which does not violate the law of conservation of energy. For simplicity, we assume unit antenna efficiency, i.e., no antenna loss. Nodes can use their antennas for directional transmission or directional reception, where the transmitter and receiver antenna gains are assumed to be the same. We also assume that each antenna is steerable so that each node can point its antenna in any desired direction. Suppose that a node i ∈ {1, · · · , n} transmits to another node k ∈ {1, · · · , n} \ i and they can beamform to each other according to our assumptions. Let I1 , I2 , and I3 denote three different sets of nodes transmitting at the same time as node i, where both nodes i1 ∈ I1 and k beamform to each other, either node i2 ∈ I2 or k beamforms to the other node (but not both), and neither node i3 ∈ I3 nor k beamforms to the other node, respectively. The received signal yk at node ∑ k ∈ {1, · · · , n} ∑ at a given time instance is given by ∑ yk = i1 ∈I1 hki1 xi1 + i2 ∈I2 hki2 xi2 + i3 ∈I3 hki3 xi3 +nk , where xi1 , xi2 , and xi3 ∈ C are the signals transmitted by nodes i1 , i2 , and i3 , respectively, and nk denotes the circularly symmetric complex Gaussian noise with zero mean and variance N0 . Here, √the channel gains are given by jϕki jϕki jϕki 2 se hki1 = Gm eα/2 1 , hki2 = Gm Gα/2 , and hki3 = Gs eα/2 3 ,
Abstract— This paper analyzes the impact and benefits of directional antennas in improving the throughput scaling law of a large wireless network in an information-theoretic perspective. More specifically, we deal with a general scenario where the beamwidth of each node can scale at an arbitrary rate relative to the number of nodes in the network. We then introduce an elastic routing protocol, which enables to increase per-hop distance elastically according to the scalable beamwidth, while maintaining a constant average signal-to-interference-and-noise ratio at the receivers. Our main results indicate that this elastic routing can exhibit a much better throughput, compared to the conventional nearest-neighbor multihop, and eventually leads to a linear throughput scaling. The gain comes from the fat that more source-destination pairs can be activated simultaneously with the boosted antenna gain. In addition, our work is extended to a hybrid network scenario using infrastructure.
I. I NTRODUCTION In [1], the sum-rate scaling was originally characterized in a large wireless ad hoc network. It was shown that for a random network having√n nodes in a unit area, the total throughput scales as Ω( n/ log n) by conveying packets in a multihop fashion. There have been further studies on multihop in the literature [2], [3]. On the one hand, the use of directional antennas [4] in large wireless networks has recently emerged as a promising technology leading to the higher spatial reuse ratio, the improved transmission distance, and the reduced interference level at a very low cost in comparison to alternative technologies. In this paper, we introduce a new routing, termed elastic routing, and analyze its throughput scaling law based on the information-theoretic approach when there are n randomly located nodes in an ad hoc network with directional antennas. Precisely, we consider a general framework in which the beamwidth of each node, θ, can scale at an arbitrary rate relative to n. Detailed descriptions and all the proofs are omitted due to the space limitation of this extended abstract.
rki
II. S YSTEM AND C HANNEL M ODELS We consider a two-dimensional wireless network consisting of n nodes that are distributed uniformly at random on a square. The nodes are assumed to be grouped into n/2 S– D pairs at random. Each node operates in half-duplex mode and is equipped with a single directional antenna. We define a hybrid antenna model whose mainlobe is characterized as
978-1-4799-6786-5/14/$31.00 ©2014 IEEE
rki
1
2
rki
3
respectively, where ϕki represents the random phase uniformly distributed over [0, 2π) and independent for different i, k, and time (transmission symbol). The parameters rki and α > 2 denote the distance between nodes i and k, and the path-loss exponent, respectively. We denote T (n) as the total throughput of the network, where each node is assumed to transmit at a rate T (n)/n on average. III. M AIN R ESULTS A. Elastic Routing Protocol We describe the elastic routing protocol with the help of directional antennas, which perform multihop (or even single-
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hop) transmission by elastically increasing per-hop distance as a function of the scaling parameter θ, which ultimately enhances the throughput performance compared to the existing multihop [1], [2]. We focus on the dense network configuration where the network area is one.1 We assume that M (n) S–D pairs, located randomly on the square, can be active simultaneously. We divide the whole area into 1/A(n) square routing cells with per-cell area A(n), where A(n) is assumed to scale as O(n) and Ω(log n). Let ¯ denote the average transmission distance at each dhop and h hop and the average number of hops respectively. ( per)S–D pair, ( ) Then, it follows that A(n) = Θ d2hop = Θ h¯12 . We draw the straight line connecting a source to its destination, termed an S–D line. While traveling along an S–D line, a certain node in each cell is arbitrarily selected as a relay forwarding the packets. Then, the antennas of each selected transmitter– receiver pair are steered so that their beams cover each other. When each routing cell with area A(n) is further divided into smaller square cells of area 2 log n, only one node in each smaller cell becomes a transmitter at the same time. Let us turn to how to decide the average per-hop distance dhop according to given beamwidth θ using the elastic routing. The parameter dhop can be elastically increased as much as possible while the average SINR at each receiver is set to Θ(1). In consequence, dhop can be determined according to the value of θ as follows: dhop
(
{
= Θ min max
{√
log n , n
(
log n n
α ) 2(α−1)
θ
−2 α−1
}
,1
})
network. The other beamwidth scaling condition bridges the √ total throughput T (n) between n and n. Our result is thus general in the sense that the achievable scheme and its throughput are shown for all operating regimes with respect to θ (i.e., for an arbitrary scaling of θ). IV. E XTENSION TO A H YBRID N ETWORK Our work is extended to a hybrid network [5], [6] consisting of both wireless ad hoc nodes and infrastructure nodes, or equivalently base stations (BSs). Suppose that the whole network area is divided into m square cells, each of which is covered by one single-antenna BS at its center. Parameters n and m are then related according to m = nβ , where β ∈ (0, 1). Furthermore, it is assumed that the BS-to-BS links have infinite capacity and these BSs are neither sources nor destinations. The elastic routing scheme utilizing BSs consists of three phases: access elastic routing, BS-to-BS transmission, and exit elastic routing. This routing essentially follows the same procedure as that in [5] except that the nearest-neighbor multihop is replaced by the elastic routing. Theorem 2: In the infrastructure-supported network using directional antennas, the total throughput achieved by elastic routing with/without BS support is given by ( { {√ ) } } α−2 2 T (n) = Ω min max n, n 2(α−1) θ− α−1 , m , n n−ϵ
w.h.p., where ϵ > 0 is an arbitrarily small constant. Using Theorem 2, it turns out that, if the inverse of the ( )α−2 α−1 4 n , then beamwidth, θ−1 , scales slower than m 2 log n the total throughput increases with the parameter m. On ( )α−2 α−1 4 n , the other hand, if θ−1 scales faster than m 2 log n i.e., the beamwidth is sufficiently narrow, then the use of infrastructure may not be helpful in improving the throughput scaling compared to the pure ad hoc mode (m = 0).
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B. Throughput Scaling Law The following theorem establishes our main result. Theorem 1: In the network with directional antennas, the total throughput achieved by elastic routing is given by (( )1/4 ) ( ) n −1 Ω n1/2−ϵ =o if θ log n (( ) )1/4 ) ( α−2 2 −ϵ n if θ−1=Ω Ω n 2(α−1) θ− α−1 log n (( T (n) = )α/4 ) n −1 and θ =o log n (( )α/4 ) ( ) n if θ−1=Ω Ω n1−ϵ log n
ACKNOWLEDGEMENT This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (MSIP) (2012R1A1A1044151). R EFERENCES [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 388–404, Mar. 2000. [2] A. El Gamal, J. Mammen, B. Prabhakar, and D. Shah, “Optimal throughput-delay scaling in wireless networks-Part I: The fluid model,” IEEE Trans. Inf. Theory, vol. 52, no. 6, pp. 2568–2592, June 2006. [3] W.-Y. Shin, S.-Y. Chung, and Y. H. Lee, “Parallel opportunistic routing in wireless networks,” IEEE Trans. Inf. Theory, vol. 59, no. 10, pp. 6290–6300, Oct. 2013. [4] P. Li, C. Zhang, and Y. Fang, “The capacity of wireless ad hoc networks using directional antennas,” IEEE Trans. Mobile Comput., vol. 10, no. 10, pp. 1374–1387, Oct. 2011. [5] A. Zemlianov and G. de Veciana, “Capacity of ad hoc wireless networks with infrastructure support,” IEEE J. Select. Areas Commun., vol. 23, no. 3, pp. 657–667, Mar. 2005. [6] W.-Y. Shin, S.-W. Jeon, N. Devroye, M. H. Vu, S.-Y. Chung, Y. H. Lee, and V. Tarokh, “Improved capacity scaling in wireless networks with infrastructure,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 5088–5102, Aug. 2011.
with high probability (w.h.p.), where ϵ > 0 is an arbitrarily small constant. 1/4 If the parameter θ−1 scales slower than (n/ log n) , then the throughput scaling does not increase even with directional antennas and the same throughput as that achieved by the pure multihop [1] in the omnidirectional mode is obtained. α/4 On the other hand, if θ−1 scales faster than (n/ log n) , then an almost linear throughput scaling is possible in the 1 The overall procedure of our protocol can be directly applied to the extended network scenario with unit node density.
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