1 Capacity analysis of mesh networks with omni or directional antennas Jun Zhang and Xiaohua Jia City University of Hong Kong

2 Outline Related work Capacity analysis for line deployment Capacity analysis for 2-dimensional deployment Numerical results Conclusions

3 Related work [Gupta 00] Per-node capacity in ad hoc networks is [Liu 03, Toumpis 04] Capacity of ad hoc networks can be O(1) by adding K base stations, [Jun 03] Capacity of mesh networks is O(1/N) (No multi-hop analysis). [Yi 03, Dai 08] Directional antennas in ad hoc networks can gain more capacity than omni ones, where αandβ are beamwidth for transmission and reception.

4 System configurations Single channel system 1 gateway node and N mesh nodes Even node distribution All traffic to/from gateway node Minimal hop routing

5 Interference model - Omni antennas

6 Interference model - Directional antennas Directional reception mode Link interference u interferes with w. u does not interfere with w. xy interferes with uv, because x interferes with v

7 Each node has traffic. Link load on l(v): |T(v)|. Collision set of l(v): I(l(v)) No two collision links can be active at the same time, thus Capacity per node is the maximal possible : Collision load of l(v): Capacity definition v l(v)l(v) T(v)

8 Capacity & Maximal Collision Load Capacity of a network is upper bounded by the maximal collision load of links. To max network capacity, we need to min the maximal collision load of links.

9 k: maximal hops to the gateway q: (interference range)/ (transmission range) Deployment Topology Capacity of omni antennas: Line deployment G 123 s1s1 456 s2s2 789 s3s s4s4 G s1s1 s2s2 s3s3 s4s

10 Collision set: Collision load: Collision load reaches max for links between S q+1 and S q+2 (both link load and collision set size reach max at this point). Capacity of omni antennas: line deployment i-q-1i+q+1 i

11 Capacity of omni antennas: line deployment Observations: 1) Capacity independents to q when k2q+3. 2) Capacity is O(1/N), decreasing as k increases. 3) Capacity is in the range of [1/N, 1/((2q+3)N)] (k = 1, and k = ).

12 Collision set: Collision load: Collision load is maximal for links between S q-1 and S q. Capacity of directional antennas (m=2): line deployment i i+q-1 i-q+1 2q-1 0 q q-1

13 Capacity of directional antennas (m=2): line deployment 1)Capacity is independent from q when k2q-1. 2)The ratio of capacity of directional antennas to omni-antennas is in the range of [1, (2q+3)/(2q-1)] (k = 1, and k = ). 3)In directional antennas, 2 radios/node, but 1 radio/node in omni antennas. The capacity is not doubled for q > 2.

14 Collision set size of a link is independent from its location (because of even node distribution). Collision load is the largest for links between R 0 and R 1, i.e., links incident to the gateway nodes. Capacity of omni antennas: 2-dimensional region deployment R0R0 R1R1 R2R2 # of nodes: 1 # of nodes: N/k 2 # of nodes: 3N/k 2 RiRi # of nodes: (2i-1)N/k 2 …

15 Capacity of omni antennas: 2-dimensional region deployment Collision set of a link between R 0 and R 1 : links in the two overlapped circles with radius qr t. Since the area of the two overlapped circles depends on the distance between two end-nodes of the link, we use one circle centered at the gateway as a lower bound of the overlapped circles. Maximal collision load: G

16 Capacity of omni antennas: 2-dimensional region deployment 1)Capacity independents to q when kq+1. 2)Capacity is O(1/N). 3)Capacity is in the range of [1/N, 1/((q+1)N)] (k = 1, and k = ). 4)The links far away from the gateway has little impact on capacity.

17 Capacity of directional antennas ( m2) : 2-dimensional region deployment Differences from omni-antennas: Since each node has only m radios, it may not be possible for gateway to link all R 1 nodes by 1-hop. R 1 nodes need multi-hops to reach the gateway. Interference area of a link is two overlapped sectors, not circles.

18 A link incident to the gateway may not have max collision load. We still use collision load of this link as a lower bound of the max one. Interference area of this link, the joint area of two sectors, is inside the circle of radius qr t, centered at the gateway. We compute the average load of all links in this circle, and then use portion of joint area of sectors as an approximation of the collision load of the link. Approximation of max collision load v G qr t

19 Max collision load: L q : total load of links with one end in circle at the gateway of radius qr t. Φ : (interference area of a link) / (area of the circle). ρ 0 : probability of a link that has an end-node inside the interference area of the link and interferes with it. Maximal collision load constraints l2l2 l1l1 v G qr t

20 Calculation of L q L q = L R1 + L R2* # of nodes 1 hop to the gateway: (1) # of nodes i th hop to the gateway: (2) h: # of hops for R 1 nodes to gateway. Since # of R 1 nodes is N/k 2, h can be obtained from the above two eqs.

21 Calculation of L q Starting from R 2, we assume all nodes in R i+1 can be directly linked to nodes in R i. (As the ring getting larger, it is more possible for all R i+1 nodes to link to R i nodes directly.) The L R2* obtained under this assumption is a close lower bound of the actual value.

22 Calculation of Φ θ : beamwidth of antennas

23 Calculation of ρ 0 Probability of a node falling into the interference sector of an antenna: ρ 0 : Probability of a link (s, t) that has an end- node, say s, inside the interference area of l(v) and interferes with l(v). It requires one of end- nodes of l(v) be inside interference sector of s: l(v)l(v) s

24 Capacity of directional antennas ( m2) : 2-dimensional region deployment 1) Capacity of directional antennas decreases with q. 2) Capacity is for m=2, for m>2, and it is bounded by. 3) The ratio of directional to omni antennas is in the range of 4) Whenθ is sufficiently small, capacity is bounded by

25 Numerical results Capacity is in unit of C/N, and q = 2 Omni antennas Line deployment 2-dimensional region deployment

26 Capacity-ratio of directional antennas to omni-antennas Line deployment

27 Impact of beamwidth on capacity-ratio 2-dimensional deployment

28 Impact of # of antennas on capacity-ratio 2-dimensional deployment:.

29 Conclusions Capacity is O(1/N). Capacity increases with transmission range. Directional antennas achieve more capacity than omni ones. The capacity increases with m, particularly whenθ is small. The capacity is higher with a smallerθ. But it is bounded by Cm/N whenθ is small enough.

30 References P. Gupta and P. R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 388–404, March B. Liu, Z. Liu, and D. Towsley, On the capacity of hybrid wireless networks, in IEEE INFOCOM,, vol. 2, San Francisco, CA, April 2003, pp – S. Toumpis, Capacity bounds for three classes of wireless networks: Asymmetric, cluster, and hybrid, in ACM MobiHoc, Tokyo, Japan, May 2004, pp. 133 – 144. J. Jun and M. L. Sichitiu, The nominal capacity of wireless mesh networks, IEEE Wireless Communications, vol. 10, no. 5, pp. 8 –14, October S. Yi, Y. Pei, and S. Kalyanaraman, On the capacity improvement of ad hoc wireless networks using directional antennas, in ACM MobiHoc, H. Dai, K. Ng, R. Wong, and M. Wu, On the capacity of multi-channel wireless networks using directional antennas, in IEEE INFOCOM, Phoenix, USA, 2008.