Capacity analysis of mesh networks with omni or directional antennas

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 aandßare 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
Capacity definition

• 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)
• 
  

l(v)
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
Capacity of omni antennas Line deployment

• k maximal hops to the gateway
• q (interference range)/ (transmission range)
• Deployment
• Topology

10
Capacity of omni antennasline deployment

• Collision set
• Collision load
• Collision load reaches max for links between Sq1
  and Sq2 (both link load and collision set size
  reach max at this point).

11
Capacity of omni antennas line deployment
Observations 1) Capacity independents to q when
k2q3. 2) Capacity is O(1/N), decreasing as k
increases. 3) Capacity is in the range of 1/N,
1/((2q3)N) (k 1, and k 8).
12
Capacity of directional antennas (m2)line
deployment

• Collision set
• Collision load
• Collision load is maximal for links between Sq-1
  and Sq.

q-1
13
Capacity of directional antennas (m2)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,
   (2q3)/(2q-1) (k 1, and k 8).
3. In directional antennas, 2 radios/node, but 1
   radio/node in omni antennas. The capacity is not
   doubled for q gt 2.

14
Capacity of omni antennas 2-dimensional region
deployment

• Collision set size of a link is independent from
  its location (because of even node distribution).
• Collision load is the largest for links between
  R0 and R1, i.e., links incident to the gateway
  nodes.

R0
of nodes 1
R1
of nodes N/k2
of nodes 3N/k2
R2

Ri
of nodes (2i-1)N/k2
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Capacity of omni antennas 2-dimensional region
deployment

• Collision set of a link between R0 and R1 links
  in the two overlapped circles with radius qrt.
• 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

16
Capacity of omni antennas 2-dimensional region
deployment

• Capacity independents to q when kq1.
• Capacity is O(1/N).
• Capacity is in the range of 1/N, 1/((q1)N) (k
  1, and k 8).
• 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 R1 nodes by
  1-hop. R1 nodes need multi-hops to reach the
  gateway.
• Interference area of a link is two overlapped
  sectors, not circles.

18
Approximation of max collision load

• 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 qrt,
  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.

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Maximal collision load constraints

• Max collision load
• Lq total load of links with one end in circle at
  the gateway of radius qrt.
• F (interference area of a link) / (area of the
  circle).
• ?0 probability of a link that has an end-node
  inside the interference area of the link and
  interferes with it.

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Calculation of Lq

• Lq LR1 LR2
• of nodes 1 hop to the gateway
  (1)
• of nodes ith hop to the gateway
  (2)
• h of hops for R1 nodes to gateway. Since of
  R1 nodes is N/k2, h can be obtained from the
  above two eqs.

21
Calculation of Lq

• Starting from R2, we assume all nodes in Ri1 can
  be directly linked to nodes in Ri. (As the ring
  getting larger, it is more possible for all Ri1
  nodes to link to Ri nodes directly.)
• The LR2 obtained under this assumption is a
  close lower bound of the actual value.

22
Calculation of F

• ? beamwidth of antennas

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Calculation of ?0

• Probability of a node falling into the
  interference sector of an antenna
• ?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

s
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Capacity of directional antennas
(m2)2-dimensional region deployment

• 1) Capacity of directional antennas decreases
  with q.
• 2) Capacity is for m2,
  for mgt2, and it is bounded by
  .
• 3) The ratio of directional to omni antennas is
  in the range of
• 4) When? is sufficiently small, capacity is
  bounded by

25
Numerical results

• Capacity is in unit of C/N, and q 2
• Omni antennas

2-dimensional region deployment
Line deployment
26
Capacity-ratio of directional antennas to
omni-antennas

• Line deployment

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Impact of beamwidth on capacity-ratio

• 2-dimensional deployment

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Impact of of antennas on capacity-ratio

• 2-dimensional deployment.

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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 when?
  is small.
• The capacity is higher with a smaller?. But it is
  bounded by Cm/N when? is small enough.

30
References

• P. Gupta and P. R. Kumar, The capacity of
  wireless networks, IEEE
• Transactions on Information Theory, vol. 46, no.
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• 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. 1543 1552.
• 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 2003.
• S. Yi, Y. Pei, and S. Kalyanaraman, On the
  capacity improvement of ad hoc
• wireless networks using directional antennas, in
  ACM MobiHoc, 2003.
• 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.