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 αandβare

beamwidth for transmission and reception.

).( NK

2)2(

).log/1( NNO

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):

))(()(

|)(|vlIul

CuT

Capacity definition

v

l(v)

T(v)

))(()())(()(

)|(| max)|(|minvlIulvvlIul

uTC

uTC

vCap

))(()(

))(( |)(|vlIul

vlI uTL

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.

))(( max vlIv

LCCap

9

k: maximal hops to the gateway

q: (interference range)/ (transmission range)

Deployment

Topology

Capacity of omni antennas: Line deployment

G 1 2 3

s1

4 5 6

s2

7 8 9

s3

10 11 12

s4

G

1

2

3

s1 s2 s3 s4

4

5

6

7

8

9

10

11

12

10

Collision set:

Collision load:

Collision load reaches max for links between Sq+1 and Sq+2 (both link load and collision set size reach max at this point).

Capacity of omni antennas:line deployment

}11,:)({)((

qijqiSuulvlI jSv i

.)/)(1(|)(|1

1

1

1))((

qi

qij

qi

qij SuvlI kNjkuTL

jiSv

otherwise.

)1)(32(

,32 if max

21

))((

Nk

qkq

qkNL

k

vlIv

i-q-1 i+q+1

i

11

Capacity of omni antennas: line deployment

otherwise ,

)1)(32(

32 if ,)1(

2

Nqkq

kC

qkNk

C

Cap

Observations:

1) Capacity independents to q when k≤2q+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 Sq-1 and Sq.

Capacity of directional antennas (m=2):line deployment

}11,:)({)((

qijqiSuulvlI jSv i

1

1

1

1))(( )/)(1(|)(|

qi

qij

qi

qij SuSvvlI kNjkuTL

ji

otherwise.

)1)(12(

,12 if max

21

))((

Nk

qkq

qkNL

k

vlIv

i i+q-1i-q+1 2q-10 qq-1

13

Capacity of directional antennas (m=2):line deployment

otherwise ,

)1)(12(

12 if ,)1(

2

Nqkq

kC

qkNk

C

Cap

1) Capacity is independent from q when k≤2q-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.

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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.

Capacity of omni antennas: 2-dimensional region deployment

tr

R0

R1

R2

tr2

# of nodes: 1

# of nodes: N/k2

# of nodes: 3N/k2

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:

]}.1,1[,:)({))((1

qiRuulvlI iRv

)1,min(

1))(( |)(|max

qk

i RvvlI

vi

vTL G

)1,min(

12

2

).)1(

1(qk

i k

iN

16

Capacity of omni antennas: 2-dimensional region deployment

1) Capacity independents to q when k≤q+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.

otherwise ,/)1(

1 if ,/

1

22

1

1

22

q

i

k

i

kiNNq

C

qkkiNkN

C

Cap

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Capacity of directional antennas (m≥2):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.

1R

2R

tqr

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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.

Approximation of max collision load

v

G

qrt

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Max collision load:

Lq: total load of links with one end in circle at the gateway of radius qrt.

Φ : (interference area of a link) / (area of the circle).

ρ0 : probability of a link that has an end-node inside the interference area of the link and interferes with it.

Maximal collision load constraints

./ max

max

))((

0))((

mNL

LL

vlIv

qvlIv

l2

l1

v

G

qrt

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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.

outint LLL

mk

N22

12

)12

immk

N（

h

i HvR

i

khNhNh

mkm

hmNNh

vTL1

2

2

2.m if 2

)1(

2 if )2(

))2/(1(

|)(|1

1

1|)(|

RvR vTL

1R

2R

tqr

21

Calculation of Lq

Starting from R2, we assume all nodes in Ri+1 can be directly linked to nodes in Ri. (As the ring getting larger, it is more possible for all Ri+1 nodes to link to Ri nodes directly.)

The LR2* obtained under this assumption is a close lower bound of the actual value.

outint LLL

1

2|)(|

*2

q

i RvR

i

vTL

q

i

k

i

kiN

qkkiN

1

22

1

1

22

otherwise. )/1(

1 if )/1( 1R

2R

tqr

22

Calculation of Φ

θ: beamwidth of antennas

q

q

qr

rqqr

t

tt

212)1(

2

2212

21 ）（

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Calculation of ρ0

Probability of a node falling into the interference sector of an antenna:

ρ0 : 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:

22220 )()1)(1(1

2

l(v)

s

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Capacity of directional antennas (m≥2):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

)))()()((

2

12,/max(

*213

412

RR LLq

qmN

CCap

)log

m

O（ ))1/log

log2 （

（m

O

NCm /

].)))(()(,/1min(

1,1[

222

12

q

qhqm

q

)./( 2kCO

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Numerical results

Capacity is in unit of C/N, and q = 2

Omni antennas

Line deployment2-dimensional

region deployment

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Capacity-ratio of directional antennas to omni-antennas

Line deployment

27

Impact of beamwidth on capacity-ratio

2-dimensional deployment

3m

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Impact of # of antennas on capacity-ratio

2-dimensional deployment:.

60 90

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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 whenθ is small.

The capacity is higher with a smallerθ. But it is bounded by Cm/N whenθ is small enough.

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References

P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEETransactions on Information Theory, vol. 46, no. 2, pp. 388–404, March 2000.

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 hocwireless 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.