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
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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
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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
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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
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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 = ∞).
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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
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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
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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
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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
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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
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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.