8th OAPS Working Paper Series
Website: http://www.oaps.hk/
Paper No. 2010-008
Capacity Analysis of Multi-Channel WSN Based on Grid
一
Partition and Interference Block
Mingjun Hua
School of Electronic, Information and Electrical Engineering
ABSTRACT
This paper studies how the capacity scales in the multi-channel wireless sensor and actor networks
(WSN). Based on the capacity analysis results,, we consider the multi-channel capacity analysis in this
work in the grid partitioned networks. It can be shown that the capacity of the network has a remarkable
increase after we use the multi-channel instead of the single -channel. When multi-channel is utilized in
every cell of one interference block, the capacity is several times of that with single-channel. When
multi-channel is utilized in every nodes in a cell, the capacity is increased in the order. Both the result is
calculated by simulation through C++and MATLAB.
KEY WORDS
WORDS:: capacity, protocol model, grid partition, multi-channel, degradation of the capacity
1 Introduction
A wireless sensor network (WSN) has recently emerged as a premier research topic,which
has a wide-range potential application such as battlefield, emergency relief, environment
monitoring, and so on .It consists of a set of sensor devices which randomly spread over a
geographical area. These sensors are able to perform processing as well as sensing and are
able to communicate with each other. The network capacity is measured in terms of
“bit-meters/sec” originally introduced by Gupta and Kumar [1] .
For a WSN, the ultimate goal is to collect sensing data from all sensors to certain sink
nodes and then perform further analysis at sink nodes. Gupta and Kuma found that with n
nodes under a random network placement, each node has a throughput capacity of Θ(W/√n
log n)[1]. Even under optimal arbitrary networks, the network could only offer a per-node
throughput of Θ(W/√n) .Xiang-Yang Li and Xinghua Shi propose a data collection method
W) which match the theoretical upper
using the grid partition[2],whose capacity is Θ(W
bounds,However,it still stays the application of single-channel, which can not increase the
capacity by orders.
Pradeep Kyasanur，Nitin H. Vaidya analyze the impact of the channels and interfaces[6],In
一
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Mingjun.Hua
Department of Automation, School of Electronic, Information and Electrical Engineering
their work,the capacity decrease when the ration of the number of channels C and the
number of interfaces m increase.There is no result of the detailed improvement analysis of
the capacity that multi-channel bring to the WSN.we further analyze the result and apply it
into the grid partition model.
This paper is to create a method to decrease the influence of the interference because the
main constraint of the capacity of wireless sensor network is the interference management.
Under the protocol model, given the transmission range r, sensors cannot transmit
Δ)r away [1] . We consider
simultaneously if there is another sensor transmitting within (1+
(1+Δ
the multi-channel communication instead of the single-channel case into the gridded WSN
so that those sensors in the interference block can transmit simultaneously.
It's not easy to apply multi-channel into grid partition directly, there is no precedent that
illustrated how to distribute multi-channel in the grid partition method. We suppose two kind
C equals the number of
of method to distribute the channels. One is to distribute C channels (C
cells in an interference) in every cell of an interference, while another is to distribute
channels in every nodes in one cell. These two methods lead to different results.the first
method could increase the capacity in a rate range from √C to C .Another method could
increase the capacity in a rate of O(log n) ,It's an increase by order and will become a large
step with the number of nodes N growing.
The rest of the paper as organized as follows: We represent our model and partition
method in Section 2. Section 3 is our contribution which is utilizing multi-channel in the grid
method to increase the capacity. In section 4, we show our way of simulation and draw the
results. We take the delay performance into consideration in Section 5. Section 6 is our
future work and other ideas as well as a conclusion.
2 Network model and partition method
2.1 Protocol model
Our work is based on the random network and protocol model. In a random scenario,
nodes are randomly located, independently and uniformly distributed. Each node has a
randomly chosen destination to which it wishes to send bits per second. The destination for
each node is independently chosen as the node nearest to a randomly located point,
uniformly and independently distributed. In this random setting, we will assume that the
nodes are homogeneous, all transmissions employ the same nominal range or power [1] .
i）The distance between X i and X j is no more than r
Xi − X j ≤ r
（2-1）
ii）For every other node X k simultaneously transmitting over the the same channel
X k − X j ≥（1 + ∆r）
（2-2）
2.2 Grid partition
In our work, we consider the grid model for the WSN.
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8th OAPS Working Paper Series Capacity Analysis of Multi-Channel WSN Based on Grid Partition And Interference Block
2.2.1 Network model
We use the interference modeled by protocol interference model in our analysis. In
protocol interference model, all nodes are assumed to have uniform interference range r.
When node v i transmits to node v j , node v j can receive the signal successfully if no other
node within a distance R is transmitting simultaneously. Here, for simplicity, we assume that
R/r is a constant α which is larger than 1.
2.2.2 Partition method
As shown in Fig. 1, the network (e.g., the unit square) is divided into a ² micro cells of the
d. Here a = 1/d
size d×d
/d..
2
Fig.2-1 Grid partition of the sensor network: a cells with cell size of d×d.
Every micro cell has at least one node with high probability which converges to 1 as n →
∞.
Randomly putting γ balls into δ bins, with probability at least 1− 1/δ , the maximum
number of balls in any bin is Ο ( γ/δ + log δ). The number of nodes inside any cell is
bounded from above by Ο (log n) with high probability [ 3] .
To promise the connection of the network ( make sure every cell should has at least one
node), We set d = 3 logn/n , r = 5 d = 15 logn/n
We also use the grid model because this method of divide ensure the connectivity of the
WSN, as well as to support a clear way to transmission,It is quite similar to the typical
application of WSN,such as industrial site,disaster prevention networks,war field and other
block area which could be divided into several cells but vertically in one layer.
3 Utilization of multiple channels in the grid model
As shown in Figure2-1, we consider the data collection of nodes from four different
directions to s1. For the purpose of analysis, we only concentrate on the direction which
has the largest number of sensors, e.g., the shaded rectangle in Figure2-1, since the sink can
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Mingjun.Hua
Department of Automation, School of Electronic, Information and Electrical Engineering
perform collection on each direction in turn and it only adds a constant 4 in the analysis. Our
collection algorithm has two phases. In the first phase (Phase I), every sensor sends its data
up to the highest cell in its column (in the pth row), and in the second phase (Phase II), all
data is sent via cells in the pth row to the sink. We define the time needed for these two
phases as T1 and T2, respectively . The number of nodes in each cell is upper bound at
Ο (log n). Every node needs one time-slot t to send one packet to its neighbor in the next
cell.The time slot t include the time of broadcasting, building routing structure and
transmission.
(a)
(b)
Fig. 3-1 Transmission method of single-channel
We first consider the situation of single-channel. Due to wireless interference, when a
node v i transmits a packet to v j , the nodes within R distance from v j can not transmit any
packets in the same time slot. Thus, every (R/d + 2) × (R/d + 1) cells (we call it an
interference block hereafter) can only have one node send a packet to its upper neighbor in
every time slot t. Other nodes should wait one after another.
In Phase I,every node sends its date to the nodes located on the top row,see Fig.3-1 (a),It
costs (R/d + 2) × (R/d + 1) Ο (log n) t for the transmission of every node in one
q×((R/d + 2) × (R/d + 1) Ο (log n) t.
interference block,and the path is q ,so T1=q
In the beginning of Phase II, all data have been already at cells of the top row and they
transmit right to the sink,see Fig.3-1 (b). The sink S lies in the same row with these cells. We
now estimate the time T2 needed for sending all data to S. Each cell in the top row has at
most q Ο (log n) nodes’ data. The interference block is 1 × (R/d + 2) now and the path is p.
That is T2=p×((R/d + 2) × q Ο (log n)t..Suppose that the transmission time between one
node to another node is t, which include the time of broadcasting, building routing structure
and transmission.Notice that the order of both p and q is the same as a ,exactly
log n
n
[ 2]
It can be hence obtained that
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上海交通大学 SJTU
8th OAPS Working Paper Series Capacity Analysis of Multi-Channel WSN Based on Grid Partition And Interference Block
⎛R
⎞⎛ R ⎞
T1 = ⎜ + 2 ⎟⎜ + 1⎟q × t × O(logn ) = O t nlogn
⎝d
⎠⎝ d ⎠
⎛R
⎞
T2 = ⎜ + 2 ⎟p × q × t × O(logn ) = O(nt )
⎝d
⎠
T = T1 + T2 = O(nt )
(
)
（3-1）
C=
nb
nb
=
= Ω(W )
T Ο(nt )
（3-2）
3.1 Utilization multiple channels in every cell within interference
block
From the formulation we can see that the interference block is one of the main reasons
which cause the delay. Our idea is utilizing multi-channel in every cell of the interference
block. We suppose that every sensor has two interfaces, one for transmission and another for
receiving. Our transmission method is following like this (Fig. 3):
（a）
（b）
Fig.3-2 Transmission method with multiple channels in every interference block
Firstly, we consider Phase I. Every sensor sends its data up to the highest cell in its
column,see Fig 3-2 （ a ） . The sensor in the lower row uses one channel to transmit its
information to the sensor in the upper cell. Simultaneously, the sensor in cell which is
receiving message can transmit to another sensor using another channel.We distribute one
channel in every cell in an interference so that every cell will have one node to send date and
one (or another) to receive date in one time slot. As a block has (R/d + 2) × (R/d + 1) cells,
so the least number of channels is (R/d + 2) × (R/d + 1) ,while in other interference blocks,
the same (R/d + 2) × (R/d + 1) channels is in the same distribution as the block marked by
the red line. It follows the protocol model [1]and will not cause an interference. Secondly, at
the beginning of Phase II, all data are already transmitted to the sensors in cells on the top
row. By using the same method used in Phase I,they transmit the date right to the sink ,see
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Mingjun.Hua
Department of Automation, School of Electronic, Information and Electrical Engineering
Fig.3-2 (b), we can know the interference block is now (R/d + 2) ×1. The sensors in every
cells at the top row send their data to the sink using (R/d + 2) channels. In every cell there is
one to send date and one to receive.
So we can get the following relations:
(
T1 = q × t × O(logn ) = O t nlogn
)
T2 = p × q × t × O(logn ) = O(nt )
T = T1 + T2 = O(nt )
（3-3）
C=
nb
nb
=
= Ω(W )
T Ο(nt )
（3-4）
It is obvious that the influence of interference block disappeared as we have no coefficient
(R/d + 2 ) or (R/d + 2) × (R/d + 1)
1)in the equation. The reason is that in every cell in the
same interference block, there is one nodes transmit simultaneously because of utilizing
multiple channels. In this case,the capacity rises in a ration of a constant between (R/d +
×1 to (R/d + 2) × (R/d + 1) .
2)
2)×
3.2 Utilization multiple channels in every node in one cell
Since we have thought about utilizing multi-channel in every cell of one interference block,
it is easy for us to have another idea that utilizing multi-channel in every node in one cell. As
we know in the former part, the number of node in every cell is bounded from above O(logn)
with high probability. So we could use only K×O(logn) channels so as to meet our
command.while in reality, O(logn) channels is not a very large figure that cannot be reached.
First, in phase I, every sensor sends its data up to the highest cell in its column. The sensor
in the lower row uses one channel to transmit its information to the sensor in the upper cell.
Simultaneously, the sensor which in the same cell can transmit to another sensor in the upper
cell using another channel. But as the protocol model constraints, in every interference block,
only nodes in one of the (R/d + 2) × (R/d + 1) cells can be in transmission.
Secondly, in the beginning of Phase II, all data are already at cells of the top row. Just like
the method used in phase I , every sensor send its data to the sink cell by cell and each path
is using O(logn) channels .
Our transmission method can be shown in Fig3-3
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上海交通大学 SJTU
8th OAPS Working Paper Series Capacity Analysis of Multi-Channel WSN Based on Grid Partition And Interference Block
(a)
(b)
Fig. 3-3 Transmission method with multiple channels in every cells
Under this method, the nodes in the same cell can transmit simultaneously as long as the
the cell which receive the information has more number of nodes. If the receiving cell has
not get enough nodes, we will let the surplus nodes wait until the receiving nodes are free.
Therefore, we get
⎛
n ⎞⎟
⎛R
⎞⎛ R
⎞
T1 = ⎜ + 2 ⎟ ⎜
+ 1⎟ × t × q = O ⎜ t
⎜ logn ⎟
⎝ d
⎠⎝ d
⎠
⎝
⎠
⎛ n
⎛R
⎞
T2 = ⎜ + 2 ⎟ × t × qp = O ⎜⎜
⎝ d
⎠
⎝ logn
⎞
t ⎟⎟
⎠
⎛ n
⎞
T = T1 + T2 = O ⎜⎜
t ⎟⎟
⎝ logn ⎠
C=
（3-5）
nb
nb
=
= Ω(log nW )
T Ο( n t )
log n
（3-6）
It is obvious that the influence of interference in every cell disappeared. The reason is that
every sensor in the same cell can transmit simultaneously because of utilizing multi-channel.
In this case, the capacity rises in a ration of the order of O(logn).
4 Simulation Study
To show the effectiveness of the proposed method, the simulation study is conducted
based on C++ . We suppose the interference block has 2X3 cells. Every node has two
interfaces for sending and receiving respectively.
4.1Simulation results
interference block
上海交通大学 SJTU
of
multi-channel
in
every
cell
within
7
Mingjun.Hua
Department of Automation, School of Electronic, Information and Electrical Engineering
(a)
N=500
(b) N=1000
（d）N=10000
(c) N=5000
Fig.4-1 Simulation results of method 1
As can be seen from the figure, our method really works. In this condition, no matter how
large the quantity of the sensors is, the capacity of multi-channel is about triple as large as
the single-channel.
And if we analyze it further, we could do some research to know how the delay rate is
decreased. After we use the multi-channel, T1 is 1/
1/((R/d + 2) × (R/d + 1) （in a 2X3 block
it is 1/6）times as the single-channel, while T2 is also 1/ (R/d + 1) （in a 2X3 block it is 1/3）
times as the single-channel. But we should not forget that the second part has a much larger
proportion so it dominates the delay rate.
S
332
344
341
Table 4-1. the time of transmission in single-channel and multi-channel
phase total phase phase total phase phase
phase
total
II
I
II
2
I:
II:
II
S/phase
S
S
M
M
M
S/M
S/M
IS
S/M
1161
1493
64
447
511 5.1875 2.5973 3.4970 2.9217
1204
1548
65
440
505 5.2923 2.7364 3.5000 3.0653
1139
1480
63
441
504 5.4127 2.5828 3.3402 2.9365
594
566
568
2565
2588
2566
phase
I
8
3159
3154
3134
95
98
95
948
949
955
1043
1047
1050
6.2526
5.7755
5.9789
2.7057
2.7271
2.6869
4.3182
4.5724
4.5176
3.0288
3.0124
2.9848
上海交通大学 SJTU
8th OAPS Working Paper Series Capacity Analysis of Multi-Channel WSN Based on Grid Partition And Interference Block
1601
1596
1604
14372
14359
14310
15973
15955
15914
244
246
245
4943
4939
4947
5187
5185
5192
6.5615
6.4878
6.5469
2.9075
2.9073
2.8927
8.9769
8.9969
8.9214
3.0794
3.0771
3.0651
S means using single-channel ；M means using multi-channel,the unit is one time slot t
From the table we can find that the in phase I the ratio between the single-channel and the
multi-channel is about 6 and in phase II the ratio is about 2.7.
T
1s
： T
2s
⎛ R
= ⎜
⎝ d
Total m = T1s ×
⎞
+ 1 ⎟
⎠
3 logn
n
T1m
T
T
+ T2s × 2m × 2s
T1s
T2s T1s
（4-1）
（4-2）
This formula explains that T 2 domain the total T. As the node increase, the delay caused by
phase II is increasing.The reason is that in phase II we should consider an extra q ,and q is
determined by the number of nodes.
4.2 Simulation results of multi-channel in every cell within cell
We assume that we have 10 channels, and the interference block is still 2X3. We still have
two interfaces, one for transmitting and another for receiving.
（a） N=500
（c）N=5000
（b） N=1000
（d） .N=10000
Fig.4-2 Simulation results of method 2
From the figure we can conclude that the method can make the capacity about 6 times as
large as the single-channel. That is because in this case,when the number of nodes is 1000 ,
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Mingjun.Hua
Department of Automation, School of Electronic, Information and Electrical Engineering
after the random distribution，the number of nodes in every cells ranges from 6 to 14,so we
use tan channels，the same oder as O(logn)
O(logn),to transmission. As the result of the calculation，
the capacity should rise the same as the number of channels . However, we should notice
that the number of channels that is really in use is bounded by the number of nodes in the
upper cells that receive the date, because a node is not able to receive more than one date
simultaneously.
As we could forecast,the rise of the capacity has positive correlation with O(logn) , and
the number of nodes n.
4.3 Rise of the capacity
Number of nodes
Table 4-2. The rise of the capacity
multi-channel in blocks
multi-channel in cells
500
2.874
1000
3.037
5000
3.062
10000
3.077
5.531
5.768
5.926
5.946
Fig .4-3 rise of the capacity
As shown in Fig4-3 ,the capacity rise 6 times when using 10 channels in every node
within a cell and 3 times when using 6 channels in every cell within a 2 × 3 interference
block.The capacity rise when the number of nodes N rise.the reason is when N is not very
large,the channels are not in full use due to the random distribution of the nodes.
5 Delay performance due to the switch of interface
It seems that the second method is better. It is true when we don’t consider about the
switching delay rate. When nodes are not equipped with a dedicated interface per channel,
then capacity degradation may occur compared to using a dedicated interface per channel.
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上海交通大学 SJTU
8th OAPS Working Paper Series Capacity Analysis of Multi-Channel WSN Based on Grid Partition And Interference Block
When a dedicated interface per channel is not available, the available interfaces can
potentially be switched among different channels to use any of the available channels. Such
an interface switching technique is often used to improve channel utilization. However,
interface switching incurs a delay, which may reduce the achievable network capacity [ 4 ] .
For the random network, if the ratio between channel and the interface is larger than
logn, the switching delay rate will occur. And the capacity after we considering the
switching delay is directly proportion to the square loot of the ratio between channel and
interface.
Now we should recall that in the first method the number of channel is 6 and the second
number in the second method is 10.
n
If c=m the upper bound of the capacity is W logn , and if the ratio between c and m is
larger than O(logn)
O(logn), we could get the equation. W
nm
n
/W
c
logn
[6]
.
Here is a table which shows how switching delay change following the change of the
number of nodes. We suppose the degradation in capacity is the ratio between the capacity
considering the switching delay and the capacity which doesn’t consider the switching delay
Table3. The impact of degradation
Number of nodes
500
1000
2000
5000
10000
20000
Degradation in capacity(c=6)
0.948
1
1
1
1
1
Degradation in capacity(c=10)
0.734
0.775
0.8125
0.86
0.894
0.927
Fig 8 Degradation in capacity
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Department of Automation, School of Electronic, Information and Electrical Engineering
6 Conclusion
In this paper, we analyze the capacity of wireless sensor network with multiple channels.
When multi-channel is utilized in every cell of one interference block, the capacity is (R/d +
2) × (R/d + 1) times of the capacity which use the single-channel. When multi-channel
is utilized in every nodes in a cell, the increase of the capacity is even more remarkable and
in an order Ο (logn) but not just by times.If consider the switching delay,and the ratio
between c and m is larger than Ο (logn) ， the rise of the second method is decrease
by W
nm
n
.
/W
c
logn
To outlook the further work, we find another way to raise the capacity---date
aggregation [ 5] ,the principal of it is to decrease the amount of date in transmission，which
doesn't conflict with multi-channel. If we apply date aggregation into the result，it could raise
the capacity by another Ο (logn).
Reference
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[0018-9448] Gupta yr:2000 vol:46 iss:2 pg:388
[2] S.Chen, Y.Wang, X.Y.Li and X.Shi, Order-Optimal Data Collection in Wireless Sensor Networks:
Delay and Capacity, Sensor, Mesh and Ad Hoc Communications and Networks (SECON)[J], 2009 6th
Annual IEEE Communications Society Conference on [1-4244-2907-2]
[3] Satish Rao, The m balls and n bins problem, Lecture Note for Lecture11[M], CS270 Combinatiorial
Algorithms and Data Structures, University of Berkeley, 2003.
[4] P. Kyasanur and N.H.Vaidya, Routing and Interface Assignment in Multi-Channel Multi-Interface
Wireless Networks,[C]IEEE Wireless Communications and Networking Conference, 2005
[0-7803-8966-2] Kyasanur yr:2005 pg:2051
[5] Changhee Joo,Jin-Ghoo Choi ,Delay Performance of Scheduling with Date Aggregation in Wireless
Sensor Networks[C], 2010 Proceedings IEEE INFOCOM [1-4244-5836-6] Joo yr:2010 pg:1
[6] P. Kyasanur ， N. H. Vaidya ,Capacity of Multi-Channel Wireless Networks:Impact of Number of
Channels and Interface[C], Proceedings of the 11th annual international conference on Mobile
computing and networking - MobiCom '05 [1-59593-020-5] Kyasanur yr:2005 pg:43
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