Small Scale Non-linear Programming (SSNLP)

In the first approach, although we can solve the problem in polynomial time if α = 2 and n_e1 : ne2 : ne3 = 1 : 1 : 1, the performance may not improved more because the proper location is only related to the traffic rate and the length of Steiner edge. What we really need is to find a proper location of merge vertex which is involved with the traffic rate, the length of Steiner edge and the ratio among ne1, n_e2 and ne3. And it is suitable for any cases. As can be seen that the reason why the complexity of Algorithm 1 is high because it considers the entire topology at the same time. But, in the previous heuristic approach, we show that we can divide the whole topology into blocks and solve them orderly. Therefore, we can modify Eq. (4.3) into

minimize

where C is a constant, e1, e2, e3 represent the three edges connected with the merge vertex, X_Aekand Y_Aek represent the locations of three vertices connected with the merge vertex, and

these three vertices can be sensor nodes, sink or the merge vertices of other blocks, and where X_{M V} and Y_{M V} are the location of merge vertex. Since we solve the relay node deployment locally, we can only calculate the ratio among ne1, n_e2 and ne3 in constraint 1. Constraint 2 ensures the proper location of merge vertex must be inside the block. Constraint 3 can guarantee all energy consumption of relay nodes on three Steiner edges of the block must be the same.

In Eq. (4.9), we only find the proper location of merge vertex, i.e., XM V and YM V, and the ratio among n_e1, n_e2 and n_e3, so the number of variables of Eq. (4.9) is only 5. The num-ber of variables is fixed and will not change due to different network topology. Although the complexity of second method is little higher than the first one, the performance of it must be better.

Now, let us summarize our heuristic approaches briefly. Before we start to execute the heuristic approaches, we first check whether the given N relay nodes can connect the Steiner tree or not by Eq. (4.1). If N relay nodes are enough to connect the Steiner tree, the heuristic approaches can be executed. Otherwise, since the given N relay nodes is less than minimum requirement, we can not find a solution for this problem. Then, after finding out all proper locations of merge vertices, we will check whether the new network topology can be connected by N relay nodes or not. If the new topology can be connected, it will be the candidate of our final solution. Then we will repeat the above steps in T times. After that, we choose the candidate with minimum energy consumption and connected network as our final solution. Our heuristic approaches can be summarized as Algorithm 2.

Algorithm 2 Heuristic

Input: s0, . . . , s_M, XS0, . . . , X_SM, YS0, . . . , Y_SM,

β1, . . . , β_M, N relay nodes, RminandRmax, method.

Output: G(V, E)

1. Construct a Steiner tree G(V, E) with heuristic approach.

2. Calculate the traffic rate λ_ek based on the G(V, E), where ek∈ E if CheckConnectivity(N, G(V, E)) then

Calculate ˆE.

for i:= 1 to T do

Partition the network topology into blocks.

switch (method)

case 1: Execute constrained heuristic

case 2: Execute small scale non-linear programming end switch

Use Eq. (1) to calculate n_e1, n_e2, . . . and calculate ˆE^′. if( ˆE^′ < ˆE) and CheckConnectivity(N, G^′(V^′, E^′)) then

Eˆ = ˆE^′;

G(V, E) = G^′(V^′, E^′);

end if end for

Use Eq. (1) to calculate n_ek for edge k, e_k ∈ E

Deploy ne_k relay nodes on edge k evenly with communication range te_k = L_ek n_ek. else

The given N relay nodes can not connect the Steiner tree.

end if

Chapter 5 Simulation

In this section, we will evaluate the performance of our algorithms by numerical analysis. Since NLP can not guarantee to solve this problem in polynomial time, we just evaluate our two heuristic approaches, i.e., CH and SSNLP. We have deployed5 to 25 sensor nodes by random distribution in a field of5000m × 5000m with the sink node settled at the center (2500, 2500).

And the normalized data rate of each sensor node is randomly chosen from (0, 1]. For each number of sensor node, we generated 10 topologies for analysis. In this network, there are only250 relay nodes for us to deploy. We set the power attenuation of relay nodes α = 2, the maximum communication range of relay nodesR^max = 500m, and minimum one R^min = 10m.

The above simulation environment is shown in Table 5.1.

For comparison, we implement two deployment approaches from [10][15], namely Connectivity-only, denote as CO, and Traffic-aware, denote as TA. CO is chosen from a state-of-the-art scheme proposed in [15], which optimizes the system performance by considering

connectivity-Table 5.1: Simulation Environment

Sensor nodes 5, 10, 15, 20, 25

Data generating rate [0.1, 0.2, . . . , 1]

Maximum communication range of relay node 500m Minimum communication range of relay node 10m

Deployment area 5000 ∗ 5000m²

Location of sink node (2500, 2500)

Strategy of sensor deployment Random distribution

1

Figure 5.1: Normalized network lifetime.

only. In [15], there are multiple versions of scheme. Here, we use the1-connectivity version and construct a Euclidean Steiner minimum tree [1] for the network topology, and CO serves as a baseline. TA is chosen from [10] which is the first study accommodate the heterogeneous traffic flows in relay node deployment for WSN.

We use network lifetime, defined as the lifetime of the first depleted relay node, as our metrics for evaluation. The first depleted relay node can serve as a good indicator for the end of the network lifetime because if the first relay node is out-of-battery, the data of some sensor nodes can not be relayed to the sink node in1-connected WSN. In the following simulation results, they are normalized by the base-line scheme CO. We use MATLAB [2] as our simulation tools, and the interior point methods[5] to solve non-linear problems. Fig. 5.1 shows the results of the network lifetime with different number of sensor nodes. When the number of sensor nodes increases, the lifetime of both SSNLP and CH increases and is higher than lifetime of TA.

Since SSNLP has considered traffic rate, length of edge and ratio of relay nodes among three edges, we can observe that the lifetime of SSNLP is always better than CH’s in Fig. 5.1. We

150

Figure 5.2: Normalized residual energy.

have also evaluated the residual energy among these approaches. Since CO can not guarantee the energy consumption of each relay node is the same, we only evaluate the residual energy of TA, SSNLP and CH when the lifetime of CO is finished. And the total residual energy can be estimated as

Eˆ_residual = X

ek∈E

[ ˆE− λe_k(L_ek

n_ek)^αT] · ne_k,

where T is the network lifetime of CO, and the result is shown in Fig. 5.2. When the number of sensor increases, the normalized residual energy of TA, SSNLP and CH increases. And because the energy consumption of SSNLP and CH is less than TA, the normalized residual energy of them is higher than TA’s.

Fig. 5.3(a) and Fig. 5.3(b) show that the improvement from TA to SSNLP and CH, respec-tively. We observe that SSNLP can always obtain8 ∼ 10% improvement with different number of number of sensor nodes, and CH can obtain5 ∼ 7% improvement. Although the averages of the improvement percentage of both SSNLP and CH are not very high, we can notice that the

0 5 10 15 20 25

5 10 15 20 25

Improvement Percentage

Number of Sensor Nodes (TA-SSNLP)/TA

(a)

0 5 10 15 20 25

5 10 15 20 25

Improvement Percentage

Number of Sensor Nodes (TA-CH)/TA

(b)

Figure 5.3: The performance improvement of CH and SSNLP with different numbers of sensor nodes.

0

Figure 5.4: The performance improvement of CH and SSNLP with different numbers of relay nodes.

be23% and the minimum can be 5% in SSNLP. How many improvement percentages obtained by SSNLP and CH is decided by the configuration of the network topology.

Next, we fix the network topology with only5 sensor nodes, and set the number of relay nodes from 100 to 1000. Fig. 5.4(a) and Fig. 5.4(b) show that the improvement from TA to SSNLP and CH, repsectively. We can observe that no matter how many relay nodes there are, CH and SSNLP can also retain a pretty good improvement. SSNLP can maintain the improvement percentage around 10 ∼ 13%, and CH can keep the improvement percentage around6 ∼ 8%.

From the above results, we realize that only considering connectivity is the worst case for relay node deployment, and only considering traffic volume based on Euclidian Steiner tree also is not the best way. The results show us that we should construct a traffic-aware network topology and the strategy of the relay node deployment must be according to the traffic volume of each edge as well.

Chapter 6 Conclusions

In this thesis, we showed some problems about deploying relay nodes on Euclidian Steiner tree. We proposed some algorithms to modify the existing Euclidian Steiner tree to adapt the traffic volume in reality. Firstly, we proposed a weighted moving algorithm. In this algorithm, we modeled the relay node deployment problem as a non-linear programming. In this pro-gramming, we could guarantee that the network is connected and also accommodate the traffic volume. But the non-linear programming could not ensure to solve the problem in polynomial time, so we had proposed two heuristic approaches, which could finish in polynomial time, to compare with the previous studies by MATLAB. According to the simulation results, we observed that the performance of these two heuristic approaches was better than all previous studies. And no matter what circumstances, they could obtain a pretty good improvement.

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Vita