(for example, it has no direct link with the BS), variable xiin (1) is bound to zero. Therefore, the existence of leader-ineligible sensors eases the scheduling work by reducing the population of leader candidates.
Round-robin leader scheduling (RR) equalizes the values of xi’s, which is generally far from optimal. In Reference [5], an improvement on RR is proposed. This approach sets up a threshold of distance, and nodes are not allowed to be leaders if their distances to their neighbours along the chain are beyond the threshold.
Instead of finding an optimal solution, we propose a simple rule called maximum residual power first (MRPF) for leader scheduling. As the name suggests, MRPF selects the node that has the maximum residual power to be the leader in each round of data collection. Residual power information can be piggybacked with data messages as part of the aggregated data. If every node attaches its own power level to data message and let the BS find the maximum value, it will incur an additional OðnÞ overhead on every message. A better approach is to let every node compare its power level with the one attached with the incoming data message (if any) and send only the larger. This is similar to existing distributed maximum-finding algorithms on rings [12–15] and the message overhead is only Oð1Þ:
Recall that the BS broadcasts the result of leader scheduling to all sensors before each data-collection round. The energy consumed in receiving broadcasts is not taken into account in the above model. If it is to be considered, a slight modification on the modelling is required.
Suppose that receiving one broadcast consumes b unit of energy. As there are P
ixi data-collection rounds in total, all sensors uniformly spend bP
ixi unit of energy on receiving broadcasts. Taking account of this quantity, (1) becomes
A x1 x2 x3
... xn
0 BB BB BB BB B@
1 CC CC CC CC CA
4 E1=b E2=b E3=b
... En=b 0 BB BB BB BB B@
1 CC CC CC CC CA
ð2Þ
This formula is essentially the same as (1) with the only exception that the initial energy of each sensor Ei is uniformly divided by b: Therefore, if hw1; w2; . . . ; wni is the optimal value for hx1; x2; . . . ; xni that maximizes P
xi subject to (1), hw1=b; w2=b; . . . ; wn=bi will be the solution that maximizes P
xi subject to (2). In other words, the consideration of energy expense on broadcasting only scales down the optimal value by a constant. It does not make the problem harder or easier to deal with.
The same conclusion also applies to other energy dissipation sources that have an equal effect on all sensor nodes. An example is the energy expense in idle mode.
4.1. Performance of chain structure
We measured network lifetime, the number of data collection rounds that can be achieved by all chain construction approaches. Sensor networks of sizes 50 50 and 100 100 were considered, with a BS located at ð50; 150Þ; ð50; 200Þ; or ð50; 300Þ: The number of nodes was set to 50, 100, and 200, respectively, with initial power of each sensor set to 50J. Round-robin leader scheduling was used in the experiments.
Figures 10–12 show the results averaged over 100 experiments. The results of direct-insertion, shortest-appending, shortest-MST, MST-insertion, and MST-reduced are nearly identical (they are ‘good’ methods) and are collectively denoted as ‘Others’ in these figures. Direct-MST generally performs better than PEGASIS does but worse than others (these two are ‘naive’
methods). These results provide insights on how well chain construction algorithms improve overall energy performance:
* Adding more sensor nodes into a bounded network increases network density, and thus decreases average distance between nodes. As a consequence, network lifetime increases as inter-sensor communication costs less power. Observe that the performance gain with good methods is nearly proportional to the number of sensor nodes. In contrast, the results of naive methods are not attractive.
* Fixing the number of sensors but increasing network size increases average distance between nodes. This is why the performance of naive methods degrades as network size increases. In contrast, good methods nearly perform the same even when network size increases.
* When the BS is further away from the network, the performance difference between good and naive schemes becomes insignificant (Figure 12). The reason is that under such condition, leader-BS communications dominate overall energy consumption. So a better chain structure does not improve network lifetime significantly in this scenario.
4.2. Performance of leader scheduling
We measured and compared the performance gains brought by several leader scheduling schemes including MRPF, RR, and RR with distance-based leader eligibility rule. A network of
0 2000 4000 6000 8000 10000 12000 14000
50 100 200
Number of nodes
Rounds
0 2000 4000 6000 8000 10000 12000 14000
Rounds
Number of nodes (a) (b)
50 100 200
PEGASIS Direct-MST Others
PEGASIS Direct-MST Others
Figure 10. Number of rounds before any node exhausts its power in: (a) 50 50 network; and (b) 100 100 network. The BS was located at ð50; 150Þ:
ENERGY OPTIMIZATION FOR CHAIN-BASED DATA GATHERING
Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (in press) DOI: 10.1002/dac
size 50 50 was considered, with a BS located at ð25; 150Þ or ð25; 250Þ: All nodes were assumed to have power 50J initially. The chains to be tested with leader scheduling schemes were produced by PEGASIS. Figure 13 shows the results, where each result is obtained by averaging 10 experiments.
It can be seen that MRPF performs slightly worse than the optimal result obtained by a linear-programming problem solver. MRPF significantly outperforms RR. When the sensor population is low, RR with distance-based leader eligibility rule (RR with threshold) results in fewer data-collection rounds than RR does, and the gap increases as the threshold value of distance decreases. The reason is that the loads on leader nodes cannot be fairly shared if only few nodes are eligible for leaders. When the sensor population is sufficiently high, RR with threshold outperforms RR. Therefore, a critical issue of using RR with threshold is to determine an appropriate threshold so that leader-eligible nodes and others fairly share the communication load, which was untold in the original paper [6].
Figure 14 shows variances of all node’s residual power when the first node dies. Observe that MRPF yields the minimal variance, meaning that it successfully equalizes power consumption among all nodes. The optimal leader scheduling does not minimize the variance of residual power but still performs good. This suggests the existence of another scheduling rule other than MRPF, which is left as our future work. The RR family does not perform well, but the results tend to be acceptable when the sensor population is getting high.
(a) (b)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
50 100 200
Number of nodes
Rounds
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Rounds
PEGASIS Direct-MST Others
50 100 200
Number of nodes PEGASIS
Direct-MST Others
Figure 12. Number of rounds before any node exhausts its power in: (a) 50 50 network; and (b) 100 100 network. The BS was located at ð50; 300Þ:
0 1000 2000 4000 3000 6000 5000 7000
Rounds
Number of nodes (b)
(a)
50 100 200
0 1000 2000 4000 3000 6000 5000 7000
Rounds
Number of nodes
50 100 200
PEGASIS Direct-MST Others
PEGASIS Direct-MST Others
Figure 11. Number of rounds before any node exhausts its power in: (a) 50 50 network; and (b) 100 100 network. The BS was located at ð50; 200Þ:
Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (in press) DOI: 10.1002/dac
4.3. Implementation issues
The implementation of the proposed method demands some capability from sensor nodes. Each sensor should be equipped with a complementary device that enables the sensor to detect its own location. The location information is reported back to the BS before any data-collection activities. After that, the locating device can be shut down to save power. Each sensor should also have the capability to measure its residual power level. As mentioned, each sensor node should have power control capability so that minimum energy is expended to reach intended recipients.
It is a challenge to apply the proposed approach in harsh communication environments.
Signal propagation problems such as interference and multi-path fading cause sensor nodes to spend more transmission power then expected for a desired signal-to-noise ratio, introducing estimation errors to our power consumption model. However, the effects of imperfect
10
(a) (b)
20 30 40 50
2 4 6 8 10 12 14 x 104
Number of Nodes
Total Rounds
10 20 30 40 50
0 2 4 6 8 10 12x 104
Number of Nodes
Total Rounds
Optimal MRPF RR
RR(Threshold = 10) RR(Threshold = 20)
Optimal MRPF RR
RR(Threshold = 10) RR(Threshold = 20)
Figure 13. Number of rounds before any node exhausts its power: (a) the BS was located at ð25; 150Þ; and (b) the BS was located at ð25; 250Þ:
10 20 30 40 50
0 50 100 150 200 250 300
Number of Nodes
Variance of Residual Energy
(a)
Optimal MRPF RR
RR(Threshold = 10) RR(Threshold = 20)
10 20 30 40 50
0 50 100 150 200 250 300 350
Number of Nodes
Variance of Residual Energy
(b)
Figure 14. Variances of residual power when the 1st node dies: (a) the BS was located at ð25; 150Þ;
and (b) the BS was located at ð25; 250Þ:
ENERGY OPTIMIZATION FOR CHAIN-BASED DATA GATHERING
Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (in press) DOI: 10.1002/dac
communications are environment dependent and the estimation errors are not easy to formulate. It is therefore a future work to include environmental factors in constructing an energy-efficient data-collection chain.