In this chapter one of the underlying assumptions of the corona model, namely that nodes always forward their packets to a node in the next inward corona, has been examined. This assumption has been used to model the number of nodes in each corona by combining it with the assumption of uniform distribution to determine that the number of nodes per corona is proportional only to the area of the corona. This is then used as a fundamental element in the analysis of the energy hole problem by Li and Mohapatra [LM05, LM07] and others including Olariu and Stojmenovic [OS06].
However, the analysis in this chapter revealed that a problem, which was termed the relay hole problem, exists whereby the underlying assumption is not true. In fact, mathematical analysis showed that there is a significant probability that the portion of a node’s reachable area that intersected the next inward corona may be empty of nodes. Moreover, even if the node was able to find a relay in the next inward corona, the relay itself may suffer from the relay hole problem which would have a knock-on effect to the node.
The analysis was confirmed through simulations and results showed that even
4.4. CHAPTER SUMMARY AND CONCLUSIONS 115
with a very high density of 100 neighbours per node (more than ten times the minimum required for guaranteed connectivity) 7% of the nodes suffer from the relay hole problem. Increasing the network density mitigates the problem to some extent but suffers from diminishing returns. Based on the mathematical analysis it is possible to show that at the incredibly high density of 1,000 neighbours per node there would still be 1.5% of nodes with the relay hole problem.
The relay hole problem not only causes an increase in latency but also makes it harder to predict the topology of the network because, despite a uniform distri-bution, not all nodes act as predicted. Whereas a node’s physical position in the network area combined with the transmission range of all nodes suggests it ought to be a certain number of hops away from the sink, the relay hole problem means that it may be more hops away than predicted.
This is important for two reasons. Firstly, the novel routing protocols proposed in this thesis rely on the nodes acting as predicted in order to achieve high inner-corona balance. Although the effect of the relay hole problem on the balance produced by the proposed protocols was not quantified, the problem is likely to have had a statistically significant impact given the proportion of nodes that are affected by it.
More widely, the relay hole problem means that the analysis of the energy hole problem that has previously been carried out is not entirely accurate. Although the energy hole problem certainly exists, the analysis which assumes a simple relationship between the number of nodes in a corona and the corona’s area needs to be revisited. Likewise, solutions to the energy hole problem that rely on this assumption also need to be revisited.
For example, the solutions that suggest a non-uniform distribution (see Section 2.2.5) all make this assumption in order to calculate the energy requirements of each corona and therefore the number of nodes needed in each to balance those requirements. However, the relay hole problem shows that simply placing a node in a physical corona does not guarantee that it will be topologically inside that corona. This calls into question the effectiveness of these solutions because in order to balance the workload they carefully calculate how many nodes must be in each corona and do so assuming that physical placement inside a corona is equivalent to topological placement. In light of the relay hole problem this approach to solving the energy hole problem needs revisiting.
The work presented in this chapter was published in [KF12a].
Chapter 5
Degree Balancing
5.1 Introduction
The main aim of this thesis is to propose fully distributed routing protocols for increasing inner-corona balance in static sensor networks. As discussed in Section 2.4, fully distributed protocols are those in which nodes select their own parent nodes based on information that originates with the neighbours they are in direct communication with. This limitation means that for distributed routing protocols, nodes cannot use the workload of their potential parents as a factor in their decision since information about workload requires gathering information from all descendants of a node. While it is certainly possible to gather the information, the cost of doing so is high as a large number of control packets must be passed through the network.
One measure that can be used is the degree of each node which is the number of neighbours that a node is directly connected to in the routing tree and is therefore local information. Since the protocols in this thesis produce a single, static routing tree, every node has only one parent and therefore a node’s degree can be replaced with the number of children it has adopted.
Degree balancing is about minimising the variation in node degree among nodes in the same level of the routing tree. For the networks considered in this thesis, it is impossible for all nodes to have the same node degree because the number of nodes in neighbouring coronas varies through the network. Macedo analysed the
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Level Children per Parent Number of Nodes in the Level
1 3 n
2 1.6666666667 3n
3 1.4 5n
4 1.2857142857 7n
5 1.2222222222 9n
6 1.1818181818 11n
7 1.1538461538 13n
8 1.1333333333 15n
9 1.1176470588 17n
10 1.1052631579 19n
Table 5.1: Values derived from equation (5.1) showing the average number of children per parent and the number of nodes in each level, where n is the number of nodes in level 1.
average number of children adopted per parent in each corona, ci of a uniformly distributed network, Ci [Mac09] and his result is shown in equation (5.1). It should be noted that according to the corona model a node’s corona number is equivalent to its level in a minimum-depth routing tree.
Ci = 2i + 1
2i − 1 (5.1)
Macedo’s analysis was designed to demonstrate that the node degree was not constant across the network; rather the average number of children per parent decreased further away from the centre. However it also shows that, except for the nodes in the first level, no node can have exactly the average node degree.
This is because, as Table 5.1 illustrates, the average number of children per parent is a fraction for all levels except the first. The best that can be achieved is to minimise the variation in node degree such that some nodes have one child and some have two but none have three or zero children.
This form of load balancing is what I call degree balancing and involves minimising the variation in node degree among nodes of the same level. In this chapter, degree balancing is analysed with two aims. The first is to show that the probability of degree balancing alone resulting in perfect inner-corona balance is extremely low. In this context, “alone” means that the routing algorithm focuses only on the degree balance of a single level of the routing tree at a time and does not