## 7 Quadratic Adaptive Replication (QAR)
### Analytical Model  
In this section we present our analytical model that will enable us to establish the optimal number of replicas so as to minimize the overall network traffic. We shall start by developing a model for the overall network traffic generated by consumption and production nodes across the network. The overall metric is the total traffic load, measured in m-bits/sec that is supported by the network, to take into account both the data rate and the traveled distance across the WSN. The locations of nodes of the wireless sensor network are fixed and modeled by a homogeneous spatial Poisson Point Process \(\Pi_{n}\)[55], _i.e._, a random set of points on the plane, with intensity \(\lambda_{n}\) nodes per unit area. Production and consumption events generated by network nodes are in turn modeled via independent homogeneous spatio-temporal Poisson Point Processes \(\Pi_{p}\) and \(\Pi_{c}\) each with intensities \(\lambda_{p}\) and \(\lambda_{c}\) events per unit time and unit area respectively. There are \(\lambda_{r}\) nodes per unit area placed in a grid fashion serving as replication nodes. Although the model corresponds to one on an infinite plane, we shall restrict attention to a fixed region \(\mathcal{A}\subset\mathbb{R}^{2}\) modeled as a convex set with area \(A=|\mathcal{A}|\), and then optimize operation on \(\mathcal{A}\), roughly ignoring edge effects. On average there are \(N_{r}=\lambda_{r}A\) replication nodes in \(\mathcal{A}\). Thus, a quadratic hierarchy level \(d\) could be defined as, \(d=\sqrt{N_{r}}\). When consumption dominates production, consumers retrieve data from the closest replication node. Then the average distance between a node and its closest replica is roughly the average distance between two nodes inside one of the replica's square cells. This distance can be modeled as \(\frac{\delta}{\sqrt{\lambda_{r}}}\) with \(\delta=0.52\) (the same used in ToW). Taking into account that the total number of consumption events in a sensornet of area \(A\), is \(\lambda_{c}A\), the next expression defines the overall network traffic associated with consumption traffic (\(T_{c}\)),  
\[T_{c}(\lambda_{r})=\alpha\lambda_{c}A\frac{\delta}{\sqrt{\lambda_{r}}}\ \ \text{ bits-m/sec},\]  
where \(\alpha\) is a proportionality constant corresponding to the average number of bits per consumption event that are exchanged between the consumer and its nearest replication node. Next we consider the replication cost when data is produced. This cost is composed of two different factors: the first one is due to a producer sending information to its closest replica and the second cost to the replication process from that closest replica to the remaining replication nodes. Therefore, the former cost is calculated in the same way than the consumption cost. The latter needs to consider all the branches in the generated replication tree. Since we rely on a grid-replication structure, the length of all the branches is exactly the same. It is the distance between the center of two neighbour square cells, that is, the distance of one side of the square cell, which is \(\frac{1}{\sqrt{\lambda_{r}}}\) in a square-unit area. The overall production traffic in a square WSN field of area \(A\), is composed of : (i) the number of branches to be traversed by the replicated information, which is the number of replicas minus 1, thus  
Figure 7: Overall network traffic generated by QAR using different number of replicas (A = 1,000 \(\times\) 1,000, N = 5,000, \(\alpha\) = 200 bits, \(\beta\) = 100 bits, \(N_{p}\) = 100; \(N_{c}\) = 2000, Iterations = 100). \(\lambda_{r}A-1\)\((N_{r}=\lambda_{r}A)\); and, (ii) the number of production events in the sensornet, \(\lambda_{p}A\).