## 3 Congestion Prediction Model
### Traffic Prediction  
#### 3.4.1 Packet Generation Prediction by Node  
This paper uses the concept of production economics to calculate packet production. When evaluating the mutual influence of all variables, relative change, (_i.e._, elasticity) is very important. Resilience analysis becomes easier when the packet product is expressed as a natural logarithm. The packets produced in the network are modeled using the Cobb-Douglas production function, which models the relationships of the inputs and outputs of the factors of production [40]. The Cobb-Douglas production function is used in economics and is often used as a utility function and is essential to reflect the modeling after analyzing the  
Figure 3: Network Modeling Concept.  
future effectiveness of the network. If the number of sensing actions and the input power are increased \(h\) times in the network modeling described in this paper, we can assume that the number of packets is also increased h times, which indicates a constant returns to scale character. The node located in the \(i\)th area produces packet \(D_{i}\) by using sensing action \(S_{i}\) and Energy \(E_{i}\). Since the production function was assumed to be a Cobb-Douglas production function, \(D_{i}\) is as follows in Equation (2):  
\[D_{i}=A(S^{n})_{i}(E^{\delta})_{i} \tag{2}\]  
Since it is a constant returns to scale function, \(\mu+\delta=1\). \(A\) is a constant with the value of the amount, and represents the number of sensing nodes.  
Sensing \(S_{i}\) is determined by the production cost per packet \(p_{i}\) and CPU operations \(U_{i}\), and energy \(E_{i}\) is assumed to be determined by the production cost per packet \(P_{i}\) and power consumption:  
\[S_{i}=\frac{\mu P_{i}}{U_{i}} \tag{3}\]  
\[E_{i}=\frac{\delta P_{i}}{r_{i}} \tag{4}\]  
This formula was not used for actual traffic prediction in this paper, but was modeled to identify the total production of packets for the total lifetime of the node depending on the node's lifetime and the number of events. There are three node types considered in this study: 100% (node 1), 70% (node 2), and 40% (node 3). When energy is 100%, 100 min of sensing coverage is possible, and when it is assumed that one node is sensed once per minute, the same number of packet nodes from 1, 2 and 3 will be produced in 40 min. However, when comparing the total product per node after 100 min, node 1 produces 1/1 s \(\times\) 1,000 s = 1,000 packets, node 2 produces 700, and node 3 produces 400. For this type of estimation, network utility analysis may be used for the future by representing the total amount of product and by using the Cobb-Douglas production function according to the amount of energy and sensing. In addition, the derived total number of packets produced can be compared with the traffic prediction of Section 3.4.2.  
Prediction of the Amount of Data by the Types of Data (Event/Periodic)  
Data transmission from the node can be divided into Periodic Data transfers \(\mathrm{F}^{\mathrm{w}}_{i\text{-sink}}\) and Event Data transfers \(\mathrm{F}^{\mathrm{v}}_{i\text{-sink}}\). Therefore, the Data node from the total throughput is \(\mathrm{F}_{i\text{-sink}}=\mathrm{F}^{\mathrm{w}}_{i\text{-sink}}+\mathrm{F} ^{\mathrm{v}}_{i\text{-sink}}\). The following \(\mathrm{F}_{i}\) in Equation (5) is the traffic volume in each traffic zone in Figure 3 showing relay node throughputs:  
\[\mathrm{F}_{i}=\sum_{\forall k,k\
eq\mathrm{sink}}\mathrm{F}_{k-\mathrm{sink }}\,,i=\mathrm{sink}\]  
\[\mathrm{F}_{i}=\sum_{i\
eq\mathrm{sink}}\mathrm{F}_{i-\mathrm{sink}}+\sum_{k =1}^{i-1}\mathrm{F}_{k-\mathrm{sink}}\,,1<i<\mathrm{sink} \tag{5}\]  
\[\mathrm{F}_{i}=\sum_{i\
eq\mathrm{sink}}\mathrm{F}_{i-\mathrm{sink}}+\sum_{k =i+1}^{\mathrm{n}}\mathrm{F}_{k-\mathrm{sink}}\,,\mathrm{sink}<i<n\]  
\[\mathrm{F}_{i}=\mathrm{F}_{i-\mathrm{sink}}\,,i=1,n\]