## 6 Simulation Evaluation of the Dynamic VWSN
### Discussion of Memory Utilization
Then, the probability that the number of \(N\)-th-tier virtual links connecting an \(N\)-th-tier module and other \(N\)-th-tier modules belonging to other \((N+1)\)-th-tier modules is \(x\), denoted by \(p\), is as follows:  
\[p=\binom{\langle Z_{CT}^{N+1}\rangle}{x}\left(\frac{1}{\langle|Sub^{N}(M^{N+1 })|\rangle}\right)^{x}\left(1-\frac{1}{\langle|Sub^{N}(M^{N+1})|\rangle} \right)^{\langle Z_{CT}^{N+1  
The entries in the routing table in the zeroth tier consist of a destination node, a forwarding node for the destination node and path weight. The entries in the routing table for the higher tier consist of an identifier of a destination module, a forwarding module for the destination module and path weight. Because of the difference of the scale of the number of nodes/modules in each tier, the memory size for the identifier of modules can be smaller in a higher tier. Therefore, we assume that the memory size of the identifier for nodes and modules are two bytes and one byte, respectively, here. We treat the path weight as the float type variable (four bytes). Then, \(b_{r}^{0}=8\) and \(b_{r}^{N}=6\) where \(N>0\).  
When node \(n\) belongs to the \(N\)-th-tier virtual module \(M_{A}^{N}\), it holds an \(N\)-th-tier connection table whose entries consist of \(N\)-th-tier neighboring module identifiers (\(M_{B}^{N}\)), the module identifier of \((N-1)\)-th-tier connected modules (\(M_{X}^{N-1}\)) belonging to \(M_{A}^{N}\) and the module identifiers of \((N-1)\)-th-tier connected modules (\(M_{Y}^{N-1}\)) belonging to \(M_{B}^{N}\). Because a zeroth-tier module denotes a node, \(b_{c}^{1}=6\) and \(b_{c}^{N}=4\) where \(N>1\).  
The entries of a long-link table consist of the end nodes of a virtual long-distance link, the next and previous forwarding nodes and a hop count from the source node of the virtual long-distance link. Because we set TTL to 50 in our evaluation, one byte is enough for a hop count from the source node of the virtual long-distance link. Then, \(b_{l}=9\).  
From the above assumptions, Equation (12) can be rewritten as  
\[B(n)=8Z_{RT}^{0}(n)+\sum_{i>0}\left(6Z_{RT}^{i}(n)\right)+6Z_{CT}^{1}(n)+\sum_ {i>1}\left(4Z_{CT}^{i}(n)\right)+9Z_{LT}(n). \tag{13}\]  
Now, we assume a two-tiered structure for one application and a situation in which \(\langle|Sub^{0}(M^{1})|\rangle=100\), \(\langle|Sub^{0}(M^{2})|\rangle=10\), \(\langle E^{M^{1}}\rangle=2000\), \(\langle k_{M^{1}}\rangle=5\) and \(\langle Z_{LT}(n)\rangle=10\). Because the expected number of nodes in a first tier module is 100 and the expected number of total first-tier modules is 10, the total number of nodes in the VWSN network is 1000. From Equations (10) and (11), the expectation of the required total memory size for tables of a node is 2152 bytes. Next, we assume a three-tiered structure for one application and a situation in which \(\langle|Sub^{0}(M^{1})|\rangle=100\), \(\langle|Sub^{1}(M^{2})|\rangle=10\), \(\langle|Sub^{2}(M^{3})|\rangle=3\), \(\langle E^{M^{1}}\rangle=2000\), \(\langle EM^{2}\rangle=20\), \(\langle k_{M^{1}}\rangle=5\), \(\langle k_{M^{2}}\rangle=2\) and \(\langle Z_{LT}(n)\rangle=20\). In this case, the total number of nodes in the VWSN network is 3000. Then, the expectation of the required total memory size for tables of a node is 2290.8 bytes. Then, we assume a four-tiered structure for one application and a situation in which \(\langle|Sub^{0}(M^{1})|\rangle=100\), \(\langle|Sub^{1}(M^{2})|\rangle=10\), \(\langle|Sub^{2}(M^{3})|\rangle=3\), \(\langle|Sub^{3}(M^{4})|\rangle=3\), \(\langle E^{M^{1}}\rangle=2000\), \(\langle E^{M^{2}}\rangle=20\), \(\langle E^{M^{3}}\rangle=3\), \(\langle k_{M^{1}}\rangle=5\), \(\langle k_{M^{2}  
management and table management. Because they can add a big overhead to nodes, how to manage device resources is a crucial viewpoint of network design. In completely centralized management systems, an overhead for collecting information of nodes explosively grows as the number of nodes in the system increases.