## 3 Problem Formulation
### Multi-Radio Switching  
We consider the WSN formed by static nodes and a sink node in [7,8] which are always powered to eliminate energy constraints, which are a major concern in WSN. The multichannel assignment will be presented in two ways. In the first way, each sink node is equipped with a single radio and can switch channels to receive data packets. In the other way, the sink node is equipped with multiple radio interfaces and has a distinct channel assign to each radio. However the nodes transmitting to the sink remain on the same channel and are not allowed to switch channels during transmission. In the case that changes or failure of any node or radio interface should occur, the sink node will update itself about these changes. The multi-channel multi-radio (MCMR) problem can be modeled as an undirected graph where vertices denoting radios comprised the wireless network and a set of unidirected edges between vertices representing node link. The rationale is to prevent nodes on the same channel to attempt to send to the same radio interface. Nodes are numbered to prevent conflicts. Transmission take place interns base on number. Unidirected graph modeling [45] has been used to model channel assignment in wireless networks. Consider a graph _G = (V, E)_ where \(V\) is the set of wireless radios at the sink and L is the set of communication links between radios and transmitting nodes. For example, there are three radio interfaces at the sink node as illustrated in Figure 2, where _R1...i_, each radio interface corresponds to one or more edge nodes _Ni...x_ but only one link can be active at any given time. The broken line represents the inactive link and only becomes active when the associated radio switches to the active channel of that node. Each transmitting node is assigned to a channel and each radio only switches to a node on the same channel. A radio can receive data packets from more than one node on same channel. We derive the radio link as: _R1_i_s_N_xn, where \(x\) and \(n\) is the node and channel number respectively. Consequently, only three links in Figure 2 can be active simultaneously, if \(D\), \(E\), and \(F\) attempt to transmit when \(A\), \(B\), and \(C\) is transmitting then a radio link conflict graph colouring problem occurs. To avoid a conflict graph colouring problem from occurring each Edge (_E_) that connected to a Vertex (_V_) is assigned a different colour. In the case where there are two \(E\) connected to each \(V\), they are given colours True and False, this means that all the True can be transmitted at the same time and all the false become inactive. The color false becomes active when the radio link switches to the inactive edge. Figure 2: Sink Node with 3 Radios Receiving from 6 Transmitting Nodes on 3 non-overlapping channels (C1, C2, C3). G represents a graph, while \(V\) represents Vertex and \(E\) represents Edge(s):  
* For this algorithm, \(V\) is a single Vertex while \(E\) can be 2 or greater (\(E\geq 2\)). * The equation \(G=(V,\,E)\) can therefore be replaced by \(G=(V,\,E_{l})\), where the subscript '\(I\)' represents the variable for the number of Edges available. The graph methodology is use to express the relationship between two communication links (represented by \(E\) in the equation) sending data to a single radio receiver (represented by \(V\) in the equation) non-simultaneously. Therefore at no time should both communication links be active to the common receiver/radio interface. The Algorithm 1 represents a system using two communication links or edges and its notation represented in Table 1. The objective is to ensure that only one communication link is active at any one time. The algorithm is laid out in a semi programming format.