## 3 Problem Formulation
### Multiple Sink Nodes  
We will increase the number of sink nodes to collect data from receiving nodes. The sink nodes will be equipped with a single radio and will be required to do channel switching in the same manner as in [7, 8]. The advantage is that all data from senders will be received by more sink nodes located in strategic positions. This will eliminate the burden encountered by a single sink node.  
Figure 1: Design overview for multi-channel multi-radio.  
We assign multiple radio interfaces in the sink node to receive data from each non-overlapping channel. The non-overlapping channels are a key technique to minimize signal interference and increase network capacity as such the multiple non-interfering channels are use to transmit data simultaneously and eliminates interference by another transmission. Each radio is assigned to a channel from each sending nodes. This will eliminate channel switching to all sending nodes by a single radio interface. Figure 1 shows modification at the MAC of the existing MACs protocol stack [46] this incorporate multiple radios, a new component is added to define the radio and radio number is set in the TCL script of the NS2. A new field is also created in the MAC into the packet header to index the channel object. This helps to achieve conflict free or reduce interference among neighboring nodes. To reduce communication interference nodes within communication range sense the network and conduct channel switching as illustrate in [7,8].  
We take into consideration that it is not practical to have same number of radios and same number of channels at all times. The practicality of this depends on the network size. A medium to large sized network may have more nodes sending data to the sink.  
By taking advantage of physical characteristics of the radio environment, the same channel can be reused by two or more nodes provided that the nodes are spaced sufficiently. To avoid co-channel interference we use non-overlapping channels. Since nodes are aware of all the channels at start-up and are able to switch channels based on a set of criteria from [7,8], the nodes sending packets to the sink are set to operate on a particular channel. All nodes are placed where they are in reach of the sink but separated by enough gaps between sending nodes. The reason for such arrangement is to ensure that radio interface switching between nodes on same channel will avoid co-channel interference.  
Formally, channel assignments problems have been modeled as:  
* Graph based [47,48], where the vertices \(V\) correspond to nodes and edges correspond to pairs of stations whose transmission areas intersect.
* Ring based [48] is considered as a form of vector where the size of the ring is a sequence of \(n\) vertices.
* Grid based [48] is considered a form of vector represent tessellations of a plane with regular polygon, where the size of the grid has row and column indexed from top to bottom and from left to right. The grid based can be classified as:
* Tree based [48], where an undirected graph _T = (V, E)_ is a free tree when it is connected and has exactly \(|V|-1\) edges.  
These assignment techniques all use various vectors coloring problems which are based on arithmetic progressions to solve the channel assignment problems.