## 3 Problem Formulation  
### Problem Description  
For the purposes of system modeling, a multicast system can be modeled as a tree. Sink node can send messages to receivers in a multicast group within the multicast tree is predefined. The sender can be viewed as a root and receivers can be viewed as leaves. Cost depends on the size and scalability of the tree, which itself depends on whether it is a single multicast tree or multiple multicast trees. Figure 4 is an example of a common case: two groups in a multicast sensor network. For each group, the root is the sink node of the multicast tree. Some sensor nodes of the multicast tree stand for the receivers in the multicast group, and others are used to forward data. In an operation sensor network, a multi-rate multicast wireless sensor network is considered. This means each sensor node can request different quality data streams. The sink node may encode these different requirements into several different layered streams for subscribers through single or multiple multicast trees. Each source represents a distinct group, which has its own end-to-end delay. However, most multiple-multicast routing problems of their objectives focus on the way of finding a set of routing trees satisfied with constraints. One of these constraints is limited bandwidth. The link capacity of the wireless sensor network might be contented by how many sensor nodes are existed and the traffic sessions are created in the routing paths of the multicast groups. Briefly, our model is designed to determine the following: (1) what multiple-multicast-group routing strategies are optimal; (2) how much capacity is allocated to the selected links used in the multicast sensor network; (3) what is the minimum end-to-end intra-delay per path in a multicast group; (4) How do various approaches towards achieving the minimum end-to-end inter-group experience delay among different multicast groups. In the model which is shown in Figure 5, we make assumptions regarding to consider the delay fairness to find the minimum end-to-end delay while dealing with constrains of a multicast WSN. Based on the purpose, listed below are our assumptions, giv  
Figure 4: Multi-rate multicast wireless sensor network. Figure 5: Multicast sensor networks. **Assumptions:**  
All routers are stationary. Each link is adopted with a M/M/1 queuing model. The delay function \(d=1/(C-F)\) subject to \(C>F(C\) means the capacity on a link, \(F\) means the aggregation flow). The aggregation flow is always less than the capacity on a link, otherwise the congestion occurs results in system crashed. When the capacity is given, the aggregated flow is finite and less than the given capacity, otherwise the extra flow has to transmit through other links by routing assignments. Once the capacity is given as a constant, the buffer size of each router does not required to be infinite. Just set an enough size, say \(\geq\)link capacity, to handle the buffer requirement. In this way, the backbone capacity is also large enough to handle the traffic. Because (1) bandwidth of the sensor network is small and much less than the backbone fiber capacity, (2) the range to calculate the end-to-end delay is set from source to sink node, the do not need to consider the backbone capacity, (3) the backbone capacity is larger than the sensor network, the backbone link will not cause congestion or delay on the wireless sensor network. **Given:**  
The set of all nodes in the network. The set of all links in the network. The set of multicast source nodes. The set of destinations for each source. The set of paths from each source to it destinations. The discrete traffic requirements for subscribers in multicast groups. The degree of importance of each multicast group. The minimum hop counts from the farthest destination node on each multicast group. **Objective:**  
To minimize the end-to-end inter-delay among multicast groups. **Subject to:**  
The flow of each link is limited by its allocated link capacity and the maximum traffic of its all sublinks.