## 3 Problem Formulation  
### Mathematical Modeling  
The environment considered here is multichannel WSNs. This paper addresses the problem of determining a good QoS with the average cross-network packet delay while taking \"user perspective\" and \"system perspective\" into consideration. The following are details of the problem identification:  
**Assumptions:**  
1. The channel assignment for each sensor node is fixed for a long period.  
2. Each sensor node is stationary.  
3. Each sensor node is equipped with multiple wireless interfaces, each of which operates on an individual and non-overlapping channel.  
4. Each sensor node can simultaneously communicate with its neighbors within transmission range without interference from the use of different channels for each link.  
5. A virtual node is added as the destination node to only connect to the sink via wired-line.  
6. All flows are transmitted to this virtual node via the sink.  
**Given:**  
1. The set of links.  
2. The set of sensor nodes.  
3. The link capacity of each link.  
4. The number of interference links of each link.  
5. The traffic requirement for each O-D pair.  
**Objective:**  
To minimize the average cross-network packet delay of the WSNs.  
**Subject to:**  
1. QoS constraints.  
2. Path constraints.  
3. Capacity constraints.  
4. Flow constraints.  
**To determine:**  
The minimum the average cross-network packet delay of the WSNs take system perspective and user perspective into consideration for each O-D pair.  
We propose a channel assignment heuristic and routing algorithm to maximize the transmission rate for each node and enhance system performance by WVSN planning. A similar formulation that aimed to minimize the average cross-network packet delay subject to end-to-end delay constraints for users was proposed by [PERSON] and [PERSON] [21]. This paper's formulation can be extended to use queue models. Once the channel is assigned to the link, two nodes will use the assigned channel to communicate with each other until the channel assignment is changed (_i.e._, static channel assignment). Every link is assumed to be fairly used at the scheduling phase, meaning the average time data transmission over each link is equal. In multichannel WVSNs, link capacity degrades because other links use the same channel in the interference range. Therefore, we divide the link capacity by the number of interference links in the following formulation (_i.e._, \(c_{l}=\frac{C_{l}}{I(l)}\)) [6].  
The following notations list the given parameters and the decision variables of our formulation, as illustrated in Tables 2 and 3.  
The objective function is illustrated as (IP) to minimize the mean delay on link \(l\). The expression of the form \(D_{l}(g)\) is a monotonically increasing and convex function with respect to \(g\), where \(g_{l}\) is the aggregate flow on link \(l\) measured in packets per second [6; 7; 8; 9].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(W\) & The set of Origin-Destination (O-D) pairs in the WSNs, where \(w\in W\). \\ \(P_{w}\) & The set of directed paths from the origin to the destination of O-D pair \(w\), where \(p\in P_{w}\). \\ \(L\) & The set of communication links in the WSNs, where \(l\in L\). \\ \(I(l)\) & The number of interference links of link \(l\). \\ \(\delta_{nl}\) & The indicator function which is 1 if link \(l\) is on path \(p\) and 0 otherwise. \\ \(C_{l}\) & (_packets/s_) The link capacity of link \(l\). \\ \(\gamma_{w}\) & (_packets/s_) The given traffic input of O-D pair \(w\). \\ \(D_{l}(g_{l})\) & The mean delay on link \(l\), which is a monotonically increasing and convex function of \\  & aggregate flow \(g_{l}\). \\ \(D_{w}\) & The maximum allowable end-to-end QoS for O-D pair \(w\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: Given parameters.