## 2 System Model and Assumptions
### Schedules and Network Throughput  
The slot scheduling matrix for the network with \(N\) nodes and a period of \(T\) is denoted as \(S_{NT}\). The element \(s_{j,t}\) in the matrix \(S_{N\times T}\) represents the state of node \(j\) at time slot \(t\). In a half-duplex transceiver configuration for unicast transmission mode, if \(s_{j,t}=i>0\), it indicates that node \(j\) sends data to node \(i\) in time slot \(t\); if \(s_{j,t}=-i<0\), it means that node \(j\) receives data from node \(i\) at time slot \(t\); and if \(S_{j,t}=0\), it signifies that node \(j\) is silent during time slot \(t\).  
Assume the communication distance of the network is denoted as \(RC=k_{1}\) and the interference distance is denoted as \(RI=k_{2}\). If node \(j\) transmits information to node \(i\) in slot \(t\), and node \(i\) receives it in slot \(D_{ij}\), the conditions for receiving the signal from node \(j\) are as follows:  
\[s_{j,t}=i\Leftrightarrow s_{i,t+D_{ji}}=-j,D_{ji}<k_{1} \tag{3}\]  
To ensure successful reception of the signal from node \(j\), it is essential that no signals from other nodes reach node \(i\) within the same time slot, as indicated in Formula (4).  
\[s_{i,t}=-j\Rightarrow s_{j,t-D_{lt}}\leq 0,D_{lt}<k_{2},l\
eq i \tag{4}\]  
Assume that the transmission rate of the link between nodes \(i\) and \(j\) is \(v\) bps, and the amount of information transmitted in one time slot is \(v\mu\) bits. The corresponding block transmission delay \(\mu\) equals the unit time slot length \(\tau\). The average network throughput \(Y\) is defined as the total amount of successfully received information by all nodes in the network within a unit of time, normalized by the link rate. The average throughput \(Y\) of the scheduling matrix \(S_{NT}\) with period \(T\) is represented by the equation  
\[Y=\frac{1}{T}\sum_{t}\sum_{j}1(s_{j,t}<0) \tag{5}\]  
where \(1(\cdot)\) is a truth function, where a value of 1 is true and 0 is false. Considering the physical layer link transmission rate, the final network throughput is \(Y_{f}=v\mu*Y\).  
In [16], the authors show that the normalized throughput of an \(N\) node unicast network does not exceed \(N/2\) when the propagation delay is non-zero.  
The following two examples illustrate the fundamental principle of time slot scheduling in the context of long propagation delay. We consider a 3-node equilateral triangle network with nodes positioned at the vertices of equilateral triangles, where the normalized communication distance and interference distance are denoted as \(RC=1\) and \(RI=1\), respectively. The transfer time for a message block is given by \(\mu=\tau\). The delay matrix D and scheduling matrix S for this network are as follows:  
\[D=\left[\begin{array}{cc}0&1&1\\ 1&0&1\\ 1&1&0\end{array}\right]\ S_{3\times 4}=\left[\begin{array}{cc}2&3&-&3&- 2\\ -3&-&1&1&3\\ -2&1&-&1&2\end{array}\right] \tag{6}\]  
For the three-node network with an equilateral triangle topology, three nodes can successfully transmit 6 units of information in the \(T=4\tau\) timeslot. The network throughput is \(Y=6v\tau/4\tau=1.5v\). When the propagation delay factor is not considered, the three nodes complete data transmission in three-time slots respectively, and the throughput is \(Y=1v\). Therefore, the throughput of the three-node network with non-zero propagation delay is 50% higher than that of the network with zero propagation delay.  
For the three-node network with an isosceles triangle topology, the delay matrix and scheduling matrix of the network are expressed as  
\[D=\left[\begin{array}{cc}0&1&2\\ 1&0&2\\ 2&2&0\end{array}\right]\ S_{3\times 8}=\left[\begin{array}{cccc}2&-&2&- 3&3&-3&3&2&-2\\ 1&-&1&3&-3&3&-3&1&-1\\ 1&2&1&2&-&2&-&1&-&2&-1\end{array}\right] \tag{7}\]  
Three nodes can successfully transmit 12 units of information in the \(T=8\tau\) time slot. The network throughput is \(Y=12v\tau/8\tau=1.5v\).