## 3 Problem Descriptions and Optimization
Therefore, Equation (7) can be restated as:  
\[T_{\mathbf{\omega}}^{\left(u\right)}\left(e_{\mathbf{0}}\right)=\lim_{K\to\infty}\frac {1}{K}\Bigg{[}\sum_{k=0}^{K-1}\big{(}g\left(\eta_{u}\left(E_{u,k}\right) \right)\eta_{u,k}\cdot\binom{U}{M}\times\prod_{i\
eq u,j}\eta_{i,k}\prod_{j \
eq u,j}\left(1-\eta_{j,k}\right)\big{)}\ |\ e_{\mathbf{0}},\mathbf{\omega}\Bigg{]}. \tag{10}\]Moreover, the average long-term throughput of the network is defined as the number of the packets successfully uploaded to the satellite, which can be represented as:  
\[T_{\mathbf{\omega}}\left(\mathbf{e_{0}}\right)=\sum_{u=1}^{M}T_{\mathbf{\omega}}^{\left(u \right)}\left(\mathbf{e_{0}}\right). \tag{11}\]  
This paper aims to design the packets transmission policies \(\mathbf{\omega}\) to maximize the throughput of the network, i.e.,  
\[\mathbf{\omega^{*}}=\operatorname*{arg\,max}_{\mathbf{\omega}}T_{\mathbf{\omega}}\left(\bm {e_{0}}\right). \tag{12}\]  
However, the design of transmission policies \(\mathbf{\omega}\) for the EH-STs in the satellite communication networks needs to consider the following trade-offs:  
* More packet replica EH-STs transmitting seems to provide a better performance of packet detection [7]; nonetheless, too many replicas will cause channel saturation easily in the heavy traffic load [35] and battery depletion occurs more frequently;
* larger transmission probability makes EH-STs transmit more often, which yields large access latencies due to packet collisions, especially for the heavy traffic load [4; 28]; hence, a lower network throughput is accrued;
* smaller transmission probability results in a lighter traffic load, which increases the packet transmission success rate; however, as Equation (1) indicated, smaller network throughput may be acquired. Thus, the optimal transmission policy \(\mathbf{\omega^{*}}\) reflects an optimal trade-off among energy consumption, access delay, and network overall throughput. In order to keep balance between the performance of packet detection and energy consumption, this paper assumes that each EH-ST can transmit \(l_{max}\) packet replicas in one frame at most. A specifically admissible policy \(\Omega\) that indicates the number of transmission packet replicas and transmission probability independent of initial energy state \(\mathbf{e_{0}}\) is defined below. **Definition 1**. : _A specifically admissible policy \(\Omega\) is defined as:_  
\[\Omega=\left\{ \begin{aligned} \eta:\begin{cases}\eta\left(0 \right)&=0,&e=0\\ \eta\left(e\right)&\in\left(0,1\right)&\text{and}\ l=e,&e<l_{ max}\\ \eta\left(e\right)&\in\left(0,1\right)&\text{and}\ l=l_{max},\ l_{ max}\leq e\leq e_{max}\end{cases}\right\}.\end{aligned} \right. \tag{13}\]  
**Remark 1**. : _Note that the set of the specific policy \(\Omega\) is made up of two parts. One is the number of transmission packet replicas \(l\) and the other one is the probability of transmitting these \(l\) packet replicas \(\eta(e)\). In fact, \(l\) can be an arbitrary positive integer, and it is no larger than the current energy level \(e\). However, considering the complexity of the analysis procedure and the algorithm design, we predefine a reasonable transmission policy in terms of the number of transmission packet replicas. Specifically, each EH-ST will transmit \(l_{max}\) replicas with probability \(\eta(e)\) if \(l_{max}\leq e\leq e_{max}\), or it will keep idle with probability \(1-\eta(e)\). If the current energy level of an EH-ST is less than \(l_{max}\), the EH-ST will transmit \(e\) replicas with probability \(\eta(e)\). Therefore, we mainly focus on the design of the optimal transmission probability \(\mathbf{\eta^{*}}\). Since there is a one-to-one mapping between the specific policy \(\Omega\) and the transmission probability \(\eta\), without loss of generality, we refer to \(\eta_{u}\) as the policy of EH-ST \(u\) in the following parts of this paper._  
Under the policy \(\mathbf{\eta}\in\Omega^{U}\), the evolution of the energy available in the battery \(\{E_{k}\}\) can be modeled as a Markov chain.