# Load-Balancing Routing for LEO Satellite Network with Distributed Hops-Based Back-Pressure Strategy
## 5 Stability Analysis
### Throughput Optimization of DHBP  
Given a queue network like the LSN, the throughput is at a maximum when the arrival rate is in a stable region. Therefore, the throughput optimization policy can be considered as a strategy that tries to ensure network stability at the upper bound of allowable \(\lambda^{c}_{n}\), i.e., to minimize the DHBP difference between \(t\), \(t+1\). We define a Lyapunov drift \(\Delta t\) to examine the difference from \(t\) to \(t+1\):  
\[\hat{Q}(t)=\sum_{n\in N}\sum_{c\in N}\hat{Q}^{c}_{n}(t) \tag{22}\]  
\[\Delta t=\sum_{n\in N}\sum_{c\in N}E\Bigg{[}\frac{\hat{Q}^{c}_{n}(t+1)^{2}- \hat{Q}^{c}_{n}(t)^{2}}{2\hat{Q}(t)}\Bigg{]}. \tag{23}\]  
The one-slot queue backlog meets  
\[\hat{Q}^{c}_{n}(t+1)\leq\max\{\big{(}\hat{Q}^{c}_{n}(t)-\hat{X}^{c}_{n}\big{)},0\}+\hat{Y}^{c}_{n}+H(A^{c}_{n}(t)), \tag{24}\]  
where \(X^{c}_{n}(t)\), \(Y^{c}_{n}(t)\) denotes the packets transmitted along \((a,b)\) and \((b,a)\) at time slot \(t\), respectively,The following inequality,  
\[(\max\{q-b,0\}+a)^{2}\leq q^{2}+a^{2}+b^{2}+2q(a-b), \tag{25}\]  
is valid if \(q\geq 0,a\geq 0,b\geq 0\). Thus, the bound of \(\Delta t\) satisfies  
\[\Delta t\leq M+\sum\limits_{n\in N}\sum\limits_{c\in N}\hat{Q}_{n}^{c}(t)\times E \big{[}\big{(}H(A_{n}^{c}(t))-\hat{X}_{n}^{c}(t)+\hat{Y}_{n}^{c}(t)\big{)}/\hat {Q}(t)\big{]}, \tag{26}\]  
where \(M\) and \(H(A_{n}^{c}(t))\) are finite constants. Thus, to minimize the above inequality, we need to maximize  
\[E\Bigg{[}\sum\limits_{n\in N}\sum\limits_{c\in n}\hat{Q}_{n}^{c}(t)\times\big{(} \hat{X}_{n}^{c}(t)-\hat{Y}_{n}^{c}(t)\big{)}/\hat{Q}(t)\Bigg{]}, \tag{27}\]  
i.e., maximize  
\[\sum\limits_{n\in N}\sum\limits_{c\in n}\hat{Q}_{n}^{c}(t)\times\big{[}\hat{X} _{n}^{c}(t)-\hat{Y}_{n}^{c}(t)\big{]}, \tag{28}\]  
where  
\[\hat{X}_{n}^{c}(t)-\hat{Y}_{n}^{c}(t)=H(X_{n}^{c}(t))-H(Y_{n}^{c}(t))=\sum \limits_{p\in X_{n}^{c}(t)}H(p)-\sum\limits_{p\in Y_{n}^{c}(t)}H(p). \tag{29}\]  
Let all packets arrive randomly; when the time span is long enough, we can get  
\[\frac{H(X_{n}^{c}(t))}{X_{n}^{c}(t)}=\frac{H(Y_{n}^{c}(t))}{Y_{n}^{c}(t)}=\frac {H(Q_{n}^{c}(t))}{Q_{n}^{c}(t)}=\frac{\hat{Q}_{n}^{c}(t)}{Q_{n}^{c}(t)}. \tag{30}\]  
Based on this assumption, we maximize  
\[ \begin{array}{l}\sum\limits_{n\in N}\sum\limits_{c\in n}\hat{Q}_{n}^{c}(t) \times\big{[}H\big{(}\hat{X}_{n}^{c}(t)\big{)}-H\big{(}\hat{Y}_{n}^{c}(t) \big{)}\big{]}\\ =\sum\limits_{n\in N}\sum\limits_{c\in n}\hat{Q}_{n}^{c}(t)\times\bigg{[}\frac {H\big{(}\hat{X}_{n}^{c}(t)\big{)}}{\hat{Q}_{n}(t)}X_{n}^{c}(t)-\frac{H\big{(} \hat{X}_{n}^{c}(t)\big{)}}{\hat{Q}_{n}(t)}Y_{n}^{c}(t)\bigg{]}\\ =\sum\limits_{n\in N}\sum\limits_{c\in n}\frac{\hat{Q}_{n}^{c}(t)}{\hat{Q}_{n} ^{c}(t)}\times\big{[}X_{n}^{c}(t)H(Q_{n}^{c}(t))-Y_{n}^{c}(t)H(Q_{n}^{c}(t)) \big{]}\\ =\sum\limits_{n\in N}\sum\limits_{c\in n}\frac{\hat{Q}_{n}^{c}(t)}{\hat{Q}_{n} ^{c}(t)}\times H(Q_{n}^{c}(t))\times\bigg{[}\sum\limits_{b\in N}\mu_{nb}^{c}(t) -\sum\limits_{a\in N}\mu_{an}^{c}(t)\bigg{]}.\end{array}  \tag{31}\]  
Since \(\hat{Q}_{n}^{c}(t)/Q_{n}^{c}(t)\) is non-negative, the above equality is equivalent to maximizing  
\[ \begin{array}{l}\sum\limits_{n\in N}\sum\limits_{c\in n}H(Q_{n}^{c}(t)) \times\bigg{[}\sum\limits_{b\in N}\mu_{nb}^{c}(t)-\sum\limits_{a\in N}\mu_{an }^{c}(t)\bigg{]}\\ =\sum\limits_{a\in N}\sum\limits_{b\in N}\sum\limits_{c\in n}\mu_{ab}^{c}(t) \times\big{[}H(Q_{a}^{c}(t))-H(Q_{b}^{c}(t))\big{]}\\ =\sum\limits_{a\in N}\sum\limits_{b\in N}\sum\limits_{c\in n}\mu_{ab}^{c}(t) \times\big{[}\hat{Q}_{a}^{c}(t)-\hat{Q}_{b}^{c}(t)\big{]}\\ =\sum\limits_{a\in N}\sum\limits_{b\in N}\sum\limits_{c\in n}\mu_{ab}^{c}(t) \times\hat{\omega}_{ab}^{c}(t),\end{array}  \tag{32}\]  
where \(\hat{\omega}_{ab}^{c}(t)\) denotes a differential DHD-based backlog of flow \(c\) of link \((a,b)\). The DHBP routing decision is equivalent to determining \(\mu_{ab}^{c}(t)\) to minimize \(\Delta t\) and maximize Equation (32). Maximization represents the routing policy of DHBP.