## 5 Heuristic Algorithm Design
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(G_{\mathrm{s}}(V_{\mathrm{r}},E_{\mathrm{s}})\) & Abstracted topology \\ \(G_{\mathrm{c}}(V_{\mathrm{c}},E_{\mathrm{c}})\) & Physical topology of GEOs and ground stations \\ \(V_{\mathrm{s}}\) & Set of nodes on the abstracted topology \\ \(E_{\mathrm{s}}\) & Set of links on the abstracted topology \\ \(V_{\mathrm{c}}\) & Set of nodes on the physical topology of GEOs and ground stations \\ \(E_{\mathrm{c}}\) & Set of links on the physical topology of GEOs and ground stations \\ \(V_{\mathrm{g}}\) & Set of nodes of the ground stations \\ \(V_{\mathrm{1}}\) & Set of nodes of the LEO satellites \\ \(V_{\mathrm{h}}\) & Set of nodes of the GEO satellites \\ \(E_{\mathrm{c}}\) & Set of links on the topology in which every GEO is linked to each ground station \\ \(E_{\mathrm{r}}\) & Set of links on the fully meshed topology of the GEOs \\ \(E_{\mathrm{r}}\) & Set of links on the topology in which every GEO is linked to each LEO \\ \(R(t_{i})\) & Set of key service requests in the time slot \(t_{i}\) \\ \(P(t_{i})\) & Set of the rest of the secret keys of QKPs in the time slot \(t_{i}\) \\ \(F_{\mathrm{\alpha}}(t_{i})\) & Set of the rest of the time resources on the link between GEOs and ground stations \\ \(F_{\mathrm{\beta}}(t_{i})\) & Set of the rest of the time resources on the link between GEOs \\ \(P_{\mathrm{d}}\) & Set of links included in the QKD relaying path \\ \(K_{\mathrm{t}}(t_{i})\) & Set of QKD key rates on the link in the time slot \(t_{i}\) \\ \(K_{\mathrm{s}}\) & Set of number of secret keys required for the key service request \\ \(r(s_{r},d_{r},k_{r},t_{i})\) & Key service request generated in the time slot \(t_{i}\), \(r(s_{d},k,t_{i})\in R(t_{i})\) \\ \(s_{r}\) & Source node of the key service request \(r\) \\ \(d_{r}\) & Destination node of the key service request \(r\) \\ \(k_{r}\) & Number of secret keys required for the key service request \(r\) \\ \(p(m,n,t_{i})\) & Rest of the secret keys of QKPs on the link \((m,n)\) in the time slot \(t_{i}\), \\ \(p(m,n,t_{i})\) & \(p(m,n,t_{i})\in P(t_{i})\) \\ \(f_{\mathrm{\alpha}}(m,n,t_{i})\) & Rest of the time resources on the link \((m,n)\) between GEOs and ground stations, \(f_{\mathrm{\alpha}}(m,n,t_{i})\in F_{\mathrm{\alpha}}(t_{i})\) \\ \(f_{\mathrm{\beta}}(m,n,t_{i})\) & Rest of the time resources on the link \((m,n)\) between GEOs, \(f_{\mathrm{\beta}}(m,n,t_{i})\in F_{\mathrm{\beta}}(t_{i})\) \\ \(v(m,n,t_{i})\) & QKD key rate (fixed in a time slot \(t_{i}\)) on the link \((m,n)\) in the time slot \(t_{i}\), \(v(m,n,t_{i})\in K(t_{i})\) \\ \(c(m,n,t_{i})\) & Boolean variable that is equal to 1 if the link \((m,n)\) is established with the quantum \\  & channel in the time slot \(t_{i}\); otherwise, it is equal to 0 \\ \(l_{\mathrm{g}}\) & Limitation of the number of quantum channels established from one LEO to the ground stations \\ \(l_{\mathrm{b}}\) & Limitation of the number of quantum channels established from one LEO to other LEOs \\ \(l_{\mathrm{h}}\) & Limitation of the number of quantum channels established from one GEO to \\  & LEOs/ground stations \\ \(\Delta\) & Interval of a time slot of the physical topology \\ \(N_{\mathrm{r}}(t_{i})\) & The number of successful key service requests in the time slot \(t_{i}\) \\ \(N_{\mathrm{h}}(t_{i})\) & The total number of relaying hops in the time slot \(t_{i}\) \\ \(\phi_{\mathrm{l}}(t_{i})\) & Number of secret keys produced by LEOs in the time slot \(t_{i}\) \\ \(\phi_{\mathrm{g}}(t_{i})\) & Number of secret keys produced by GEOs in the time slot \(t_{i}\) \\ \(\tilde{\omega}\) & Average keys for one request \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and Definitions. The average number of relaying hops (\(AH\)) for the successful key service requests within the simulation time can be calculated as follows.