Abbreviations
AWGN	Additional Gaussian White Noise
BER	Bit Error Rate
BH	Beam-Hopping
BQP	Binary Quadratic Programming
CDMA	Code Division Multiple Access
CRDSA	Contention Resolution Diversity Slotted Aloha
CSA	Coded Slotted Aloha
DLL	Double-Loop Learning
DSA	Diversity Slotted Aloha
ECEF	Earth-Centered Earth-Fixed
E-SSA	Enhanced-SSA
FBHCA	Flexible Beam-Hopping Control Algorithm
FEC	Forward Error Code
GA	Genetic Algorithm
GEO	Geostationary Earth Orbit
IRSA	Irregular Repetition Slotted Aloha
IoT	Internet of Things
ISSA	Improved Spread Spectrum Aloha
LEO	Low Earth Orbit
MADRL	Multi-Agents Deep Reinforcement Learning
maxUSWG	maximal User Service Weight Gain
MPMM	Multiplier Penalty and Majorization Minimization
MWC	Maximum Weighted Clique
NOMA	Non-Orthogonal Multiple Access
NRT	non-RT
PDF	Probability Density Function
QoS	Quality of Service
QPSK	Quadrature Phase Shift Keying
RA	Random Access
RT	Real-Time
SA	Slotted Aloha
SDP	Semi-Definition Programming
SIC	Successive Interference Cancellation
SINR	Signal to Interference plus Noise Ratio
SS	Spread Spectrum
SSA	Spread Spectrum Aloha
ST	Satellite Terminal
TDMA	Time Division Multiple Access
Notations
$\mathcal{E}$	Set of edges in auxiliary graph
$\mathcal{G}$	Auxiliary graph
$\mathcal{H}$	Set used to store the beam-hopping pattern
$\mathcal{I}$	Set of slots
$\mathcal{J}$	Set of beams
$\mathcal{K}$	Set of cells
$\mathcal{N}\left(v_k\right)$	Set of all adjacent vertices for $v_k$
$\mathcal{P}$	Set used to store the proportion of online STs
$\mathcal{Q}$	Clique in the auxiliary graph
${\mathcal{Q}}_m$	Maximum weighted clique
$\mathbf{Q}$	Clique set based on auxiliary graph
$\mathcal{T}$	Set used to store the number of allocated slots to each cell
$\mathcal{V}$	Cell set used to store the vertices in auxiliary graph
$\mathcal{W}$	Weight set used to store the weights of all cells’ or vertices’
$\mathrm{\Theta }$	Beam angle set
A	Channel gain matrix
B	Beam bandwidth
$D_k$	Number of online STs in cell k
$G_r$	Receiving antenna gain matrix
$G_t$	Transmitting antenna gain matrix
$G_T\left(\theta \right)$	Gain of phased array antenna with off-axis angle $\theta$
$H_i$	Backward channel gain matrix at slot i
I	Number of slots in a BH cycle
J	Number of beams
K	Number of cells
M	Number of chips in preamble
N	Number of phased array elements
$N_0$	Power of white Gaussian noise
$N_{pre}$	Number of chips in preamble
$P_i$	Transmitting power matrix at slot i
$R_{i,k}$	If cell k still needs service at slot i, $R_{i,k}=1$; Otherwise, $R_{i,k}=0$
S	Throughput expressed in packets/users per beam-hopping slot
T	Number of STs in a cell whose signal arrived at satellite in a BH cycle
$T_{max}$	Maximum throughput in a parameter setting
X	Beam occupancy condition
$g_{k,j}^t$	Transmitting antenna gain from cell k using beam j
$g_j^r$	Receiving antenna gain at satellite on beam j
$h_{k,j}$	Backward link gain from ST in cell k to on-board receiver using beam j
$p_{k,j}$	Total transmitting power in cell k using beam j
$p_0$	Packet recovery probability
$v_k$	Vertex in auxiliary graph used to represent cell k
$w_k$	Weight of cell k or vertex $v_k$
$x_{i,j,k}$	Illuminating state, if beam j illuminate cell k at slot i, $x_{i,j,k}=1$; Otherwise, $x_{i,j,k}=0$
$x_{k,j}$	The illuminating state of cell k using beam j
${\mathrm{\Gamma }}_{i,j,k}$	SNIR of receiving signal at on-board receiver from cell k using beam j at slot i
${\alpha }_j$	Channel gain of beam j over backward link
$\gamma$	Receiving SNR before hybrided at satellite
${\gamma }^{\prime }$	Receiving SNR after hybrided at satellite
$\eta$	Spectrum Spread factor
${\eta }^{\prime }$	Antenna efficiency
$\theta$	Off-axis angle of phased array antenna
${\theta }_c$	Angle of interference avoidance threshold
${\theta }_{k_m,k_n}$	Angle between two beams illuminate cell $j_m$ and $j_n$
${\theta }_{i,k_m,k_n}$	Angle between two beams illuminate cell $j_m$ and $j_n$ at slot i
$\lambda$	Poisson arrival rate
$\tau$	Beam-hopping cycle denoted as the number of slots
${\tau }^{\prime }$	Minimum number of slots allocated to a cell in a BH cycle