# Resource- and Neighbor-Aware Observation Transmission Scheme in Satellite Networks
## 4 Constrained Stochastic Game
### Transition Probability  
The transition probability \(P[S^{\prime}_{i}\mid S_{i},A_{i}]\) means the probability of transiting to the next state \(S^{\prime}_{i}\), given the current state \(S_{i}\) and action \(A_{i}\) of observation satellite \(i\). The state \(M_{i}\) is affected by the chosen action (e.g., when observation satellite \(i\) transmits its observation, \(M_{i}\) becomes 0), whereas the state \(G_{i}\) is not influenced by the chosen action. In addition, these states change independently of each other. Therefore, \(P[S^{\prime}_{i}\mid S_{i},A_{i}]\) between the current state \(S_{i}=[M_{i},G_{i}]\) and the next state \(S^{\prime}_{i}=[M^{\prime}_{i},G^{\prime}_{i}]\) can be described as  
\[P[S^{\prime}_{i}\mid S_{i},A_{i}]=P[M^{\prime}_{i}\mid M_{i},A_{i}]\times P[G^ {\prime}_{i}\mid G_{i}]. \tag{5}\]  
When it is assumed that the inter-observation time of observation satellite \(i\) follows the exponential distribution and its mean is \(1/\lambda^{M}_{i}\), the probability that observation satellite \(i\) observes the target during the decision epoch \(\tau\) can be calculated as \(\lambda^{M}_{i}\tau\)[20; 21]. Thus, \(P[M^{\prime}_{i}\mid M_{i}=0,A_{i}]\) can be represented as  
\[P[M^{\prime}_{i}\mid M_{i}=0,A_{i}]=\left\{ \begin{array}{ll}1-\lambda^{M}_{i }\tau,&\text{if }M^{\prime}_{i}=0\\ \lambda^{M}_{i}\tau,&\text{if }M^{\prime}_{i}=1\\ 0,&\text{otherwise.}\end{array} \right. \tag{6}\]  
Meanwhile, if observation satellite \(i\) does not transmit its observation (i.e., \(A_{i}=0\)), the elapsed time from the observation increases one by one until \(M_{i}\) reaches the maximum elapsed time. Therefore, the corresponding transition probabilities can be described as  
\[P[M^{\prime}_{i}\mid 0<M_{i}<M_{\text{max}},A_{i}=0]=\left\{ \begin{array}{ll}1,&\text{if }M^{\prime}_{i}=M_{i}+1\\ 0,&\text{otherwise}\end{array} \right. \tag{7}\]  
and  
\[P[M^{\prime}_{i}\mid M_{i}=M_{\text{max}},A_{i}=0]=\left\{ \begin{array}{ll}1,&\text{if }M^{\prime}_{i}=M_{i}\\ 0,&\text{otherwise.}\end{array} \right. \tag{8}\]  
When observation satellite \(i\) transmits its observation (i.e., \(A_{i}\
eq 0\)), \(M_{i}\) will be always 0. Therefore, \(P[M^{\prime}_{i}\mid M_{i}\
eq 0,A_{i}\
eq 0]\) can be represented as  
\[P[M^{\prime}_{i}\mid M_{i}\
eq 0,A_{i}\
eq 0]=\left\{ \begin{array}{ll}1,&\text{if }M^{\prime}_{i}=0\\ 0,&\text{otherwise.}\end{array} \right. \tag{9}\]  
We assume that the duration of the disconnection (connection) between the ground station and observation satellite \(i\) follows the exponential distribution with mean \(1/\lambda^{D}_{i}\) (\(1/\lambda^{C}_{i}\)). Thus, the probability that observation satellite \(i\) connects with the ground station during \(\tau\) is \(\lambda^{C}_{i}\tau\)[20; 21]. In addition, the probability that observation satellite \(i\) loses the connection with the ground station during \(\tau\) is \(\lambda^{D}_{i}\tau\)[20; 21]. Therefore, \(P[G^{\prime}_{i}]G_{i}=0,A_{i}]\) and \(P  
four cases: (1) observation satellite \(i\) transmits its observation to the relay satellite without the compression (i.e., \(A_{i}=1\)); (2) observation satellite \(i\) transmits its observation to the ground station without the compression (i.e., \(A_{i}=2\)); (3) observation satellite \(i\) transmits its observation to the relay satellite after the compression (i.e., \(A_{i}=3\)); and (4) observation satellite \(i\) transmits its observation to the ground station after the compression (i.e., \(A_{i}=4\)). For each case, the delay to deliver the observation to the destination can be calculated. For the first and third cases, because the observation is delivered through the relay satellite, we consider the latency between observation satellite \(i\) and the relay satellite and the latency between the relay satellite and the the ground station. In addition, the average queuing delay in the relay satellite should be considered.