## 5 Markov Model and Calculations
### Calculations  
In this section, several main parameters of the network are calculated. Without specification, parameters are defined to describe the contention access process during the CSMA period. To avoid confusion, at the end of this article, Table 11 is given and detailed description of the notations can be found. (A) Throughput  
Let \(N_{slots}\) denote the number of the slots needed for a successful transmission, and let \(L\) be the length of the packet. We then have:  
\[N_{slots}=nL\tau(1-\tau)^{n-1}p \tag{21}\]  
Here we define \(T_{CSMA}\) as the throughput of the CSMA period of the network:  
\[T_{CSMA}=n\tau(1-\tau)^{n-1}p \tag{22}\]  
As mentioned earlier, the superframe of the hybrid network consists of three periods, and there beacons are included, while there are no packets transmission during the inactive period. Let \(T_{TDMA}\) and \(T_{CSMA}\) be the throughput of the TDMA and CSMA period, and the \(L_{TDMA},\ L_{CSMA}\) be the lengths they occupy, then we obtain the throughput of the entire network \(T_{*}\):  
\[T_{*}=\frac{L_{TDMA}T_{TDMA}+L_{CSMA}T_{CSMA}}{L_{TDMA}+L_{CSMA}+L_{Inactive}+3} \tag{23}\]  
(B) Transmission Probability \(T_{c}\)  
For a particular node, transmission happens only when the channel is idle and the node performs a CCA. Then we define its transmission probability:  
\[P_{tx_{o}}=L\tau(1-p) \tag{24}\]  
For the CSMA period, the transmission probability will be:  
\[P_{tx_{*}}=L\tau(1-(1-\tau)^{n})(1-p) \tag{25}\]  
(C) Collision probability  
For a single node, the per-node collision occurs when the node is in CCA, while there are other nodes performing the CCA as well. We define probability of such collision as \(P_{c_{o}}\). Then we get:  
\[P_{c_{o}}=1-(1-\tau)^{n-1} \tag{26}\]  
For the whole CSMA period, a collision occurs when one node is transmitting, while there are other transmitting nodes at the same time. Then we have the collision probability \(P_{c_{*}}\). \[P_{c_{*}}=1-\frac{n\tau(1-\tau)^{n-1}}{1-(1-\tau)^{n}} \tag{27}\](D) Packet discard probability  
For a particular attempt, Let \(\overline{p}_{suc}\), \(p_{col}\) and \(p_{suc}\) denote the probabilities of failure, collision and success, respectively, and we have:  
\[\left\{ \begin{array}{l}p_{fail}=p^{m+1}\\ p_{col}=p_{c_{o}}(1-p_{fail})\\ p_{suc}=(1-p_{c_{o}})(1-p_{fail})\end{array} \right. \tag{28}\]  
There are two reasons for a packet to be discarded: collision and channel access failure. We define \(p_{d}\) as the probability of the packet discard, and \(p_{dc}\), \(p_{df}\) as the probability of packet discard due to collision and failure. We can derive:  
\[p_{d}=p_{dc}+p_{df}, \tag{29}\]  
where:  
\[p_{dc}=p_{col}^{R+1}, \tag{30}\]  
and:  
\[p_{df}=\sum\limits_{i=0}^{R}{p_{fail}{p_{col}}^{i}}=p_{fail}\frac{1-p_{col}^{R +1}}{1-p_{col}}, \tag{31}\]  
Then we have:  
\[p_{d}=p_{dc}+p_{df}=p_{col}^{R+1}+p_{fail}\frac{1-p_{col}^{R+1}}{1-p_{col}} \tag{32}\]  
(E) Retransmission Probability  
For a particular node, no ACK will be sent or received when collision happens. In this case, retransmission will be executed unless the packet has been retransmitted \(R\) times, in which \(R\) represents the maximum times of retransmission. We have the probability of retransmission when a collision happens:  
\[p_{retx}=1-\frac{p_{retx}^{R}}{\sum\limits_{i=0}^{R}{p_{retx}^{i}}}=1-\frac{ \left((1-p_{fail})p_{c_{o}}\right)^{R+1}}{\sum\limits_{i=0}^{R}{\left((1-p_{ fail})p_{c_{o}}\right)^{i}}} \tag{33}\]  
(F) Power Consumption  
Due to the limited power of an on-orbit satellite, the power consumption of the wireless network should be given more attention. Let \(\overline{E}_{CSMA}\) be the average energy consumption during the contention period, and let \(\overline{n}_{B}\) and \(\overline{n}_{C}\) be the average slot number required in backoff and CCA for each attempt. Also, let \(P_{rx}\), \(p_{tx}\) and \(P_{id}\) denote the power consumption in packet receiving, transmitting and idle sate.