# Scheduling Sparse LEO Satellite Transmissions for Remote Water Level Monitoring
## 2 Materials and Methods
### Uplink Transmission Energy Model  
Energy consumption modeling of communication interfaces is a complex issue, and we have relied on the existing Iridium satellite communication model [21], as well as a model for long-range terrestrial network Sigfox [22], as the closest detailed model that similarly to us relies on the published energy consumption values from the datasheets. To determine the average power consumption, we introduce the unified uplink transmission energy model. Since the stochastic nature of transmission success prohibits the derivation of a deterministic model, a probabilistic model is created to give an estimate of average power consumption. There are two key causes of transmission non-determinism: (1) whether a transmission will succeed for a given pass, and (2) if it does succeed, how long the modern will be in Receive Mode before it is able to transmit. Figure 6: Routine for selecting the next satellite pass to attempt transmission. To build the model, let \(t_{SL}\) be the mean time that the modem is in Sleep Mode, \(t_{GPS}\) be the mean time the modem is in GPS Acquisition Mode, and \(t_{RX}\) be the mean time the modem is in Receive Mode before transmission is successful. With typical modem power consumption values \(P_{SL}\), \(P_{GPS}\), and \(P_{RX}\), the total energy usage in these modes over a single transmit attempt cycle, \(E_{attempt}\) is:  
\[E_{attempt}=P_{SL}t_{SL}+P_{GPS}t_{GPS}+P_{RX}t_{RX}+E_{TX}N_{pkt} \tag{4}\]  
where \(N_{pkt}\) is the number of packets transmitted in a given pass. Depending on satellite pass selection and/or previous transmission attempt successes, \(N_{pkt}\) may be 1 or larger. In the case of an unsuccessful attempt, \(N_{pkt}\) is 0. An expression for non-zero \(N_{pkt}\) is:  
\[N_{pkt}=\frac{r_{pkt}}{p_{success}r_{atempt}} \tag{5}\]  
where \(p_{success}\) is the transmission success probability, \(r_{atempt}\) is the mean transmission attempt rate, and \(r_{pkt}\) is the rate at which fully bundled packets are generated. Since \(r_{atempt}\) is smaller or equal to \(r_{pkt}\), \(N_{pkt}\) is guaranteed to be 1 or greater because successful transmission of one packet entails successful transmission of all queued packets. In Equation (4), also note that, while \(P_{SL}\), \(P_{GPS}\), \(t_{GPS}\), \(P_{RX}\), and \(E_{TX}\) (at least for full packets) are constant, \(t_{SL}\) and \(t_{RX}\) are variable. Here, \(t_{SL}\) represents the mean time in the Sleep Mode before making a transmission attempt, approximated as: \(t_{SL}=\frac{1}{r_{atempt}}\). The value of \(t_{RX}\) depends on how long the modem waits until it receives a packet and begins the transmission, or the pass is over. For a successful transmission, the quickest case is to transmit immediately after exiting GPS Acquisition Mode. The slowest success case is to transmit at the very end of the satellite pass. The worst failure case is the modem reaching the end of a given satellite pass in Receive Mode, with no transmission. In terms of \(t_{RX}\), this case and successful transmission at the very end of the pass would be approximately equal. All three cases depend on the mean pass duration, denoted as \(t_{pass}\). \(E_{attempt}\) can take two forms, depending on the transmission attempt success. A success is expressed in Equation (6), where \(\epsilon_{pass}\) represents the proportion of a satellite pass spent in Receive Mode before receiving a packet from the satellite and is able to transmit. For pessimistic and optimistic models, \(\epsilon_{pass}\) can be treated as either 1 or 0, as these serve as the upper and lower bounds of the time in Receive Mode for a given satellite pass. \[E_{success}=P_{SL}\frac{1}{r_{atempt}}+P_{GPS}t_{GPS}+\epsilon_{pass}P_{RX}t_{ pass}+E_{TX}\frac{r_{pkt}}{p_{success}r_{atempt}} \tag{6}\]  
If the attempt is a failure, the model is represented by Equation (7). Note that there is no \(\epsilon_{pass}\) value and no \(N_{pkt}\), as the system will wait out a full pass without transmissions.