# Analytical Model for the Duty Cycle in Solar-Based EH-WSN for Environmental Monitoring
## 4 Analytical Energy Consumption Model
### Analytical Model
#### 4.2.1 Packet Generation and Transmission  
As it can be noticed from Figure 3, there are some relevant time intervals in addition to the already defined \(T_{I}\) (DUTY_ON_TIME), \(T_{slp}\) and DELAY_AFTER_RECEIVE:  
* \(T_{c}\): This is the transmission cycle, which is the time spent by a node in the transmission of a single non-successful packet. During this period, three standard operations are performed in sequence: (1) clear channel assessment (CCA) and random back-offing (\(T_{CCA}\)); (2) packet transmission (\(T_{plt}\)); and (3) waiting for acknowledgement (\(W_{ack}\)). In general, the first time component is an unpredictable variable that depends on the amount of workload and its spatio-temporal distribution over the network. Thus, it is usually characterized by its average. However, if transmissions from all nodes are sufficiently randomized, as we have postulated in our assumptions, the channel is likely to be found idle every time a node senses the channel. Hence, the incurred time is just for clear channel assessment and thus deterministic and of small value (\(T_{CCA}\) constant and small). * \(T_{c}^{\prime}\): This is a modified transmission cycle that corresponds to the transmission of a successful packet. It is very similar to the previous one, but instead of including a waiting time for acknowledgement, it contains the actual time to receive such acknowledgement. * \(T_{ack}\): This is the duration of acknowledgements packets. In order to ensure that a receiving node, upon its radio activation, detects the start of a packet transmission regardless of the asynchrony with the sender, it must be accomplished that \(T_{l}>T_{c}\). From Figure 3 we can notice that in general the transmitter needs to send its packet repeatedly until the receiver catches it entirely after waking up from its sleep period. Depending on the number of tries, the transmitter wastes more or less energy. Thus, we first characterize the number of tries (\(k\)) required by the transmitter to receive an acknowledgement from the receiver. Table 1 shows the characterization of this random variable by taking the backend of the receiver duty period as reference (see Figure 3). Note that the actual position of such backend is measured with regard to the time origin set up in the figure, which is the start of the transmitter duty period. Accordingly, for every outcome  
Figure 3: Time diagram for packet generation and transmission (node A) and packet forwarding (node B). The transmission process starts at the beginning of a duty period in the sending node A. This period is extended to the total time required to complete the transmission, i.e., until an ACK is received from node B. Packet reception by node B generally involves detection of a packet fragment before the complete packet is received and acknowledged. Then, the node forwards the packet through its next-hop link. Note also that the two nodes spend an extra time in reception mode before switching again to sleep mode. of the number of tries, the table shows the time interval to which the backend belongs, as well as its probability. As noticed, the case of single try corresponds to the union of two time intervals: in the first one, the level of asynchrony between transmitter and receiver is very small, whereas in the second one such asynchrony is sufficiently large so as to make the next receiver duty period come into scene. Also, it is worth observing that the overall distribution is not uniform because not all time intervals are equally long; in particular, the probability of a single try is precisely the duty cycle of the LPL mechanism.