## 3 System Model and Medium Access Control Protocols  
### System Model  
We consider a single-hop wireless network composed of one coordinator surrounded by \(n\) end-devices in its transmission range. The coordinator collects data from the end-devices by periodically initiating data collection rounds (DCR). Each DCR is initiated when the coordinator broadcasts a request for data (RFD) packet. In every DCR, each end-device has a number \(L\) of data packets ready to transmit to the coordinator, where \(L\) is an exponentially distributed random variable. The value of \(L\) is independent of other end-devices and has a new realization in every DCR. The assumption of exponentially-distributed message lengths is common in the analysis of communication networks [25] and has been traditionally used to model random data processes related to different sources of data.  
We assume that the end-devices are in a low-power listening mode [26] in which they periodically wake up and turn on the radio transceiver for a short period of time to detect communication requests from the coordinator. We assume that all of the end-devices are synchronized and listen to the channel when the coordinator transmits an RFD. After decoding an RFD, the end-devices start contending for the channel to transmit their \(L\) data packets within a sequence of time frames. Each frame is composed of a fixed number \(m\) of slots. An end-device transmits each one of its data packets in one slot of one frame according to the rules of the adopted MAC protocol, as detailed in Section 3.2. The data packets have a fixed size and fit within the duration of a slot.  
The end-devices transmit without performing carrier sensing; thus, a given slot can be in one of three states: (i) empty, _i.e._, no data packet has been received by the coordinator in the slot; (ii) success, _i.e._, a data packet has been received and decoded by the coordinator; or (iii) failure, _i.e._, one or more data packets have been transmitted in the slot, but none have been decoded by the coordinator. This later case includes both collisions and channel errors. At the end of each frame, the coordinator transmits a feedback packet (FBP) to inform about the data packets successfully received in every frame. Once an end-device succeeds in transmitting its \(L\) data packets, it becomes inactive for the remainder of the DCR to save energy.  
In order to focus on the contention process, we consider a communication scenario where all packets are always transmitted without transmission errors induced by the wireless channel. In addition, we assume that when two or more data packets collide, none of them can be decoded by the coordinator, _i.e._, there is no capture effect. The inclusion of the capture effect and transmission errors, and their impact on the overall network performance, constitutes part of our ongoing and future work.  
Regarding energy consumption, the coordinator and end-devices can be in five different modes of operation: (i) transmitting a packet; (ii) receiving; (iii) idle listening; (iv) standby; or (v) sleeping. The associated power consumptions are \(\rho_{tx}\), \(\rho_{rx}\), \(\rho_{\sigma}\), \(\rho_{stby}\) or \(\rho_{sleep}\), respectively. The model is general for any power-saving mechanism and any value of the power consumptions in different modes of operation. We assume that the energy required to switch between inactive (_i.e._, standby, sleep) and active modes (_i.e._, transmitting, receiving, idle listening) is negligible. In sleep mode, the radio interface is fully disabled, and thus, end-devices consume the lowest power consumption.