## 2 Heterogeneous Scenarios and APT Transmission Scheme
### Network Assumptions  
As a simplified example, the scenario under study is depicted in Figure 1. Two classes of SNs coexist in the WSN, C1 and C2 SNs. C1 SNs have priority over the C2 SNs for medium access. For modeling convenience, we arbitrarily select one SN from each class and refer to them as _reference nodes_, RN1 and RN2. We focus on a single cell, where SNs are one hop away from each other. However, multiple cells together may form a larger network. SNs transmit packets towards a single destination node that we refer to as the _sink_. We assume that the _sink_ node behaves as a packet absorption node, i.e., it only receives packets and never transmits them. The network deploys the PSA-MAC protocol with the common CSMA/CA-based \(RTS/CTS/DATA/ACK\) packet handshake. SN that uses the PSA-MAC protocol perceive the time as partitioned into cycles, and each cycle is further divided into three parts (_periods_): _sync_, _data_ and _sleep_. Also, we refer to the _active_ period as the period of time composed by the aggregation of the _sync_ and _data_ periods. Backoff timers of active SNs that govern medium access are reset at each cycle initiation. In PSA-MAC, a SN goes to _sleep_ until the next _data_ period when: (i) it loses the contention (hears a busy medium before its backoff timer expires); (ii) it encounters an _RTS_ collision; and (iii) after a successful transmission. With aggregated packet transmission (APT), nodes might transmit a batch of _DATA_ packets in a single cycle instead of a single one. We refer to the set of consecutive packets that will be transmitted together in the same cycle as a _packet frame_ (PF). The maximum number of _DATA_ packets that can be aggregated in a PF is constrained by the cycle duration \(T\), the maximum frame length of the wireless link, by the number of _DATA_ packets in the queue of the SN that gains access to the channel, and by a configurable parameter denoted by \(F\), the maximum number of _DATA_ packets that can be aggregated in a PF. SNs only transmit one PF per cycle. As an example of the operation of the AT scheme, assume that \(q\) is the number of _DATA_ packets in the RN\(c\) queue, \(c=1,2\). When \(q\leq F\), a successful PF transmission will empty its queue. However, when \(q>F\), a successful PF transmission will leave \(q-F\)_DATA_ packets in its queue. SNs are equipped with a battery, but energy harvesting is not supported [25]. In practical deployment scenarios, it has been shown that one or two retransmissions are sufficient to successfully send a PF [26, 27]. Then, we consider an infinite retransmission model instead of the more complex finite retransmission one. We assume that the number of _DATA_ packets that arrive at the buffer of an SN follow a discrete Poisson distribution of mean \(\lambda T\), where \(\lambda\) is the _DATA_ packet arrival rate and \(T\) is the cycle duration. However, the proposed analytical model is sufficiently general to accommodate any alternative discrete distribution. Each SN has a finite buffer size to hold up to \(Q\)_DATA_ packets, Packets are served according to a FIFO discipline. From the MAC layer perception, we also assume that the channel is ideal (error-free). Active SNs randomly select a backoff time with equal probability from the set \(\{0,...,W-1\}\), where \(W\) is the length of the contention window, measured in backoff time slots. Consider that RN\(c\) from cluster \(Cc\) is active, \(c=1,2\). It transmits a PF successfully (without collision) if the other contending SNs from \(Cc\) selected backoff times greater than the one chosen by the RN\(c\). A PF transmitted by RN\(c\) will fail (collide) when more than one contending SN from \(Cc\) select the same backoff time as the RN\(c\), and the backoff time is the smallest among all contending SNs.