## 2 Background
### Related Work  
In recent years, multiple research works have been presented addressing some of the most relevant challenges concerning cluster-tree WSNs, including network formation schemes, communication mechanisms, MAC protocol configuration, energy-efficiency, scalability, admission and congestion control, and beacon scheduling. Each of these issues has its own special considerations. Within the context of this paper, we are particularly interested in works that address throughput in IEEE 802.15.4 cluster-tree WSNs, using CAP communication mechanisms. For this topic, modelling the main constraints of the IEEE 802.15.4 MAC protocol and defining adequate communication periods are important mechanisms to avoid known problems such as packet drops due to buffer overflows, network congestion and high end-to-end communication delays, which may lead to undesired operation of the network [27]. Several works in the literature show that the configuration of IEEE 802.15.4 MAC parameters has a direct impact on the performance of WSNs, in what concerns energy efficiency, wide-scale deployments and time-sensitive message transfers [15; 18]. In this context, we point out a set of works [28; 29; 30; 31] that provide analytical models for the timing analysis of the contention access period of the IEEE 802.15.4 MAC protocol. [PERSON] et al. [28] present an  
Figure 1: IEEE 802.15.4 Superframe structure. accurate analytical model to evaluate the behaviour of the IEEE 802.15.4 MAC protocol with periodic traffic, which is a common scenario for WSN-based monitoring applications. The authors consider the probabilities of CCA (Clear Channel Assessment) failures and transmission collisions, considering the standard characteristics of retransmissions and the double CCA of CSMA-CA protocol. Although the authors point out that this model can be used to define adequate active period durations, no scheme or guidelines are provided. Several works encompass analytical models for the contention period of IEEE 802.15.4 MAC protocol, based on Markov chains [29; 30; 31]. Basically, each of these analytical models considers a specific set of parameters and characteristics of the CSMA-CA protocol. [PERSON] et al. [29] provide a new IEEE 802.15.4-based MAC protocol named _Variable CCA MAC protocol_. The idea behind this proposal is to change the number of CCAs that a node must perform before transmitting a data packet (by default, this parameter is defined as 2). The authors model this new protocol using Markov chains and demonstrate its accuracy and capability of predicting its behaviour through simulation. However, this new protocol has a negative performance regarding channel utilisation, communication delays, reliability and energy consumption. In fact, increasing the number of CCAs leads to the increase of the contention window, which can generate higher delays and energy consumption. Furthermore, performing a higher number of CCAs does not avoid collisions, as nodes are not linearly spread along the time. [PERSON] et al. [30] proposed an analytical model based on Markov chains, considering several node classes, by setting different values for the CSMA-CA protocol, but without considering inactive periods. Instead, [PERSON] et al. [31] provided two Markov chains to model the behaviour of IEEE 802.15.4 MAC protocol, considering both CAP and CFP. This analytical model considers the main parameters of CSMA-CA protocol. A major drawback of these approaches is that they do only consider star networks and do not address the characteristics of cluster-tree networks. Thus, they are limited to specific environments and can not be applied to wide-scale applications. In addition, other weakness of these works is that they provide analytical models for the IEEE 802.15.4 MAC protocol without considering any scheme to adequately allocate the active period durations, in order to improve the network throughput. Other works addressed analytical models encompassing cluster-tree networks [32; 33; 34]. [PERSON] et al.