## 2 Background
### Related Work
[32] proposed an analytical framework to model the behaviour of the IEEE 802.15.4 MAC protocol based on Markov chains. An important requirement assumed by the authors is the finite buffer queues for the nodes. The proposed model is quite simple, where the node traffic is always generated by nodes and forwarded toward the sink node (PAN coordinator), but it does not describe the complex features of this network type. [PERSON] et al. [33] used network calculus theory to model cluster-tree WSNs according to several network parameters, such as: depth, maximum number of child nodes and maximum number of child routers. In that work, the authors provide a worst-case behaviour evaluation for upstream and downstream data flows, considering important constraints such as: buffer and bandwidth requirements, flow directions and end-to-end communication delays. However, they consider only contention-free periods (GTS), where the limitation of a maximum of seven GTS restricts the number of data flows in the network. Moreover, data traffic generated by typical monitoring scenarios was not considered in this analysis. [PERSON] et al. [34] provided several mathematical models for the timing analysis of IEEE 802.15.4 CSMA-CA mechanisms and MAC operation and verified the proposed models through simulations. However, only models were provided and no further schemes were proposed. Moreover, we highlight other set of works presented by [9; 35; 36], which provide performance assessments of the MAC protocol by setting different values for its parameters. [PERSON] et al. [9] analysed the impact of MAC parameters upon the network communication behaviour and proposed an adaptive cross-layer framework to minimise the energy consumption for single and multihop WSNs. The authors show that changing the _macMinBE_, _macMaxCSMABackoffs_ and _macMaxFrameRetries_ parameters may increase the probability of winning the contention for the wireless channel. However, this approach is valid just up to a certain threshold. They also show that macMaxCSMABackoff_ parameter has a higher impact over the energy consumption than _macMinBE_ parameter. [PERSON] et al. [PERSON] et al. (2017) proposed the TRADIF methodology, which enables traffic differentiation during the contention-access period, by setting different values of CSMA-CA parameters for critical messages (such as alarm reports and management messages), in order to provide higher priority and quality-of-service for this type of traffic. [PERSON] et al. [PERSON] et al. (2017) provided a performance evaluation of IEEE 802.15.4 star networks through simulation. Differently of [PERSON] et al. (2017), authors focused on the selection of SO and BO parameters and their impact upon different industrial scenarios. However, only star topologies were evaluated and no allocation schemes were proposed. Some works have used other techniques to increase the bandwidth for message streams, such as: beacon scheduling [PERSON] et al. (2017); [PERSON] et al. (2017); [PERSON] et al. (2017) and superframe duration adjustment schemes [PERSON] et al. (2017); [PERSON] et al. (2017); [PERSON] et al. (2017). [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) presented a Time-Division Cluster Scheduling (TDCS) mechanism to meet end-to-end deadlines of time-bounded message streams. This mechanism employs a pure time-division scheduling approach, avoiding the inter-cluster collision problem. The authors formulated the TCDS approach as a cyclic extension of the Resource Constrained Project Scheduling with Temporal Constraints (RCPS/TC), which defines a feasible schedule considering temporal and resource constraints for a set of tasks. After modelling this problem, they used an integer linear programming algorithm to solve the scheduling problem. [PERSON] et al. [PERSON] et al. (2017) proposed a dynamic cluster scheduling scheme to provide QoS for different traffic flows in cluster-tree networks.