## 3 System Model  
In this work, we assume a set of \(N_{nodes}\) sensor nodes organised according to a cluster-tree topology and randomly deployed along a wide-scale environment. Moreover, we consider that the network formation procedure ensures that all monitored environment is covered (no orphan nodes). The PAN coordinator (root of the tree and sink node) is a special FFD with a powerful central processing unit and an unlimited power source. It is responsible to trigger the network formation and acts as cluster-head for the first cluster, according to the IEEE 802.15.4 standard [41] and the ZigBee [6] specification. Moreover, the cluster-tree network is composed of \(N_{CH}\) coordinator nodes, acting as cluster-heads (CHs) of their clusters and periodically sending beacon frames to synchronise their child nodes. Each CH (except the root) belongs to two clusters, once as a child and once as a parent (i.e., a cluster-head). We assume a set of static clusters, where there are no mobile nodes. Figure 2 illustrates an IEEE 802.15.4 cluster-tree wireless sensor network deployed in a specific wide-scale environment. Within this context, we consider that cluster-heads share the same beacon interval, but a specific superframe duration is defined for each one of them. The evaluation of the superframe durations is performed through an offline scheduling approach, settled up during the network formation procedure. Therefore, the set of cluster-heads is characterized by:  
\[CH_{j}=(SD_{j},BI),\quad\text{for }1\leq j\leq N_{CH}, \tag{1}\]  
where \(SD_{j}\) is the Superframe Duration of the cluster-head \(CH_{j}\) and \(BI\) is the Beacon Interval for all clusters. To avoid inter-cluster collisions caused by overlapping clusters, we assume that all active periods have been previously scheduled according to a time division beacon scheduling approach. For this, two different beacon scheduling schemes may be adopted:  
Figure 2: IEEE 802.15.4 / ZigBee cluster-tree network. * Top-down beacon scheduling, where superframe durations are ordered in a top-down direction. Firstly, the PAN coordinator, then clusters of the depth 1, and later clusters of the following depths are sequentially scheduled. This scheduling scheme prioritises downstream traffic. * Bottom-up beacon scheduling: superframe durations are ordered in a bottom-up direction. Firstly, the deepest clusters (max depth of the tree) are scheduled, then clusters of next lower depth, following depth-by-depth until reaching the PAN coordinator. This scheduling scheme prioritises upstream traffic. Figure 3 illustrates a top-down scheduling, while Figure 4 illustrates a bottom-up scheduling for the network presented in Figure 2. Note that for the latter, clusters are scheduled using the reverse order of the top-down scheme. For the sake of simplicity, no data aggregation or data fusion operations are performed by the cluster-heads. We also assume that the active portions of the clusters are composed only of contention access periods. Finally, we consider that whenever there are error sources affecting the wireless communication, these sources are statistically distributed along the communication environment. As the purpose of using a balanced bandwidth allocation scheme is to guarantee a fair distribution of the available communication resources (bandwidth), we do not consider the error behaviour of the communication channel as we consider that error sources equally affect all the network clusters. For the message traffic model, we assume that sensing nodes periodically send messages to a _sink_ node (PAN Coordinator) through the tree path routing (upstream traffic).