## 1 Introduction
### Objectives and Contributions of This Paper  
In this paper, we define a holistic approach to deal with the problem of how to allocate superframe durations to multiple cluster-heads in a wide-scale WSN. This paper extends the work previously presented in [19], where it was defined a simple protocol constraint for cluster-tree networks and proposed a superframe allocation scheme based on the traffic load generated by sensor nodes. We define a set of boundary equations, that act as upper-bounds for the protocol operating behaviour of IEEE 802.15.4 cluster-tree networks. Based on these boundary equations, we propose a set of superframe duration allocation schemes to be applied at system design time (offline scheduling). These schemes are able to improve the network throughput in wide-scale WSN deployments. They distribute the available network bandwidth among the different network clusters, according to the associated traffic. As a consequence, they enable the reduction of some of the typical problems of cluster-tree networks, such as: network congestion around the PAN coordinator, high message communication delays and a high number of discarded messages due to buffer overflows. Basically, the proposed superframe duration allocation schemes consider: (1) the message load imposed by sensor nodes; (2) the number of descendant nodes of each cluster; and (3) the number of child nodes belonging to the cluster itself. This type of bandwidth allocation schemes is based on earlier work done for Fieldbus networks [20] and for FDDI networks [21] and, more recently, for FlexRay networks [22; 23].  
We focus our study on ZigBee-based cluster-tree topologies, due to some of its specific features, such as suitability to deploy wide-scale networks with energy-efficiency Quality of Service (QoS), which are common requirements for typical WSN monitoring applications. We envisage the use of the proposed allocation schemes in real world monitoring applications, such as those that can be found in precision agriculture and environmental monitoring. One of the major requirements of this type of applications is the capability to deal with periodic data traffic generated by a large number of widely deployed sensor nodes. In precision agriculture applications, WSN-based technologies are responsible for automating information collection processes about the crop production, enabling actuating strategies to maximise the crop yield and quality, and to optimise the environmental resources [24; 25; 26].  
The main contributions of this paper can be summarised as follows:  
* A set of boundary equations for the operating behaviour of IEEE 802.15.4 cluster-tree networks, defining a set of constraints that must be fulfilled to guarantee the adequate timing behaviour of supported applications.
* A set of guidelines to define transmission opportunities (superframe durations) and buffer sizes for each cluster, in a cluster-tree network. These guidelines are based on both the traffic requirements and the network topology, including the number of child nodes and the depth of the cluster-heads in the cluster-tree network. The use of such guidelines significantly increases the network throughput, avoiding or reducing the congestion of the cluster-tree network.
* A simulation assessment that highlights the advantages of using the proposed superframe duration allocation schemes for wide-scale deployments of wireless sensor networks.