Sensors_**2011**, _11_, 3611-3639; doi:10.3390/s110403611  
_Series of_  
**ISSN 1424-8220**  
www.mdpi.com/journal/sensors  
_Article_  
**Optimal Rate Allocation in Cluster-Tree WSNs**  
**[PERSON] \({}^{1,\star}\), [PERSON] \({}^{1}\), [PERSON] \({}^{2}\), [PERSON] \({}^{3}\) and**  
**[PERSON] \({}^{1}\)**  
\({}^{1}\) Telecommunications and Systems Engineering Department, Universitat Autonoma de Barcelona,  
UAB Campus, 08193 Cerdanyola del Valles, Spain; E-Mails: [EMAIL_ADDRESS] (J.L.V. );  
[EMAIL_ADDRESS] (G.S.-G.)  
\({}^{2}\) IT, Multimedia and Telecommunications Department, Universitat Oberta de Catalunya, Rambla  
Poblenou, 156, 08018 Barcelona, Spain; E-Mail: [EMAIL_ADDRESS]  
\({}^{3}\) World Sensing S.L., Baixada Gomis, 1, 08023 Barcelona, Spain;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{\star}\) Author to whom correspondence should be addressed; E-Mail: [EMAIL_ADDRESS];  
Tel. : +34-93-586-82-39. _Received: 26 January 2011; in revised form: 21 March 2011 / Accepted: 23 March 2011 / Published: 25 March 2011_  
**Abstract:** In this paper, we propose a solution to the problem of guaranteed time slot allocation in cluster-tree WSNs. Our design uses the so-called Network Utility Maximization (NUM) approach as far as we aim to provide a fair distribution of the available resources. From the point of view of implementation, we extend here the authors' proposed Coupled-Decompositions Method (CDM) in order to compute the NUM problem inside the cluster tree topology and we prove the optimality of this new extended version of the method. As a result, we obtain a distributed solution that reduces the total amount of signalling information in the network up to a factor of 500 with respect to the classical techniques, that is, primal and dual decomposition. This is possible because the CDM finds the optimal solution with a small number of iterations. Furthermore, when we compare our solution to the standard-proposed First Come First Serve (FCFS) policy, we realize that FCFS becomes pretty unfair as the traffic load in the network increases and thus, a fair allocation of resources can be considered whenever the price to pay in terms of signalling and computational complexity is controlled. **Keywords:** wireless sensor networks; contention free access; fair time slot allocation; distributed optimization; reduced signallingIntroduction  
Wireless Sensor Networks (WSN) have attracted the attention of the scientific community in the last years. This interest is mainly driven by the large amount of industrial applications that have appeared thanks to the cost reduction of the sensors. In this kind of networks, the standard IEEE 802.15.4 is usually adopted for the PHY and MAC layers [1,2]. This is because this standard has been specially designed with the aim of providing low cost devices with long battery life. The IEEE 802.15.4 standard, in particular, offers two operating modes: the non-beacon mode and the beacon mode. In the first mode, nodes in the network are not synchronized and access the channel via unslotted Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA). In the beacon-enabled mode, on the other hand, the network is synchronized by means of the beacons transmitted by a node acting as network coordinator and transmissions are organized in time slots. Apart from allowing the nodes to access the channel via contention (with slotted CSMA/CA in this case), this mode also provides guaranteed service to the nodes by allocating Guaranteed Time Slot (GTS) by means of a scheduling strategy. The adoption of GTS is included in order to assure QoS requirements in situations where CSMA/CA is less effective (e.g., under high traffic load). In that sense, new versions of the IEEE 802.15.4 standard such as IEEE 802.15.4e [3] or other coming standards such as WirelessHART [4] or ISA 100.11a [5] evolve with a clear definition of resources in order to manage the Quality of Service (QoS) of the ongoing connections. For example, IEEE 802.15.4e will be mostly based on GTS.