## 5 Conclusions and Future Work  
This paper presented a novel method for increasing the lifetime of fixed or mobile wireless sensor networks. The method dynamically controls the so-called communication activity levels of the sensor nodes in a way that the fractions of time or time-frequency slots in a frame are optimally assigned to the sensor nodes to communicate with the sink node, at the same time saving energy. \begin{table}
\begin{tabular}{c c} \hline \hline
**Weights \(w_{1},w_{2}\)** & **Growth Order \(a\)** \\ \hline \(1,0\) & \(1.32\) \\ \(0,1\) & \(1.44\) \\ \(1,1\) & \(1.32\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average empirical time complexity of the proposed lifetime optimization method. The column on the right shows the growth order \(a\). The proposed method uses the solution of a convex optimization problem in which the objective function to be minimized is the weighted sum of the maximum residual energy and the maximum discounted residual energy, where the discount acts in favor of assigning higher activity levels to sensor nodes that have higher residual energies and better channels to the sink node. The net result is a balance between the minimization of wasted energy, the maintenance of throughput and the minimization of the energy spent for reporting sensory information to the sink node. Since the optimization problem is convex, with the possibility of being transformed into a linear programming problem, mature and fast solvers available in practice can be efficiently applied. An empirical complexity analysis was carried out and unveiled that our optimization method has a time complexity \(\mathcal{O}(N^{1.44})\), meaning that it has potential for implementation in real networks. Moreover, the proposed method is flexible enough to allow for controlling the frequency in which the optimization tasks are performed, according to the expected mobility of the nodes. Another merit with significant practical appeal of the proposed method is that it only uses past information on the residual energies and possible recharges reported by the sensor nodes, thus not demanding any real-time information. Comparisons with a greedy algorithm that uses real-time channel state and residual energy information unveiled that the method proposed in this paper is capable of achieving comparable or even superior performances in most of the situations and that it is consistent with solid guidelines for the development of lifetime improvement strategies. Some interesting investigations that represent opportunities for future work are:  
1. The modeling of the communication task through queuing theory and its aggregation to the proposed optimization model is a natural first deployment. From this aggregation, one could analyze the influence of the optimization strategy in the data traffic associated with a realistic network model. 2. The adoption of a mobility model for the nodes is an important step towards the construction of an overall application-dependent lifetime analysis, as this model will have a great impact on the energy consumption pattern of the nodes. 3. The implementation of a real network operating under the proposed optimization method comes next, and here there is room for MAC and routing strategies that might benefit from the method and from possible connections with the Internet of Things (IoT), for instance in light of [45]. 4. The application of the present optimization model to the design of a network level model in order to consider relaying or multi-hop communication is also an interesting point to be considered. This might improve even more the network lifetime by reducing the high energy consumption caused by the communication with distant central nodes in the centralized architecture, but would demand smarter sensor nodes to cope with the protocols associated with multi-hop communication. 5.