## 2 Problem Statement and Formulation  
We consider an MWSN consisting of a number of sensor nodes whose communication tasks are controlled by a sink node or any other central node in a time-frequency-division basis. Figure 1 illustrates a possible topology adopted for the MWSN. The sensor nodes that can be directly controlled are those inside the sink node coverage area. Nonetheless, indirect control of all sensor nodes or a large portion of them can be achieved by means of multi-hop communication. Our lifetime optimization approach applies to both situations, but in the second one it must be combined with some suitable routing protocol. When multiple hops are considered, two well-known network architectures can be defined, namely [21]: flat ad hoc and hierarchical ad hoc. Under the flat architecture, sensor nodes relay each other's data to the sink node, whereas in the hierarchical architecture cluster heads send to the sink node the aggregated data from clustered sensor nodes. When direct communication with a mobile sink node is considered, it is said that the network is working under the SENMA (sensor network with mobile access) architecture. Although many existing WSN protocols also handle routing to improve the network lifetime [11; 23; 24], here we consider the SENMA architecture so that the performance of our optimization approach can be decoupled from any routing protocol that could be working in parallel. Nonetheless, the potential lifetime improvements brought by our method can be assessed without loss of generality. A similar architecture is considered for instance in [25], and fits in applications such as intelligent transportation, environmental surveillance and wild life monitoring. The decoupling between the lifetime improvement method and routing is also considered in [21]. The sensor nodes, the sink node or both are allowed to move and, as a consequence, the distances from the sensor nodes to the sink node vary, thus varying the energy consumption during communication. The sensors use batteries that cannot be replaced, but in our model they are supposed to be sporadically charged by means of a wireless power transmission from a charge node (which can be the sink node itself). It is assumed that each sensor node spends most of its energy during communication events with the sink node. We associate the term activity level to the amount of time or time-frequency resources allocated to these events. Since the communication process is typically organized in frames (or superframes) and each frame is divided into time or time-frequency slots, then the activity levels will define the fraction of slots in a frame that each sensor is allowed to use for communication purposes. Due to the node mobility, the problem is to determine dynamically the optimum activity levels assigned to the sensor nodes so that the network lifetime is maximized. In this context, the proposed sensor node activity optimization model can be in principle applied to any wireless sensor network that employs slotted operation, especially those that employ a centralized manager to maintain the network schedule, e.g., WirelessHART [26], ISA100.11a [27; 28] and IEEE 802.15.4e [29; 30]. Before the presentation of the lifetime optimization method, some parameters, variables and the system model need to be defined:  
* As depicted in the frame structure shown in Figure 2, an optimization event \(k=1,2,\ldots,K\) spans \(F\) frames indexed by \(f=1,2,\ldots,F\), meaning that a lifetime optimization event acts on groups of \(F\) frames. The number of time or time-frequency slots in a frame is defined to accommodate the planned number of sensor nodes in the network. We call \(F\) the optimization span. Low mobility or fixed WSNs will demand less frequent optimization events, i.e., large \(F\), whilst WSNs with high mobility nodes will need frequent optimizations, i.e., small \(F\) or even \(F=1\).