## 1 Introduction
It has been successfully applied to several topologies, like a grid mesh of sensors [15] or in a linear network [16], by means of dynamic linear programming. Nevertheless, these methods, based on a sequential decision problem slot by slot, are far from easy to implement in networks with a medium/large number of nodes. A new function to obtain a figure of merit of the vector of the states, denoted reward in [13], must be optimized by iteration for each network size. In these networks, the main drawback is the need to ascertain an efficient and simple algorithm to determine the optimal schedule with low complexity. In the case of a regular spaced linear network, another approach to find an optimal schedule is to analyze the constraints to avoid collisions within the dual space-time domain (location of nodes and time slots) in a geometric 2D lattice chart [17]. We now show in this work that it is possible to use the 2D space-time lattice to find the best possible schedule (optimal) for a linear network. A key issue in UWSN is the energy consumed by the nodes. In [18], a procedure to avoid retransmissions is pointed out by means of transmitting duplicate data through different paths of the network (routing). Nevertheless, this topic is not the focus of the present work. Another key point in UWSNs is the physical topology. We consider a static monitoring network with fixed nodes anchored to the seabed, which is a realistic assumption for a monitoring or surveillance UWSN [16]. According to the number of nodes, and the area to cover for sensing, we can find the formation ranging from simple isolated linear networks (called string or chain networks) up to clusters of subnetworks linked through special nodes acting as master nodes. In this work, we have chosen a simple case, the multi-hop string network, but under the worst operation conditions: the sink node is located at one end of the string. This fact may create a bottleneck that needs to be overcome. As a consequence, multi-hop TDMA network scheduling has two components: 1) A time schedule to assign transmission slots to nodes and 2) a packet service policy to determine the origin of the packet to be transmitted in a particular time slot. In this paper, scheduling is proposed so that the network performance is optimal in terms of throughput and fairness. In order to obtain the packet end-to-end delay, two services policies are compared: FIFO and Round-Robin. Probabilistic wireless networks can be modeled and analyzed using process calculus [19; 20] describes a compositional theory for networks with static topology assuming that communication between nodes is reliable, and applies the theory to routing protocols. [PERSON], et al. [21] defines a well-formed network as a network which is node-unique, connected, exposure-consistent, and transmission-consistent. [PERSON], et al. [22] describes a framework for the analysis of mobile ad-hoc networks (MANET) and proposes the evaluation of communication interference based on preorders; this work also models the concept of a scheduler in wireless networks. The analysis is carried out for well-formed networks in which active receivers are in the range of exactly one transmitter and transmitters sense the channel beforehand. A similar probabilistic approach is used in [23] to address connectivity and energy consumption in mobile wireless sensor networks. Networks like the one analyzed in this work can be modelled as an open Jackson network using queueing theory. Each node is considered a service station that receives as input the packets from the previous node plus its own generated packets [24]. In our network, however, we can simplify those more general approaches as nodes generate traffic in a deterministic way, and we use spatial TDMA multiplexing [12] with fixed scheduling. This means that the arrival times at each node are deterministic, as well as the instants of the generation of packets in each node. Thus, we can see our network as a special case of the previous analysis frameworks.