## 3 Results: Network Performance
### Delay  
One of the most important parameters of the performance of a multi-hop network is the delay of the packets delivered by the nodes to the Gateway, which should be as low as possible. In order to compute the delay, the network has been simulated with a Matlab(r)-based proprietary discrete events simulator. Two measures have been considered: the maximum and the average end-to-end delay \(\overline{D}\), defined by:  
\[\overline{D}=\frac{1}{P_{tot}}\left(\sum_{j=2}^{N}\sum_{i=1}^{p_{j}}D_{j,i} \right)\,, \tag{18}\]  
with \(P_{tot}\) being the total number of packets delivered, \(p_{j}\) the amount of packets delivered from node \(j\), and \(D_{j,i}\) the end-to-end delay of the \(i\)-th packet from node \(j\). In a transmission state, the service policy of the node has to choose which packet to transmit: its own generated packet or a packet from another node (relayed packets). As seen in Figure 10, several strategies for the service can be implemented independently of the scheduling scheme adopted. In order to measure the packet delay, we have considered two strategies: FIFO and Round-Robin. In the case of FIFO policy, we need to include two queues, one for the packets received from the up-stream nodes and another one for the self-generated packets. For the sake of clarity, we will denote the former to be the queue and the latter to be the buffer. For the buffer, we will assume the packet rate in Equation (14). In the case of Round-Robin policy, node \(j\) has \(N-j\) different queues that store packets coming from the up-stream nodes (nodes \(j+1\) to \(N\)) and the buffer for its own generated packets. In every transmission slot, a different queue is selected by rotation to send a packet. If the selected queue is empty, the next queue in the sequence is chosen. If none of them have packets, it will be the turn of node \(j\), which will send a packet from its buffer. Both policies, FIFO and Round-Robin, are examples of polling systems. Systems with several queues attended by a single server based on a pre-scheduled rule belong to the family of vacation queues with a gated operation and a time limited service (one single time slot) [26; 27]. However, since the transmissions are scheduled, the arrivals process is deterministic and can be studied without resorting to queueing theory. Both delays are shown in Table 1 for a set of network sizes varying from \(N=4\) to \(N=100\). It can be seen that the delays are almost equal in both cases. The FIFO service policy is then preferable for large \(N\) because of an easier implementation since there are only two queues per  
Figure 10: Scheme for two service policies in a string network: FIFO (above) and Round-Robin (below). node. Last but not least, the initial conditions different from perfect synchronism, i.e., all the nodes start the counter to transmit from their own buffer simultaneously, can slightly affect the delay shown in Table 1.  
A significant improvement can be obtained when initial conditions are taken into account in the FIFO service policy. In every Tx-slot, a counter is increased and when it reaches a preset threshold, a packet of the buffer is transmitted and the counter will be reset. This means that in FIFO, the packet is transmitted in that slot, while in Round-Robin, the new packet is stored in the buffer, waiting for its turn to be transmitted. In both cases, if the threshold is set to a different value, the offered traffic of the node will also be different. At this point, it is convenient to define a set of variables for the Tx-slot counter \(W\):  
\(W_{j,t}\): Tx-slots elapsed for the node \(j\) (\(j\)= \(2,..,\mathrm{N}\)) in the slot \(t\) since the last reset;  
\(W_{j}^{max}\): preset threshold for \(W_{j}\) (when \(W_{j,t}=W_{j}^{max}\) a new packet is generated in slot \(t\) and \(W_{j,t}=0\));  
\(W_{j}^{ini}\): initial value for the counter of the node n (\(W_{j,0}=W_{j}^{ini}\));  
\(W^{ini}\): vector containing \(W_{j}^{ini}\) values for all nodes: \(W^{ini}=\left[