# An Analysis of the Optimum Node Density for Ad hoc Mobile Networks

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3 Number of Number of Random Waypoint Random Direction Modified Random Direction Random Waypoint Random Direction Modified Random Direction Time (sec) Fig. 2. Average per Node at 1 m/s Mobility Time (sec) Fig. 3. Average per Node at m/s Mobility Initial Configuration Random Waypoint at 4 sec Random Direction at 4 sec Modified Random Direction at 4 sec 8 Number of Pairs Fig. 4. Simulated Movement in the Random Direction Mobility Model. Distance (m) Fig.. Number of Node Pairs at a Given Distance. distribution of user-specified speeds. The node then travels to its selected destination at the selected speed. Onceit reaches the destination, it is stationary for some pre-defined pause time. At the end of the pause time, it selects a new destination and speed combination, and then resumes movement. This model causes continuous changes in the topology of the network. While running the simulations, an interesting behavior of the mobility model was noticed. The average number of neighbors seen at a given node periodically increases and decreases as the simulation progresses, where the frequency of the change is relative to the speed of the nodes. The solid lines in Figures 2 and 3 illustrate the average number of neighbors per node in a 1 node network during 1 seconds of simulated time. This fluctuation is due to the inherent characteristics of the mobility model. Because a node must select a destination in the simulation area, the node is most likely to travel in the direction in which there are the most destinations from which to choose. This predisposes nodes to choose destinations that are either in the middle of the simulation area, or that they reach by traveling through the middle. This characteristic creates a situation in which nodes converge in the center of the area, and then disperse, and then re-converge, etc., resulting in density waves. Because a fairly constant number of neighbors per node was necessary throughout the simulation, we developed our own mobility model, called the Random Direction model. Instead of selecting a destination within the area, the nodesselect a direction in which to travel, where a direction is measured in degrees. At the beginning of the simulation, each node selects a degree between and 39, and then finds a destination on the boundary in this direction of travel. It then selects a speed, and travels to that destination at the given speed. Once it reaches the destination, it rests for the given pause time, and then selects a new degree between and 18. The direction is limited because the node is already on the boundary, and nodes do not pass through the boundary. Thus the degree determines a direction relative to the wall of the boundary area on which the node is located. The node then identifies the destination on the boundary in this line of direction, selects a new speed, and resumes travel. This model can be thought of as examining a microcell of a larger area, as illustrated in Figure 4. A node travels to the boundary and then selects a new direction, which is similar to a node that continues past the boundary, while a new node enters the microcell at that location traveling in a different direction. The average number of neighbors per node for this model is illustrated in Figures 2 and 3. As is evident from the figures, the Random Direction mobility model causes many fewer fluctuations in the node distribution than the Random Waypoint model. Also in these figures is the average number of neighbors per node using a variation

4 4 3.4 Packets Delivered (x 1 3 ) m/s 1. m/s. m/s 1. m/s Normalized Throughput Kleinrock s Result m/s 1. m/s. m/s 1. m/s Fig. 6. Number of Packets Delivered Fig. 7. Normalized Throughput. Path Length (hops) m/s 1. m/s. m/s 1. m/s Fig. 8. Average Path Length. Probability of Initial Connection Establishment m/s 1. m/s. m/s 1. m/s Fig. 9. Probability of Being Able to Establish Initial Route. of the Random Direction model, called the Modified Random Direction model. In this model, nodes select a direction degree as before, but they may choose their destination anywhere along that direction of travel. They do not need to travel all the way to the boundary. Figure illustrates the average number of pairs of nodes with the given distance of separation. The figure represents nodes moving at 1 m/s. Initially, the nodes are evenly distributed throughout the simulation area. As the simulation progresses, the characteristics of the Random Waypoint model show that after 4 seconds of simulation time, the node density is much higher than the initial configuration. However, the Random Direction model is able to maintain a node density that is similar to the initial node distribution. B. Results The number of data packets received by their respective destination is shown in Figure 6. The figure shows that for small transmission radius and low connectivity, few data packets are delivered due to lack of a route. As the connectivity increases, however, the number of packets delivered rapidly increases, until the curves level off. As the transmission power continues to increase, the network becomes saturated, and consequently there is a gradual decline in the number of data packets delivered. Network saturation results in an increased number of collisions, as well as reduced channel access which leads to buffer overflows. There does not appear to be a global optimal number of neighbors for all mobilities. For m/s mobility, the results show this optimum to be around seven or eight neighbors per node, which differs only slightly from what Kleinrock proved for a stationary network [6]. As mobility increases, however, the optimum shifts to higher connectivity. The faster nodes move, the more frequently link breaks occur. Hence, even though the effective bandwidth seen at individual nodes suffers due to the increased transmission power and collisions, the number of packets delivered still increases, relative to shorter transmission ranges. This is because link breaks are less frequent and routes are maintained for longer periods of time. Figure 7 represents Kleinrock's normalized throughput achieved by the simulations, and compares these values to an approximation derived by Kleinrock in Figure 3 of his paper [6]. The network throughput is given by the number of successful transmissions divided by the average path length, = n Neh where n is the total number of nodes, N is the average node degree, e is the natural logarithm, and h is the average path length. To normalize, the value p n is plotted.

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