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Map-based mobility. In all the models introduced so far, nodes are free to move within any subregion of the deployment region R. However, in many realistic scenarios, nodes are constrained to move within specified paths. This is the case, for instance, of cars moving on a freeway, or people moving on sidewalks, and so on. Map-based models have been used to model these situations.
The first step in the definition of map-based models is the map setup, that is, the definition of the paths within which nodes are allowed to move. Then, a certain number of nodes are randomly located on the paths, and they start moving according to scenario-specific rules.
An instance of map-based mobility is the Freeway mobility model (Bai et al. 2003), used to mimic the movement of cars on freeways. In this model, several freeways are located in the deployment region. Each freeway is composed of a varying number of lanes in both directions. Nodes are randomly located on a freeway, and they move with a random velocity, which is temporally dependent on its previous velocity. If two nodes on the same lane are within a certain minimum distance (safety distance), the velocity of the following node cannot exceed the velocity of the preceding one.
Another instance of map-based mobility is the Manhattan mobility model (Bai et al.
2003), which is used to emulate urban movement scenarios. First, a Manhattan-like map, composed of horizontal and vertical streets, is generated. Nodes can move along the streets in both directions. When a node arrives at an intersection, it randomly chooses whether to continue moving along the same direction, or to take a left or a right turn. Similar to the Freeway model, the velocity of a node at a certain instant of time depends on the node velocity at the previous time step.
MODELING AD HOC NETWORKS
Figure 2.4 Examples of RWP (a) and random direction mobility (b). In case of RWP mobility, nodes tend to cross the center of the deployment region (border effect).
A third instance of map-based mobility is the Obstacle mobility model introduced in (Jardosh et al. 2003). In this model, a map is first generated by adding obstacles (buildings) to the environment. The obstacle-generation phase can be either random or based on real maps. Once the buildings are deployed, a number of pathways connecting different buildings are generated, and nodes are assumed to move along these pathways. An interesting feature of this model is that obstacles are accounted for also when simulating the radio signal propagation in the environment: in other words, it is assumed that the wireless signal is obstructed by the obstacles.
Group-based mobility. All the models described so far resemble individual mobility. However, in many situations, nodes are expected to move in groups (for instance, groups of tourists moving in a city). Group-based mobility has been introduced to model these situations.
In group-based models, a small subset of the network nodes is defined as the set of group leaders. The remaining nodes are randomly assigned to one of the leaders, thus forming groups. Initially, the leaders are randomly distributed in the deployment region R, and the members of each group are randomly located in the neighborhood of the leader. Then, the group leader moves according to one of the previous mobility models, such as RWP or random direction. The other group members ‘follow’ the leader, having a speed and direction that are a random perturbation of those of the leader. When two groups cross, any group member can leave its group and join the other with a certain probability. Group-based mobility models have been used in (Hong et al. 1999; Wang and Li 2002).
Examples of RWP and random direction mobility are shown in Figure 2.4, while Figure 2.5 reports examples of map-based mobility.
2.5 Asymptotic Notation
Before concluding this chapter, we recall the standard notation regarding the asymptotic behavior of functions.
MODELING AD HOC NETWORKS
Figure 2.5 Examples of map-based mobility: the freeway model (a) and the Manhattan mobility model (b).
Let f and g be functions of a certain parameter x. We are interested in characterizing the asymptotic behavior of f and g as x ^-<x>.
Definition 2.5.1 We say that f(x) has order at most g(x), denoted as f(x) e O(g(x)), if there exist constants c and x0 such that, for any x > x0, f(x) < c ? g(x). We say that f(x) has order at least g(x), denoted as f(x) e U(g(x)), if g(x) e O(f(x)). We say that f(x) and g(x) have the same order, denoted as f(x) e &(g(x)), if f(x) e O(g(x)) and f(x) e ^(g(x)). We will sometimes use the notation f(x) & g(x) also to indicate that f(x) and g(x) have the same order.
Definition 2.5.2 We say that f(x) is asymptotically smaller that g(x), denoted as f(x) ^ g(x), ;/lim,^co = 0. We say that /(.x) is asymptotically larger than g(x), denoted as f(x) » g(x), if g(x) « f(x).
3.1 Motivations for Topology Control
In Chapter 1, we have briefly described the many challenges that the ad hoc and sensor network designer must face. In this chapter, we start focusing on two of these challenges, which have motivated researchers to study the realm of topology control techniques.