wherePnew(k) is defined in Section 2.4.1. The PMF of the path duration,fD(k, h),
is then
fD(k, h) =Pnew(k−1, h)− Pnew(k, h). (2.31)
In this section we have derived exact expressions for the mobility metrics using a probability transition matrix derived from the PDF of the node separation after one epoch.
2.4.3
Application to Mobility Models
Our framework does not directly apply to deterministic mobility models such as trace-based models. Indeed, the mobility metrics examined do not make sense with respect to such models. However, for deterministic models, equivalent average mo- bility metrics may be of interest. These can be calculated using our framework by replacing the PDF fLm+1|Lm(lm+1|lm) with the network average movement between epochs.
In the next chapter, we will apply these expressions to the Random Walk Mobil- ity Model. It should be noted that our framework can be applied to any statistical mobility model where nodes move in an i.i.d. manner and node separation evolu- tion relies only on the previous relative position. Therefore, it can also be applied to other models, such as the Random Waypoint Mobility Model, since it is shown in [92] that Random Waypoint is Asymptotic Mean Stationary.
2.5
Conclusions
Frequent changes in network topology caused by mobility in mobile ad hoc networks impose great challenges for developing efficient routing algorithms. The theoretical analysis framework presented in this chapter allows new insights into network be- haviour under mobility and some fundamental work on the issue of path stability. The Markov chain model used in this chapter, has enabled us to derive exact ex- pressions for a series of mobility metrics. Further, we have presented intuitive and simple expressions for the relationship between link change rate and link duration, just given the network node degree.
The main contributions of this chapter are:
• Introduction of notion of link (path) persistence and its calculation method.
• Expressions for the expected link (path) duration and its PMF.
• Expressions for the expected link (path) residual time and its PMF which are derived by using a random mobility model rather than a non-random travelling pattern (straight-line mobility model).
38 Mobility Metric Analysis
• An exact expression for link (path) availability, unlike the approximation analysis employed in [61].
• An explicit relationship between link change rate and link duration.
We believe that these calculations are useful for comparison of artificial mobility behaviours with actual network implementation scenarios. The analytical results can be readily applied to various adaptive routing protocols that use corresponding mobility metrics, such as the adaptive caching strategies and clustering algorithms to augment existing on-demand routing protocols, considered in the following chap- ters. A case study of the analysis in this chapter will be investigated in the next chapter, with respect to a Random Walk Mobility model.
Chapter 3
Mobility Metrics in Random
Walk Mobility Model
The random walk mobility model (RWMM) is probably the most mathematically tractable mobility model in use. It describes the basic node mobility parameters, velocity and direction of travel, in terms of known probability distributions. In this chapter, we use the RWMM to illustrate the use of the MCM derived expressions for the mobility metrics, from Chapter 2.
Based on the node separation analysis of Section 3.1, in Section 3.2 we develop an expression for the PDF of the separation distance between an arbitrary pair of mobile nodes after one epoch, with respect to constant node speed. The PDF calculation is extended to variable node speed in Section 3.3. In Section 3.4 we present closed-form approximations of link residual time and link duration. In Section 3.5, we compare our theoretical results with simulation results. Finally, we present conclusions and further work in Section 3.6.
3.1
Node Separation After One Epoch
We assume that each mobile node moves with a velocity uniformly distributed in both speed, V ∼ U[vmin, vmax], and direction, Φ ∼ U[0,2π]. Both the speed
and direction change in each epoch but are constant for the duration of an epoch, and are independent of each other. The speed has mean v = 1
2(vmin+vmax), and
variance, σ2
v = 121(vmax −vmin)2. This random mobility model is widely used to
analyze route stability in multi-hop mobile environments [6, 82].
We saw in Section 2.3 of Chapter 2 that the movement-related PDF required for the MCM is fLm+1|Lm(lm+1|lm) where lm is the separation distance between a pair of nodes at epoch m. Only after we obtain the PDF fLm+1|Lm(lm+1|lm) can we construct the transition matrices in (2.8) and (2.13), because the PDF determines the transition probabilities aij in (2.12). By considering the relative movement X
between two nodes after one epoch, we can consequently determine the distance
40 Mobility Metrics in Random Walk Mobility Model X m L X m+1 L m+1 L Vi Vj Vj Node at epoch m Node at epoch j i Node at epoch m+1 j Θ Node at epoch m+1 i m ψ
Figure 3.1: Relationship between the node movement vectors −→V i and
− →
V j of nodes
i and j, respectively, relative movement vector, −→X, separation vector at epoch m, −
→
Lm, and separation vector after one epoch,
− →
Lm+1. Solid lines indicate actual
vector positions and dashed lines indicate vectors shifted for illustration purposes. The dotted circles indicate the loci of possible positions for nodes i andj at epoch
m+ 1.
between them, Lm+1, after this movement, given the distance between them, Lm,
prior to it. Figure 3.1 illustrates the relationship between the node movement vectors −→V i and
− →
V j of nodes i and j, respectively, relative movement vector,
−→
X, separation vector at epoch m, −→Lm, and separation vector after one epoch,
− →
Lm+1.