## 4 Analysis
### Route Sub-Optimality
#### 4.1.1 Uniform Jitter  
In the case of uniform jitter, \(T_{j}\equiv\text{Uniform}\left[0,J_{m}\right]\). Let \(P_{U}\) and \(D_{U}\) denote the probability of inversion with uniform jitter and the inter-path delay difference with uniform jitter, respectively. Then, the probability of inversion \(P_{U}\) has the expression detailed in Proposition 1.  
**Proposition 1**.: _The probability of inversion with uniform jitter, \(P_{U}\), in two paths of length \(n\) and \(m\), has the following expression:_  
\[P_{U} = \frac{1}{(n+m)!}\sum_{k=0}^{n+m}{\left[(-1)^{k}{n+m\choose k} \times\left((n+2m-k)^{n+m}-(m-k)_{+}^{n+m}\right)\right]} \tag{5}\]Figure 3a illustrates the theoretical values for the probability of inversion for different values of \(n\) and \(m\), _i.e._, the probability that a path of \(m\) hops performs faster forwarding than a path of length \(n\). Figure 3b displays the same probability for different values of path length \(m\), for cases in which \(n<m\). Both the theoretical values and the results from a discrete-event simulation (each point corresponding to the averaged value over 200 samples) are displayed. Figure 4 illustrates the evolution of the probability of delay inversion between paths with a constant hop difference \(\Delta=m-n\) as the absolute length of these paths (in number of hops) increases. Note that both Figures 3b and 4 show bidimensional cuts of the surface presented in Figure 3a; these cuts result from the intersection of this surface with planes \(\pi_{1}:\{m=ct.\}\) and \(\pi_{2}:\{\Delta=m-n=ct.\}\), respectively.  
Expression (5) indicates that the delay inversion occurs, under the conditions specified in RFC 5148 [9], with a significant probability. For the topology presented in Figure 2a, for instance (\(n=2\), \(m=3\)), this probability is \(Pr(D_{U}^{(2,3)}>0)=0.225\), implying that the RREQ traversing the longer path will reach the destination first in almost one out of four cases.  
Two aspects can be highlighted from this analysis: (i) from Equation (5), the probability of inversion does not depend on the length of the jitter interval, \(J_{m}\), meaning that it cannot be addressed by modifying the jitter interval length; and (ii) from Figures 3b and 4, the probability of inversion does not only depend on the difference between path lengths, \(\Delta=m-n\), but also on the absolute values of path lengths \(n\) and \(m\): as paths become longer, more random jitter values are assigned to an RREQ message, and it is more likely that delay inversions occur.  
The fact that delay inversion is more frequent in long paths (in terms of number of hops) is due to the fact that the range in which total jitter values are possible (adding all per-hop jitter values) has a linearly growing upper bound (\(nJ_{m}\), where \(n\) is the path hop length) and a fixed lower bound set to zero. Recall that, while a single uniform jitter value within \([0,J_{m}]\) has mean \(\frac{J_{m}}{2}\) and variance \(\frac{J_{m}^{2}}{12}\), the sum of \(n\) uniform jitter values converges in law (by using the Central Limit Theorem, for \(n\longrightarrow\infty\)) to a Gaussian distribution with mean \(n\frac{J_{m}}{2}\) and variance \(n\frac{J_{m}^{2}}{12}\): longer paths (with \(n\) hops) thus lead to more variant total jitter distributions.