### 6.1 Cumulative probability calculations

In order to investigate the effect of beacon frequency, we also need to look at the cumulative probability of a successful packet reception, in addition to calculating the probability of a successful packet reception for an individual packet at a given time ‘t’. Since we know the single packet reception probability using Equation 8 from the simulation, the cumulative probability can be calculated.

Therefore, if *P* is the probability of a successful reception, then the cumulative probability for a sequence of *N* receptions is given by:

P+(1-P)P+{(1-P)}^{2}P+\dots \dots +{(1-P)}^{N-1}\mathrm{P.}

(8)

In probability theory, *P* is constant and cumulative probability (CP) tends to 1 as *N* tends to infinity. In this case, it means that successful reception of the beacon is guaranteed once the CP reaches 1. But in this scenario because the vehicle is moving towards the RSU, *P* increases for every sequence. Therefore, for *N* receptions, the CP is

\text{CP}={P}_{1}+\left(1-{P}_{1}\right){P}_{2}+\left(1-{P}_{1}\right)\left(1-{P}_{2}\right){P}_{3}+\dots \phantom{\rule{0.3em}{0ex}}.,

(9)

where, *P*_{
N
} is greater than *P*_{N-1}….=1.

Since *P* is increasing because the vehicle is moving towards the RSU, hence the cumulative probability reaches ‘1’ long before infinity and therefore affects the successful reception of the beacon. This analysis applies when the vehicle enters the network.

For exit times, we consider the probability of not receiving the packet *P*_{
n
}=1-*P* from the RSU as we drive away, i.e., the negative cumulative probability. If *P* is the probability of successful reception the negative cumulative probability (CP_{
n
}) is given by:

{\text{CP}}_{n}=\left(1-{P}_{1}\right)+P1\left(1-{P}_{2}\right)+P1\mathrm{.P}2\left(1-{P}_{3}\right)+\dots \phantom{\rule{0.3em}{0ex}}.

(10)

For the exit scenario *P* the probability of the successful reception decreases as we move away from the RSU; hence, 1 - P is increasing. Once the vehicle does not hear the beacon after the period *T*, the inverse of the beacon frequency, it immediately hands over to the next RSU. Our results consider the effect of the cumulative probability on entrance and exit regions of RSU coverage.

### 6.2 Cumulative probability effect on beacon frequency

To understand the effect of frequency in determining the NDTr, the CP of the packet reception rate reaching ‘1’ (i.e. vehicle moving towards RSU) has been calculated for different frequency and different sizes of beacon for the entry region. The CP of this entry region for two different velocities (10 m/s and 30 m/s) of a vehicle is shown in Figures 14 and 15, respectively. The result presented as graphs in Figures 14 and 15 show that the cumulative probability reaches ‘1’ long before the probability of an individual successful beacon reception, and therefore, this parameter better explains the relationship between beacon frequency and successful reception and not the individual probability.The graph in Figure 15 shows that as the frequency increases, the CP is reaching ‘1’ much before the actual probability. Here, after 10 Hz to 15 Hz, there is not much decrease in the time. This shows the impact of the frequency on the NDT. The first result shows that the cumulative probability reaches ‘1’ long before the probability of an individual successful reception, and therefore, it is the parameter that explains the relationship between beacon frequency and successful reception and not the individual probability.

The negative cumulative probability is the cumulative probability of no longer hearing (i.e. vehicle driving away from RSU) the beacon as the vehicle exits the coverage area. It could be thought of as the opposite of the positive cumulative probability when the vehicle enters the area. So the negative cumulative probability is the cumulative probability of (1 -*P*) where *P* is the probability of successful packet reception. The graphs in Figures 16 and 17 show the negative cumulative probability and single packet reception probability for two velocities of vehicle.The exit graph shown in Figure 17 also depicts exit times for different frequencies and different sizes so it clearly shows that the size of the packet affects the exit times due to fact that the probability of error and hence not hearing the packet increases with packet size and so the larger the packet size, the lower the exit times.

To obtain the NDT from our model, we subtract the exit times from the entry times of CP reaching ‘1’ and this NDT is called as cumulative probability NDT (*N* *D* *T*_{
CP
}) (i.e. NDT derived from CP). This means that NDT is being calculated based on the cumulative and single packet reception probabilities (*N* *D* *T*_{
P
}) with the above results and depicted as a graphs in Figures 18 and 19.

The graphs in Figures 20 and 21 show NDTr, NDTi, *N* *D* *T*_{
P
} and *N* *D* *T*_{
CP
} for two different sizes of beacon. It is clear that these values are affected by the sizes of beacon. For relatively small beacon sizes, *N* *D* *T*_{
CP
} is greater, but for much larger beacon sizes, the trend seems to be reversed. For beacon sizes around 723 bytes the *N* *D* *T*_{
CP
} and *N* *D* *T*_{
P
} are almost equal. This indicates that for handover, where predictability is important, maximum beacon sizes around 600 to 800 bytes (approx.) could give the best chance for seamless communication.

### 6.3 The change in probability of successful beacon reception (*Δ* *P*)

#### 6.3.1 The change (*Δ* P) at entry

For the entry region the rate of change in *P*, i.e. probability of successful beacon reception is shown in the Equation 11.

\Delta {P}_{\text{ENTRY}}={P}_{N}-{P}_{N-1}

(11)

*Δ* *P* is significant because the SNR changes more rapidly with the increased velocity of the vehicle. Hence, *Δ* *P* increases significantly as the velocity of the vehicle increases. Where, *P*_{
N
} is the probability of packet reception of an individual packet ‘N’ and ‘N - 1’ is the previous packet. *Δ* *P* is calculated until *P* reaches 1.

#### 6.3.2 The change (*Δ* P) at exit

For the exit region the rate of change in *P* is as shown in the Equation 12.

\Delta {P}_{\text{EXIT}}={P}_{N}-{P}_{N+1},

(12)

where *P*_{
N
} is the probability of packet reception of an individual packet ‘N’ and ‘N + 1’ is the next packet. *Δ* *P* is calculated until *P* reaches 0.

The change in probability of successful beacon reception, i.e. *Δ* *P* vs SNR (dB) for beacon sizes 300 bytes and 723 bytes is illustrated as a graph in Figure 22. The graph is generated using the Equation 13 which do not take into account the velocity of vehicle. We know that *Δ* *P* for second packet (i.e N + 1) with respect to the first packet (i.e *N*) can be calculated as

\Rightarrow \mathrm{\Delta P}={P}_{2}-{P}_{1}.

We know the formula for *P*, i.e.,

\begin{array}{cc}\mathrm{\Delta P}=& \phantom{\rule{0.3em}{0ex}}{\left[1-1.5\mathit{\text{erfc}}\left(0.45\sqrt{{\text{SNR}}_{2}}\right)\right]}^{L}\\ -{\left[1-1.5\mathit{\text{erfc}}\left(0.45\sqrt{{\text{SNR}}_{1}}\right)\right]}^{L}\end{array}

(13)

The simulation experiments were conducted to analyse the change in *P* with respect to different velocities and different beacon frequencies. Due to vast amounts of results collected from the simulation, therefore, these results are available on request. These results clearly show the effect of size of beacon, velocity of vehicle and frequency of beacon. If a formula is being modelled based on these results, then for a given velocity of vehicle and for a given beacon size and frequency, the rate of change of *P* can be calculated using the modelled formula. With this rate of change being known the *P* and CP at any point can be calculated, which in turn can be used to predict the NDTr more accurately.