#### 4.1. Subsection Transmission Opportunity Under EDCA

The EDCA can be modeled as a two-phase protocol for theoretical analysis, which includes a sensing period and a contention period. The flow chart of EDCA protocol is shown in

Figure 3.

In the sensing period, each vehicle senses the channel before transmitting, and a set of vehicles are preselected only if their detecting maximum signal power is lower than the threshold ${\theta}_{c}$. In the contention period, after querying the priority mechanism table to obtain the contention window value and arbitration inter frame space, the preselected vehicles wait for an AIFS and start the back-off process to avoid collisions among others. The value of the back-off timer is a random value in the range [0, CW]. The CW initial value is CW_{min}. The value of CW increases as a collision occurs, and remains constant when added to CW_{max}. By monitoring the medium, a pre-selected vehicle decides to start its transmission if no contender is detected to the run out of its back-off timer, otherwise it defers. That is to say, a preselected vehicle in the contention period is allowed to transmit only if it has the minimum back-off timer among all the contenders.

The probability that a vehicle is preselected is denoted by ${Q}_{q}$, and the probability that the preselected one successfully transmits is denoted by ${Q}_{c}$. Therefore, the transmission opportunity under the EDCA protocol is expressed as $Q={Q}_{q}{Q}_{c}$.

In the sensing period, vehicles sense the channel before transmitting at the beginning of each time slot. We first calculate the probability that a vehicle is preselected during the sensing period. Let

${\mathsf{\Pi}}_{t}$ denotes the set of active transmitters occupy the channel at the same time, and the density of

${\mathsf{\Pi}}_{t}$ is assumed as

${\lambda}_{t}$. The received signal power from the

${i}_{th}$$\left(i\in {\mathsf{\Pi}}_{t}\right)$ active transmitter occupying the channel at an arbitrary vehicle

$X$ is given by

where

${Y}_{i}$ is the coordinate of the

${i}_{th}$$\left(i\in {\mathsf{\Pi}}_{t}\right)$ active transmitter, and

${H}_{i}$ is the power coefficient of the fading channel between

${Y}_{i}$ and

$X$. Then, the maximum received signal power at

$X$ can be denoted as

During sensing period,

$X$ is selected as an preselected transmitters only if its detecting maximum signal power is lower than the threshold

${\theta}_{c}$, thus,

${Q}_{q}$ can be calculated as follow:

According to the definition of the indicating function, it can be obtained as

where (a) follows by the channel is Rayleigh faded, i.e.,

$H$ is exponentially distributed with unit mean, that is

${f}_{H}(x)=\mathrm{exp}(-x)$.

**Lemma** **1.** During the sensing period in EDCA, the probability that a vehicle is selected as a preselected transmitter is given by During the competitive period, the preselected vehicle needs to wait for an Arbitration Inter Frame Space (AIFS) after the channel detection is idle. The representation of AIFS is shown in (11).

For different priorities of data, the values set for

$\mathrm{SIFS}$ are the same, and the values set for

$\mathrm{AIFSN}$ are not the same. Thus, according to Formula (11), different AIFS can be obtained by setting different values of AIFSN. After waiting for an AIFS, the eligible vehicle turns on the back-off timer, and the back-off time is expressed as

where the initial value of CW is set to CW

_{min}, and the value of CW increases as

$(CW+1)\times 2-1$ for each collision, and remains constant when added to CW

_{max}. After the data transfer is completed, it returns to the initial value CW

_{min}. Combined with Formula (11) and Formula (19), the total time that the eligible vehicle in the competitive phase needs to wait is expressed as

After an AIFS, the preselected vehicles compete with each other, and only the one which has the minimum timer is selected as the active node and allowed to transmit.

As

${Q}_{q}$ is obtained, the density of the point process formed by the eligible transmitters is given by

${\lambda}_{q}={\lambda}_{t}{Q}_{q}$, and

${\mathsf{\Phi}}_{q}$ denotes the point process formed by the eligible transmitters. Consider two arbitrary eligible transmitters

$X$ and

${Z}_{i}$(

$X,{Z}_{i}\in {\mathsf{\Phi}}_{q}$), we define

${Z}_{i}$ as the contender of

$X$, only if

$PH{\left|{Z}_{i}-X\right|}^{-\alpha}\ge {\theta}_{q}$, where

${\theta}_{q}$ is the predefined carrier sensing threshold in the contention period. Thus, the density of the contenders around

$X$ is computed as

${\lambda}_{c}={\lambda}_{q}\mathrm{Pr}\left(\frac{PH}{{r}^{\alpha}}\ge {\theta}_{q}\right)$$={\lambda}_{t}{Q}_{q}\mathrm{exp}(\frac{-{\theta}_{q}{r}^{\alpha}}{P})$. We assume

${\mathsf{\Pi}}_{c}^{X}$ is the set of all the contenders around

$X$, and

${m}_{i}$ is the total waiting timer of the contender

${Z}_{i}$. Then, the probability that the preselected vehicle is ultimately allowed to transmit is derived as

In VANET, the application messages are divided into four access categories according to the delay sensitivity. We assume that the probability of occurrence of different priority services is the same as 1/4. Since the calculation process is similar, we only analyze the transmission opportunities for the highest priority message. In Formula (21), the value of SIFS is fixed, so we simplify the total waiting time $t$ in the contention phase to ${t}^{\prime}=AIFSN+CW\times Random(0,1)$. Set $AIFS{N}_{i}$ and $C{W}_{i}$ as the priority of arbitration inter frame space and contention window.

For the highest priority message (

${t}^{\prime}\in [AIFS{N}_{1},AIFS{N}_{1}+C{W}_{1}]$), we have

Then, under the highest priority,

${Q}_{c}^{h}$ is calculated as

For the lowest priority message (

${t}^{\prime}\in [AIFS{N}_{8},AIFS{N}_{8}+C{W}_{8}]$), we have

Then, under the lowest priority,

${Q}_{c}^{l}$ is calculated as

**Lemma** **2.** When ${Q}_{q}$ and ${Q}_{c}$ are derived as above, the transmission opportunity of an arbitrary vehicle under the EDCA protocol is thereby characterized as $Q={Q}_{q}{Q}_{c}$.

#### 4.3. Outage Probability

Because of the complexity of the distribution of the nodes in VANET, it is difficult to calculate the statistical distribution of all the interference nodes in the network. According to the bound effect of interference power in the wireless network, the first layer of interferers can only be considered instead of the aggregated simultaneous transmitters [

25].

As shown in

Figure 4, the worst interfered case is considered where the vehicles transmitting concurrently are just outside the competition region (

$CR$). Assume the reference receiver

${R}_{0}$ locate at the origin, and its corresponding transmitter is denoted by

${T}_{0}$, and the distance between

${T}_{0}$ and

${R}_{0}$ is expressed as

${D}_{s}$. Denote the right and left interfering nodes of

${R}_{0}$ as

${T}_{1}$ and

${T}_{2}$, and the distances between

${T}_{1}$ and

${R}_{0}$,

${T}_{2}$ and

${R}_{0}$ are expressed as

${D}_{{I}_{1}}$ and

${D}_{{I}_{2}}$, respectively. Since the boundary of

$CR$ uniformly locates between two adjacent vehicle nodes, the distance between the right boundary of

${T}_{0}$’s

$CR$ and

${T}_{1}$ is expressed as

${R}_{G}=X\cdot U$, where

$X$ is the distance between any two adjacent vehicle nodes, and

$U$ is uniformly distributed within [0,1]. Therefore, the cumulative distribution function (CDF) and probability density function (PDF) of

${R}_{G}$ can be calculated as

Duo to the symmetry feature,

${D}_{2}$ has the identical PDF of

${D}_{1}$. Given

${D}_{s}={d}_{s}$, the conditional PDF of

${D}_{1}$ and

${D}_{2}$ is derived, respectively, as follow

According to Formula (4), the outage probability

$\tau $ is derived as follows

where

${I}_{1}=H{D}_{{I}_{1}}$, and

${I}_{2}=H{D}_{{I}_{2}}$ denote the interference experiences by

${R}_{0}$ from

${T}_{1}$ and

${T}_{2}$ respectively.

**Lemma** **3.** When the transmission opportunity $Q$ and the outage probability $\tau $ are derived, the transmission capacity ${C}_{T}={\lambda}_{t}Q\left(1-\tau \right)$ of the linear VANET under the EDCA protocol is thereby characterized.