Performance Analysis for Downlink Transmission in Multi-Connectivity Cellular V2X Networks

Fig. 1 shows the downlink multi-connectivity scenario in C-V2X networks. The vehicles are randomly distributed on an urban freeway segment, and the DBSs are densely distributed along the road. To simplify the analysis, we make the assumption that both the DBSs and vehicles utilize a single antenna, and denote the height difference between the antenna of the DBS and the vehicle as $h$.

For the tractability of the downlink performance analysis, we consider a 1-D scenario on a road including vehicles, DBSs, and interference DBSs as shown in Fig. 1, as in [9, 6]. From a statistical perspective, the spatial distributions of vehicles and DBSs conform to 1-D PPP distributions [13], and we use the 1-D PPPs ${\varphi_{V}},\varphi_{D}$ with density ${\lambda_{v}},\lambda_{d}$ to denote the locations of vehicles and DBSs on the road, respectively, where ${\varphi_{V}},\varphi_{D}$ can be expressed as

All of the vehicles and DBSs are distributed along a road with length $l$. As per Slivnyak’s theorem, the distribution of point processes remains unchanged even after adding a node at the origin [28], and in order not to lose generality and eliminate segmentation due to boundary effects, we place the typical vehicle at the origin $v_{o}=(0,0)$, i.e. which represents the center of the road [9].

In relation to the formation of virtual cells, we assume that each vehicle is connected to the $n$ nearest DBSs on a Euclidean plane. The 1-D distance between the typical vehicle $v_{o}$ with the $i$-th ($i\leq n$) DBS is $r_{i}$, thus the actual distance $x_{i}$ between the transmit antenna of DBS to receive antenna of the typical vehicle is

We adopt a common power-law pathloss and Rayleigh fading model with a decay rate of $x^{-\alpha_{d}}$, where $x$ denotes the distance between the DBS and the typical vehicle. The downlink pathloss exponent parameter is denoted as $\alpha_{d}$ $(\alpha_{d}>2)$. $g_{d}$ is used to denote the power gain of Rayleigh fading and it is modeled by an exponential distribution with a mean of $1/\mu$. Therefore, we have $g_{d}\sim exp\left(\mu\right)$. The distribution function of $g_{d}$ is

Furthermore, we use random variable ${\chi_{d}}$ to model the effects of shadowing between the DBS and the typical vehicle in the downlink, and ${\chi_{d}}$ follows a log-normal distribution given by $10log_{10}\chi_{d}\sim\aleph\left(\omega_{d},\delta_{d}^{2}\right)$, where $\omega_{d}$ represents the mean of the logarithm of $\chi_{d}$ (i.e., the geometric mean of $\chi_{d}$), while $\delta_{d}^{2}$ represents the variance of the logarithm of $\chi_{d}$ [29]. Hence, the received signal power of the typical vehicle from the $i$-th DBS in the downlink is [30]

where $P_{d}$ is the transmitting power of the DBS and assumed to be the same for all DBSs.

The typical vehicle is assumed to be connected to $n$ nearest DBSs by measuring all the receiving power from the nearby DBSs, finding the DBSs with the maximum receiving power (MRP) in turn [9]. Since the received power $P_{r,v}$ is not exponentially distributed for the modeling of the shadow fading [16], the lemma of random displacement theorem is considered to solve this issue [31]. Thus, $P_{r,v}(x_{i})={P_{d}}{g_{d}}{\chi_{d}}{{x_{i}}^{-{\alpha_{d}}}}$ can be transformed to $P_{r,v}(y_{i})={P_{d}}{g_{d}}{{y_{i}}^{-{\alpha_{d}}}}$, where $y_{i}={\chi_{d}^{-\frac{1}{\alpha_{d}}}}x_{i}$. The 1-D PPP transformed converges to a 1-D homogeneous PPP and the intensity $\lambda_{d}$ is transformed to $E\left[{\chi_{d}^{-\frac{1}{\alpha_{d}}}}\right]\lambda_{d}$, and the intensity of the 2-D PPP is $E\left[{\chi_{d}^{-\frac{2}{\alpha_{d}}}}\right]\lambda_{d}$ after executing the procedure of random displacement [13]. Specifically, $E\left[{\chi_{d}^{-\frac{1}{\alpha_{d}}}}\right]\lambda$ can be calculated as

Then we use 1-D PPP $\varphi_{{}_{D}}^{t}$ to denote the transformed set of DBS and $\lambda_{D}=E\left[{\chi^{-\frac{1}{\alpha_{d}}}}\right]\lambda_{d}$ denotes the transformed DBS intensity. To facilitate performance analysis in the following sections, we use the symbol $\varphi_{D}^{t,d}$ to denote the set of distances between the DBSs and the typical vehicle,

where $x_{i}$ denotes the distance between the typical vehicle $v_{o}$ and the $i$-th nearest DBS $\in\varphi_{D}^{t}$. Thus the candidate serving DBSs is changed to the $n$ nearest DBSs $\in\varphi_{D}^{t}$ in turn, and this can be expressed as

where we use $\varphi_{c}^{d}=\{x_{1},x_{2},\dots,x_{m}\}$ to denote the set of the distances between the connected collaborative DBSs and the typical vehcle $v_{o}$ at the origin, $m$ is the number of DBSs that the typical vehicle has already connected to, and $x_{i},{i\in\{m+1,m+2,\cdots\}}$ denotes the distance between the DBS outside $\varphi_{c}^{d}$ and the typical vehicle. Let $\varphi_{c}$ to denote the set of the connected collaborative DBSs. This means that to expand the set $\varphi_{c}$, we need to find the nearest DBS among DBSs in $\varphi_{D}^{t}\backslash\varphi_{c}$.

In the collaboration DBSs set $\varphi_{c}$, all DBSs will transmit the control and data signals simultaneously on the same subband [7]. Since the signal components of the DBSs are within the cyclic prefix, the resulting multi-connectivity SINR experienced by the typical vehicle $v_{o}$ in the downlink is defined as follows:

where $\sum\limits_{i\in\varphi_{c}}{{P_{d}}{g_{d}}x_{i}^{-{\alpha_{d}}}}$ represents the sum of received signal power from the DBSs in $\varphi_{c}$. We use $\sigma_{d}^{2}$ to denote the power of the additive white Gaussian noise (AWGN) [14]. $I_{D}$ is the power of aggregate interference from the DBSs outside of $\varphi_{c}$ and $I_{D}$ can be expressed as

In order to enable advanced C-V2X applications such as automated driving applications and stream media [4, 16], it is crucial to ensure that the downlink transmission is both reliable and capable of transmitting data at a high rate. This is important not only from the perspective of a single vehicle but also from the perspective of the whole C-V2X network. To this end, this paper conducts an analytical evaluation of two performance metrics, i.e. coverage probability and spectral efficiency as follows.

• •

  The coverage probability of the typical vehicle $v_{o}$ in downlink, is defined as the probability that the received SINR outperforms a predetermined threshold $t$ [32]. It can be expressed as

  $\displaystyle{\mathbb{P}_{{\mathop{cov}}}}\left(t\right)=\mathbb{P}\left({S\!I\!N\!{R_{D}}>t}\right).$

  (9)

  It can also be calculated as the proportion of vehicles that have the received $S\!I\!N\!{R_{D}}$ above a threshold $t$, i.e., establish a successful connection with the DBSs in $\varphi_{c}$, among all vehicles in the simulation scenario. Since the cumulative distribution function (CDF) of $S\!I\!N\!{R_{D}}$ is ${\mathbb{P}_{{\mathop{cov}}}}\left(t\right)=\mathbb{P}\left({S\!I\!N\!{R_{D}}<t}\right)$, the coverage probability can also be expressed as the complementary cumulative distribution function (CCDF) of the $S\!I\!N\!{R_{D}}$ at the typical vehicle from the DBSs.

• •

  The spectral efficiency of the typical vehicle $v_{o}$ is the amount of data transmitted per unit of bandwidth [33]. According to the Shannon Theory, the spectral efficiency of the downlink is

  $\displaystyle\tau_{D}=\mathbb{E}\left[{\ln\left({1+S\!I\!N\!{R_{D}}}\right)}\right],$

  (10)

  where $\mathbb{E}(\cdot)$ is the expectation function. The spectral efficiency describes the likelihood of a wireless communication system achieving a specific information amount within a certain time period and space range during actual use [23]. It can help evaluate the performance of C-V2X in a multi-connectivity environment and determine whether system optimization or adjustments are needed [33].

Since the typical vehicle is connected to the $n$ nearest DBSs in multi-connectivity, no other DBSs are closer than distance $x_{n}$. And it also means that all interference DBSs are farther than $x_{n}$. The above definition can be expressed by $f\left({{x_{1}},{x_{2}},\cdots,{x_{n}}}\right)$, and we call it joint distance distribution for ${{x_{1}},{x_{2}},\cdots,{x_{n}}}$.

To compare the joint distance distribution $f\left({{x_{1}},{x_{2}},\cdots,{x_{n}}}\right)$ and the PDF of $x_{n}$, we provide the PDF of $x_{n}$ in Eq. (18) as

where $\Gamma(n)=(n-1)!$ when $n$ is a positive integer.

A general expression for the coverage probability of multi-connectivity in C-V2X is calculated in this subsection.

This subsection derives the expression of spectral efficiency for the downlink by using the tools of stochastic geometry for C-V2X in multi-connectivity. We computed the spectral efficiency in units of $nats/s/Hz$ ($1~bit=ln(2)=0.693~nats$) for the typical vehicle.

In order to compare the performance with multi-connectivity in C-V2X, this subsection focuses on the calculation of the coverage probability and spectral efficiency in a cellular single-connectivity scenario, which represents the most basic approach. In this scenario, the typical vehicle associates with the cellular base station (CBS) whit the MRP.

As the single-connectivity is a special case of multi-connectivity, we model the similar channel model as in multi-connectivity scenario, and use $\lambda_{C}$ to denote the transformed intensity $\lambda_{c}$ of CBS $\varrho_{{}_{C}}^{t}$ after executing the procedures of random displacement, and $\lambda_{C}>\lambda_{D}$. As the CBS are distributed along the road following a 1-D PPP, the PDF of distance distribution is

where $x$ is the distance between the nearest CBS and the typical vehicle.

The coverage probability of the downlink in cellular single-connectivity is

where $j=\mu t{x^{{\alpha_{d}}}}/{P_{d}}$, the $S\!I\!N\!R_{c}^{D}$ is

where the interference $I_{d}^{c}$ is

The Laplace transform of $I_{d}^{c}$ is

The spectral efficiency of cellular single-connectivity for downlink is

where

where $j=\mu({e^{t}}-1){x^{{\alpha_{d}}}}/{P_{d}}$, the ${\zeta_{I_{d}^{c}}}\left(j\right)$ is

Fig. 3(a) shows the distance distribution of $x_{1}$, $x_{2}$, and $x_{3}$. We can see that the peak is gradually moving away from the origin from $x_{1}$ to $x_{3}$. Fig. 3(b) and Fig. 3(c) depict the joint distance distribution for $f(x_{1},x_{2})$ and $f(x_{1},x_{2},x_{3})$. We can see that the peak of $f(x_{1},x_{2},x_{3})$ is closest to the origin, followed by $f(x_{1},x_{2})$, and the furthest is $f(x_{1})$. Compared with distance distributions in Fig. 3(a), the peak of joint distance distribution has a huge boost. The closer the distance between the peak and the origin, the better the performance. It can be observed that in single-connectivity, $f(x_{1})$ exhibits better performance. Compared to a single-connectivity, a greater number of DBSs connections in multi-connectivity lead to a more significant performance improvement.

The coverage probability variation of downlink with threshold $t$ is illustrated in Fig. 4. It is apparent that the simulation values closely match the theoretical values, which further verifies the validity of the theoretical derivation results. The density of BS $\lambda_{c}$ in single-connectivity is set as 3 nodes/km, and the density $\lambda_{d}$ of DBSs in multi-connectivity is set as 6 nodes/km. Though $\lambda_{c}>\lambda_{d}$, we can see that the dual-connectivity and triple-connectivity still have a greater coverage probability than single-connectivity. This suggests that multi-connectivity performs better than cellular single-connectivity in C-V2X and multi-connectivity enhances the coverage area of communications.

Fig. 5 illustrates the coverage probability as a function of path loss exponent $\alpha_{d}$. It can be seen that the Monte Carlo simulation data and analytical data fit well. Considering the dense deployment of DBSs in the simulation, vehicles are in an interference-limited state. At this moment, the interference power will decrease in accordance with the increase of $\alpha_{d}$, which in turn lead to a promotion of SINR. Therefore, the coverage probability will be improved even if the channel gain decrease.

For a better investigation of the impact of path loss exponent $\alpha_{d}$ on the coverage probability under different densities of DBSs, we plot Fig. 6 in a dual-connectivity scenario. As shown in Fig. 6, when the density of DBSs is in a dense deployment, the system is an interference-limited network. The distance between the signal DBSs $\in\varphi_{c}$ and the interference DBSs is close to the typical vehicle, so the increase of $\alpha_{d}$ leads to a greater impact on the interference signal power, resulting in an increase in the coverage probability. However, when the density is low enough, the system can be considered as a noise-limited network. Both the signal DBSs and the interference DBSs are far away from the typical vehicle, so the increase of $\alpha_{d}$ has a greater impact on the receiving signal power, leading to a continuous decrease in the coverage probability. When the density $\lambda_{d}$ is at an appropriate size, such as $\lambda_{d}$ $=0.005~nodes/km$, the coverage probability first increases and then decreases with the increase of $\alpha_{d}$.

The coverage probability of all cases decreases as the threshold $t$ increases, while the difference between single-connectivity and multi-connectivity first increases and then decreases in Fig. 7. This is mainly because the coverage probability is respectively high and low at small and large thresholds, respectively. Only when the threshold value is in the middle range, the difference in coverage probability is large, and the advantage of applying multi-connectivity is also demonstrated. It can be observed that increasing the number of connected DBSs does not result in a proportional increase in the coverage probability gain. Hence, at a particular threshold, there exists a balance tradeoff between the number of corporation DBSs that are connected and the associated cost.

As shown in Fig. 8, the coverage probability varies with the density $\lambda_{d}$ of DBSs. The coverage probability goes up first and then stays relatively constant for both cellular single-connectivity and multi-connectivity when the density $\lambda_{d}$ increases. It can also be seen that the growth rate of multi-connection is faster than that of single-connection, and it also reaches a stable point later. At the same time, it can be seen that the coverage probability does not increase significantly when comparing triple-connectivity to dual-connectivity. Therefore, increasing the number of cooperative BSs can improve communication coverage area, but it may also incur high costs. By observing the horizontal axis, it can be noted that the change in DBS density $\lambda_{d}$ is relatively small in magnitude. This implies that the coverage probability is sensitive to changes in $\lambda_{d}$, and suggests that simply increasing $\lambda_{d}$ does not necessarily lead to an improvement in coverage probability.

The spectral efficiency varies with the path loss exponent $\alpha_{d}$ is illustrated in Fig. 9. The simulation data is represented by the points and the theoretical data is represented by the dashed line in Fig. 9. We can observe that the simulation data matches the theoretical data well, demonstrating the correctness of the theoretical derivation. The increase of path loss exponent leads to a gradual increase in spectral efficiency and an approximately linear relationship. The reason is that, as the dense deployment of DBSs, vehicles are in a interference-limited state, with the increase of path loss exponent $\alpha_{d}$, the interference signals weaken more than the signals transmitted by the cooperative BSs, resulting in an increase in SINR and consequently spectral efficiency. It appears that there exists a close-to-linear relationship between the path loss exponent $\alpha_{d}$, mainly because after dividing the received signal power ${{\sum\limits_{i\in\varphi_{c}}{{P_{d}}{g_{d}}x_{i}^{-{\alpha_{d}}}}}}$ into the denominator of Eq. (9), since the thermal noise is much smaller than the received signal and thus $\sigma_{d}^{2}/{{\sum\limits_{i\in\varphi_{c}}{{P_{d}}{g_{d}}x_{i}^{-{\alpha_{d}}}}}}\approx 0$, then after logarithmic calculation, it approximates to a linear relationship. Fig. 12 depicts the spectral efficiency of dual-connectivity as a function of $\alpha_{d}$ in different densities of DBSs. Since the spectral efficiency is mainly affected by SINR, it can be seen that the trend of spectral efficiency is similar to that of coverage probability in Fig. 6 when the DBSs layout changes from extremely dense to extremely sparse.

Fig. 11 plots the spectral efficiency improvements between the multi-connectivity and single-connectivity. It can be observed that the application of multi-connectivity technology greatly improves the spectral efficiency in C-V2X. The improvement achieved by dual-connectivity can reach up to 40%, and that achieved by triple-connectivity can increase to more than 75%. With the increase of path loss exponent $\alpha_{d}$, the performance improvement of spectral efficiency does not increase significantly. This indicates that multi-connectivity technology has a stable performance gain.

Fig. 12 illustrates the spectral efficiency varies with the DBS density $\lambda_{d}$. With the increase of $\lambda_{d}$, the spectral efficiency first improves and then remains stable. Moreover, the improvement of multi-connectivity is greater than that of single-connectivity, and the stable point is also further back. This means that multi-connectivity has a larger range of performance improvement. Similar to Fig. 8, adding too many DBSs does not continuously improve spectral efficiency. Additionally, it can be observed that the stable point of spectral efficiency arrives earlier than the stable point of coverage probability. Thus, when increasing the density of DBSs $\lambda_{d}$, it is necessary to comprehensively consider the demands between spectral efficiency and coverage probability.