Ultra-Reliable and Low-Latency Vehicular Transmission: An Extreme Value Theory Approach

I Introduction

Vehicle-to-vehicle (V2V) communication is one of the most promising enablers for intelligent transportation systems in which latency and reliability are prime concerns . Nevertheless, the vast majority of the existing V2V literature does not address latency and reliability while some others focus on the coverage probability of radio signal transmission . To ensure ultra-reliable low latency communication (URLLC), queuing latency plays a pivotal role when the traffic arrival and service rates are dynamic and non-deterministic. Particularly in V2V communication, the quality of wireless links varies significantly due to vehicles’ high mobility. The authors in take into account the dynamics of queue length and aim at bounding the average queue length within a finite value. While interesting, focusing only on average performance metrics (e.g., average queue length and average delay) is not sufficient to enable URLLC, which instead requires looking into the higher-order statistics or the tail behavior of the distribution. To this end, we define a new reliability measure in terms of maximal queue length among all vehicle pairs and characterize its statistics. Analyzing the statistics of the network-wide maximal queue length provides key insight for the URLLC system design. The studied problem is cast as a power minimization problem subject to statistical constraints on the network-wide maximal queue length. However, to get the network-wide maximal queue length, all vehicles and the roadside unit (RSU) need to exchange queue state information (QSI) which can incur significant signaling overhead in V2V communication. To alleviate this issue, we leverage principles of extreme value theory (EVT) to locally characterize the maximal queue length, which is incorporated as a constraint into the stochastic optimization problem. Our proposed solutions include one semi-centralized and one distributed extreme queue-aware power allocation approaches for V2V communication. Numerical results show the effectiveness of using EVT for the study of ultra-reliable and low-latency vehicular communication.

II System Model

for all VUE pairs $k\in\mathcal{K}$. Additionally, since the RBs are reused by distant VUE transmitters in multiple groups, we treat the aggregate interference power as a constant term $I$ and approximate the transmission rate as R_{k}(t)\approx\sum\limits_{n\in\mathcal{N}_{k}}W\log_{2}\big{(}1+\frac{P_{k}^{n}(t)h_{kk}^{n}(t)}{N_{0}W+I}\big{)}.

III Extreme Queue-Aware Power Allocation

with $\mathbf{P}(t)=(P_{k}^{n}(t),k\in\mathcal{K},n\in\mathcal{N}_{k})$ satisfying (1) and $\bar{B}_{\rm th}=2(\kappa-\bar{M}_{\rm th})/\delta$. To solve problem (2), we use tools from Lyapunov stochastic optimization to dynamically allocate VUEs’ transmit power. In order to ensure (2b) and (2c), we respectively introduce two virtual queues which evolve as follows:

Due to space limitations, we skip the rest of the derivations related to the Lyapunov optimization. The interested readers please refer to for the details. Here, we directly show the results after applying Lyapunov optimization. In each slot $t$, each VUE pair $k\in\mathcal{K}$ solves the convex optimization problem,

with $P_{k}^{n}(t)$ satisfying (1) and J_{k}(t)=WT_{\rm c}\big{[}Q^{(M)}(t)+\big{(}2Q^{(B)}(t)+1\big{)}\big{(}Q_{k}(t)+\lambda_{k}(t)\big{)}+2\big{(}Q_{k}(t)+\lambda_{k}(t)\big{)}^{3}\big{]}. Here, the parameter $V\geq 0$ trades off the power cost optimality and queue length reduction of (2). Applying the Karush-Kuhn-Tucker (KKT) conditions to (5), the VUE transmitter finds a transmit power $P_{k}^{n*}(t)>0,\forall\,n\in\mathcal{N}_{k}$, which satisfies $\frac{J_{k}(t)h_{kk}^{n}(t)}{(N_{0}W+I+P_{k}^{n*}(t)h_{kk}^{n}(t))\ln 2}=V+\eta$, if $\frac{J_{k}(t)h_{kk}^{n}(t)}{(N_{0}W+I)\ln 2}>V+\eta$. Otherwise, $P_{k}^{n*}(t)=0$. Moreover, the Lagrange multiplier $\eta$ is 0 if $\sum\limits_{n\in\mathcal{N}_{k}}P_{k}^{n*}(t)<NP_{\max}$, and we have $\sum\limits_{n\in\mathcal{N}_{k}}P_{k}^{n*}(t)=NP_{\max}$ when $\eta>0$. Note that given a small value of $V,$ the derived power $P_{k}^{n*}(t)$ provides a sub-optimal solution to problem (2) whose optimal solution is asymptotically obtained by increasing $V.$ After sending data, the VUE pair $k$ updates $Q_{k}(t+1)$ for the next time slot $t+1$. The information flow diagram of the RSU-aided power allocation scheme is shown in Fig. 1. Note that to obtain $J_{k}(t)$ at the VUE, the RSU requires all VUEs’ QSI in each time slot to calculate $M(t)$, update (3) and (4), and feed $Q^{(M)}(t)$ and $Q^{(B)}(t)$ back to all VUE pairs. However, frequent information exchange between the RSU and VUEs incurs significant overhead. To address this issue, we propose a solution based on EVT to locally characterize the distribution of the network-wide maximal queue length.

III-B EVT-Based Power Allocation

Considering that VUE pairs are uniformly distributed on the lanes, we can assume that VUEs’ transmission rates are i.i.d. since $R_{k}(t)$, approximately, does not vary with the other VUEs’ transmit power. The traffic arrivals are also i.i.d. among VUE pairs. Thus, we deduce that $Q_{1}(t),\cdots,Q_{K}(t)$ are i.i.d., and $M(t)$ converges to a GEV distributed RV as $K\to\infty$. Referring to the support of $M(t)$, we focus on VUE pair $k$’s queue length conditioned on $1+\xi(Q_{k}(t)-\mu)/\sigma\geq 0$. In other words, we consider the situation in which VUE pair $k$ is likely to achieve the largest queue length in the network. Subsequently, imposing the constraints on the mean and second moment of the conditional queue length, i.e.,

each VUE pair $k$ locally focuses on the power minimization problem which is modeled as follows:

In (6) and (7), the VUE requires the parameters $\mu$, $\sigma$, and $\xi$ of the network-wide maximal queue length $M(t)$, which are unknown beforehand. To deal with this, we introduce the following Theorem and then specify a local and empirical estimation mechanism for these parameters.

with $\psi\approx 0$, and $\hat{F}_{Q_{k}}$ is the empirically estimated cumulative distribution function (CDF) of $Q_{k}$. Analogously to Section III-A, we solve problem (8) using the Lyapunov optimization by introducing two virtual queues,

IV Numerical Results

Let us first verify the accuracy of using EVT to characterize the network-wide maximal queue length $M$ in the EVT-based scheme. Specifically, in Fig. 3, we plot the CCDFs of $M$ obtained numerically in the EVT-based scheme as well as theoretically using Theorem 1. When $K=20$, there is a gap since the number of VUE pairs is not sufficient to have a converged GEV approximation. However, when $K\geq 40$, numerical values match well with the theoretical approximation. Thus, even though the number of VUE pairs is moderate, EVT still provides a powerful framework to characterize the network-wide metric without resorting to $K\to\infty$. If there are more VUEs sharing resources, the incurred lower rate results in higher queue length. Next, we consider $K=80$ in the following simulations. In Fig. 4, we show the throughput-latency (i.e., power-delay since throughput increases with transmit power) tradeoffs of our proposed queue-aware approaches and the baseline. At $V=0$, the VUE aims to boost the transmission rate as per (5), yielding the highest average throughput with lowest maximal queue length. On the other hand, the optimal solutions to the power minimization problems (2) and (8) are asymptotically achieved by increasing $V$ in (5). Since the average throughput to maintain system stability is minimized as $V\to\infty$ (via power minimization), the queue length increases dramatically. Additionally, given that the VUE can increase its transmit power with a tighter requirement on $\bar{M}$ and $\mbox{Var}(M)$, the VUE can estimate the statistics of $M$ locally and find the transmit power without global QSI exchange with the RSU. If the VUE has lower power budget, using the RSU for exchanging the global QSI helps to alleviate the maximal queue length albeit increasing signaling overhead. In contrast with the baseline, our two proposed approaches achieve performance enhancement since the former is oblivious to the queue value. At low average throughput whereby higher gains are attained, resource scheduling helps to deliver data efficiently. Subsequently, we consider the RSU-aided scheme with $V=0$ owing to its highest throughput and lowest queue length performance.

Note that due to the high mobility feature in V2V communication, the small time slot length $T_{\rm c}$ (i.e., coherence time) restricts the codeword length (or blocklength) in each transmission. This hinders vehicles from achieving the Shannon rate with an infinitesimal decoding error probability. Taking into account this practical concern in finite blocklength transmission, we consider the transmission rate $R_{\rm f}=\log_{2}(1+\gamma)-\frac{\sqrt{2\gamma(\gamma+2)}{\rm erfc}^{-1}(2\epsilon)}{\sqrt{L}(1+\gamma)\ln 2}$ which incorporates the blocklength $L\ll\infty$ and a block error probability $\epsilon>0$ with the inverse complementary error function ${\rm erfc}^{-1}(\cdot)$ . Additionally, the performance of the system design in Section III can be generalized by letting $\epsilon=0.5$. Based on $R_{\rm f}$, we investigate the average throughput, denoted by $\bar{R}(L,\epsilon)$, and average queuing latency versus the blocklength for various block error probabilities in Figs. 5 and 6, where $L$ is varied by changing the coherence time $T_{\rm c}$ (i.e., vehicle speed ). For a given $L$, decreasing the average throughput allows for more reliable communication, i.e., lower $\epsilon$, as per $R_{\rm f}$. On the other hand, lower throughput increases the queue length, resulting in longer average queuing latency. Next we vary $L$ while fixing $\epsilon$. Although decreasing $L$ lowers the transmission rate, the average queuing latency can be further alleviated due to the smaller transmission time period $T_{\rm c}$. At $\epsilon=0.5$, $R_{\rm f}=\log_{2}(1+\gamma)$ is not explicitly affected by $L$. However, as $L$ (or $T_{\rm c}$) is increased, more traffic arrivals require higher power (i.e., higher throughput) whereas the average latency increases with $L$. As the blocklength increases, the average throughout curves converge to the capacity-achieving bound, i.e., $L\to\infty$ (unbounded latency) and $\epsilon\to 0$. Furthermore, using the Shannon rate-based design in the finite blocklength transmission, i.e., $\epsilon=0.5$, reliable communication is obtained at the expense of significant throughput loss in the low signal-to-noise ratio case (i.e., large VUE pair distance). Finally, Table II shows throughput ratios as a function of different VUE pair distances.

V Conclusions

This letter has studied the problem of transmit power minimization subject to high-order constraints on the maximal queue length among all vehicles. We have proposed a semi-centralized and a distributed dynamic power allocation solutions by marrying tools from Lyapunov stochastic optimization and EVT. Simulation results have shown the effectiveness of extreme value theory in designing URLLC systems as well as the performance improvements of our proposed approaches.

References