Power Allocation Based on Finite-Horizon Optimization for Data Transmission in Vehicle-to-Roadside Communications

Abstract

In this paper, we study the power allocation strategy in a drive-thru scenario, where several access points are installed along the highway to provide Internet services to vehicles on the highway. We consider single-hop vehicle-to-roadside communications for a vehicle that aims to upload data within a hard deadline, where the bandwidth allocated to it is time-varying, and the size of the data is known upon it enters the area. The data bits over a time slot are correctly received if the instantaneous channel capacity $r_t$ is greater than or equal to a threshold $R_t$, and corrupted otherwise. The vehicle has to consume an amount of transmission powers for data transmission according to the power consumption at each time slot no matter whether the data bits are correctly received or not. The target is to complete the transmission of the traffic demand volume with the minimal cost. First, we consider an optimal slotted power allocation strategy with random vehicular traffic arrivals. We formulate it as a finite-horizon sequential power allocation problem. Then we solve the problem using dynamic programming and find the optimal power allocation strategy. The existence of the optimal value of the cost-to-go function is proven. Simulation results show that our proposed strategy achieves less cost than the heuristic strategy. The impacts of different deadlines, traffic demand volumes, mean vehicle speeds, and outage probabilities on total cost are also analyzed.

Appendix: Proof of Theorem 1

Proof

Denote $[p_{out}J_{t+1}(\widetilde{D}_t,\frac{B_t}{u})+ (1-p_{out})J_{t+1}(\widetilde{D}_t-R_t(p_t)\triangle t,\frac{B_t}{u})]$ as $\varphi (\widetilde{D}_t,b_t,p_t)$, namely,

$$\begin{aligned} \quad \varphi (\widetilde{D}_t,b_t,p_t)=\left[ p_{out}\times J_{t+1}\left( \widetilde{D}_t,\frac{B_t}{u}\right) + (1-p_{out})\times J_{t+1}\left( \widetilde{D}_t-R_t(p_t)\triangle t,\frac{B_t}{u}\right) \right] \end{aligned}$$

(52)

Let $\triangle p_t > 0$, then we have

$$\begin{aligned}&\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)\nonumber \\&\quad =(1-p_{out})\left[ J_{t+1}\left( \widetilde{D}_t-R_t(p_t+\triangle p_t)\triangle t,\frac{B_t}{u}\right) - J_{t+1}\left( \widetilde{D}_t-R_t(p_t)\triangle t,\frac{B_t}{u}\right) \right] \nonumber \\ \end{aligned}$$

(53)

$\widetilde{D}_t-R_t(p_t+\triangle p_t)\triangle t \le \widetilde{D}_t-R_t(p_t)\triangle t$ for $R_t(p_t)=b_tlog_2(\frac{-p_t l_t ln(1-p_{out})}{N_0 b_t}+1)$ is a nondecreasing function of $p_t$. From Lemma 1, we have:

$$\begin{aligned} \left[ J_{t+1}\left( \widetilde{D}_t-R_t(p_t+\triangle p_t)\triangle t,\frac{B_t}{u}\right) - J_{t+1}\left( \widetilde{D}_t-R_t(p_t)\triangle t,\frac{B_t}{u}\right) \right] \le 0 \end{aligned}$$

(54)

Suppose $p_{t1}> p_{t2}$. Since $R_t(p_t)=b_tlog_2(\frac{-p_t l_t ln(1-p_{out})}{N_0 b_t}+1)$ is a nondecreasing concave function in $p_t$, we have:

$$\begin{aligned} \left( \widetilde{D}_t-R_t(p_{t1}+\triangle p_t)\triangle t\right) \le \left( \widetilde{D}_t-R_t(p_{t2}+\triangle p_t)\triangle t\right) \end{aligned}$$

(55)

and we have:

$$\begin{aligned}&\left[ (\widetilde{D}_t-R_t(p_{t2}+\triangle p_t)\triangle t)- (\widetilde{D}_t-R_t(p_{t2})\triangle t)\right] \nonumber \\&\quad \le \left[ (\widetilde{D}_t-R_t(p_{t1}+\triangle p_t)\triangle t)- (\widetilde{D}_t-R_t(p_{t1})\triangle t)\right] \nonumber \\&\quad \le 0 \end{aligned}$$

(56)

We know $J_t(\widetilde{D}_t,b_t)$ is a nondecreasing convex function in $\widetilde{D}_t$, $\forall b_t \in \mathcal B, t\in \mathcal T$ from Lemma 1, then we have (55).

$$\begin{aligned}&\lim \limits _{\begin{array}{c} \triangle p_t > 0,\\ \triangle p_t \rightarrow 0 \end{array}}\frac{\left[ J_{t+1}\left( \widetilde{D}_t-R_t(p_{t2}+\triangle p_t)\triangle t,\frac{B_t}{u}\right) - J_{t+1}\left( \widetilde{D}_t-R_t(p_{t2})\triangle t,\frac{B_t}{u}\right) \right] }{\triangle p_t}\nonumber \\&\quad \le \lim \limits _{\begin{array}{c} \triangle p_t > 0,\\ \triangle p_t \rightarrow 0 \end{array}}\frac{\left[ J_{t+1}\left( \widetilde{D}_t-R_t(p_{t1}+\triangle p_t)\triangle t,\frac{B_t}{u}\right) - J_{t+1}\left( \widetilde{D}_t-R_t(p_{t1})\triangle t,\frac{B_t}{u}\right) \right] }{\triangle p_t}\nonumber \\&\quad \le 0 \end{aligned}$$

(57)

Denote $[J_{t+1}(\widetilde{D}_t-R_t(p_{t}+\triangle p_t)\triangle t,\frac{B_t}{u}) - J_{t+1}(\widetilde{D}_t-R_t(p_{t})\triangle t,\frac{B_t}{u})]$ as $\triangle J_{t+1}(\widetilde{D}_t$ $-R_t(p_{t}+\triangle p_t)\triangle t,\frac{B_t}{u})$, then we have:

$$\begin{aligned}&\lim \limits _{\begin{array}{c} \triangle p_t > 0,\\ \triangle p_t \rightarrow 0 \end{array}} \frac{\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)}{\triangle p_t}\nonumber \\&\quad =\lim \limits _{\begin{array}{c} \triangle p_t > 0,\\ \triangle p_t \rightarrow 0 \end{array}} (1-p_{out})\frac{\triangle J_{t+1}\left( \widetilde{D}_t-R_t(p_{t}+\triangle p_t)\triangle t,\frac{B_t}{u}\right) }{\triangle p_t}\nonumber \\&\quad \le \quad 0 \end{aligned}$$

(58)

and $\lim \limits _{\begin{array}{c} \triangle p_t > 0,\\ \triangle p_t \rightarrow 0 \end{array}} \frac{\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)}{\triangle p_t}$ increases with $p_t$.

Suppose $\triangle p_t<0$. By the same way, we have:

$$\begin{aligned} \lim \limits _{\begin{array}{c} \triangle p_t < 0,\\ \triangle p_t \rightarrow 0 \end{array}} \frac{\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)}{\triangle p_t} \le 0 \end{aligned}$$

(59)

and $\lim \limits _{\begin{array}{c} \triangle p_t < 0,\\ \triangle p_t \rightarrow 0 \end{array}} \frac{\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)}{\triangle p_t}$ increases with $p_t$.

From (58) and (59), we have

$$\begin{aligned} \lim \limits _{\begin{array}{c} \triangle p_t \rightarrow 0 \end{array}} \frac{\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)}{\triangle p_t} \le 0 \end{aligned}$$

(60)

and $\lim \limits _{\begin{array}{c} \triangle p_t \rightarrow 0 \end{array}} \frac{\varphi (\widetilde{D}_t,b_t,p_t+\triangle p_t)-\varphi (\widetilde{D}_t,b_t,p_t)}{\triangle p_t}$ increases with $p_t$.

Thus, $\varphi (\widetilde{D}_t,b_t,p_t)$ is a nonincreasing convex function on $p_t$.

Denote $\sum _{u=1}^{2\rho S_j} [\frac{\lambda _j^u}{u!\phi _j(n_{t+1})} \varphi (\widetilde{D}_t,b_t,p_t)]$ and $\sum _{u=1}^{2\rho S_{j+1}} [\frac{\lambda _{j+1}^u}{u!\phi _{j+1}(n_{t+1})} \varphi (\widetilde{D}_t,b_t,p_t)]$ as $A(\widetilde{D}_t,b_t,p_t)$ and $C(\widetilde{D}_t,b_t,p_t)$, respectively. We know that $A(\widetilde{D}_t,b_t,p_t)$ is a nonincreasing convex function of $p_t$ for $\frac{\lambda _j^u}{u!\phi _j(n_{t+1})}\ge 0$. By the same way, we know that $C(\widetilde{D}_t,b_t,p_t)$ is a nonincreasing convex function of $p_t$ for $\frac{\lambda _{j+1}^u}{u!\phi _{j+1}(n_{t+1})}\ge 0$.

From (41), we have

$$\begin{aligned}&\psi _t(\widetilde{D}_t,b_t,p_t)\nonumber \\&\quad =\mu (p_t)+Pr(h_t\in \mathcal H)A(\widetilde{D}_t,b_t,p_t)+[1-Pr(h_t\in \mathcal H)]C(\widetilde{D}_t,b_t,p_t) \end{aligned}$$

(61)

Since $Pr(h_t\in \mathcal H)\ge 0$ and $1-Pr(h_t\in \mathcal H) \ge 0$, we know that $\psi _t(\widetilde{D}_t,b_t,p_t)$ is a convex function on $p_t$ for the convexity of $\mu (p_t)$, $A(\widetilde{D}_t,b_t,p_t)$ and $C(\widetilde{D}_t,b_t,p_t)$. So $\psi _t(\widetilde{D}_t,b_t,p_t)$ has a minimal value when $0 \le p_t \le \frac{N_0b_t}{-ln(1-p_{out})l_t}(2^{\frac{\frac{\widetilde{D}_t}{\triangle t}}{b_t}}-1)$. Thus we proved Theorem 1. $\square$