Rate allocation and relaying strategy adaption in wireless relay networks

Abstract

In this paper, we propose a dynamic rate allocation and relaying strategy adaption scheme for data transmission over wireless relay networks, where autonomous nodes expose node-selfishness, i.e. being unwilling to forward other nodes’ data packets, for saving their energy resources. Aided by an incentive mechanism, we develop a virtual node-selfishness queue (VSQ) to portray the node’s dynamic selfish characteristic in terms of its energy resource and incentives. Then, a stochastic optimization model is employed to maximize the average network throughput whilst keeping the network stable and bounding the node-selfishness. The problem of stochastic optimization is further decomposed into two subproblems via the Lyapunov optimization theory, which corresponds to the rate allocation for source and the relaying strategy adaption for autonomous relay, respectively. After that, a joint rate allocation and relaying strategy adaption algorithm is developed to accommodate the wireless network only according to the current network state information, i.e. the data queue sate information, VSQ information at relays and the channel state information of the links. The explicit tradeoff between network throughput and average data transmission delay is theoretically proven. Simulation results validate the theoretical analysis of our proposed scheme.

Appendices

Appendices

1.1 Appendix 1: Proof of Lemma 1

Followed from Eq. (8), we have

$$\begin{aligned}&L({\varvec{\Theta }}^{t+1})-L({\varvec{\Theta }}^{t})=\frac{1}{2}(\max [\mathbf {Q}^{t}-\mathbf {R}^{t},0]+\mathbf {F}_m^{t})^{H}(\max [\mathbf {Q}^{t}-\mathbf {R}^{t},0]+\mathbf {F}_m^{t})\\&\quad +\,\frac{1}{2}(\max [\mathbf {T}^{t}-\mathcal {R}^p_m\mathbf {F}_n^{t},0]+\mathcal {R}^g\mathbf {R}^{t}+\mathbf {S}^{t}P^T+P^R\mathbf {F}_m^{t}))^{H}(\max [\mathbf {T}^{t}\\&\quad -\,\mathcal {R}^p_m\mathbf {F}_n^{t},0]+\mathbf {S}^{t}P^T+P^R\mathbf {F}_m^{t}+\mathcal {R}^g\mathbf {R}^{t})-\frac{1}{2}(\mathbf {Q}^{t})^{H}\mathbf {Q}^{t}-\frac{1}{2}(\mathbf {T}^{t})^{H}\mathbf {T}^{t}, \end{aligned}$$

where we have used the queue evolution in Eqs. (1) and (3).

Based on the inequality property that $(\max [Q-r,0]+f)^{2}\le Q^{2}+r^{2}+f^{2}+2Q(f-r)$, we obtain

$$\begin{aligned}&L({\varvec{\Theta }}^{t+1})-L({\varvec{\Theta }}^{t}) \le \frac{1}{2}(1+(\mathcal {R}^g)^2)(\mathbf {R}^{t})^{H}(\mathbf {R}^{t})+\frac{1}{2}(\mathbf {F}_m^{t})^{H}(\mathbf {F}_m^{t}) +(\mathbf {F}_m^{t}-\mathbf {R}^{t})^{H}\mathbf {Q}^{t}+\frac{1}{2}(\mathcal {R}^p_m\mathbf {F}_n^{t})^{2} \nonumber \\&\qquad +\,\frac{1}{2}\mathbf {S}^{2}(P^T)^2+\frac{1}{2}(P^R\mathbf {F}_m^{t})^2+2(\mathbf {S}^{t}P^T-\mathcal {R}^p_m\mathbf {F}_n^{t}+\mathbf {F}_m^{t}P^R+\mathcal {R}^g\mathbf {R}^{t})^{H}\mathbf {T}^t \nonumber \\&\quad \le B+(\mathbf {F}_m^{t}-\mathbf {R}^{t})^{H}\mathbf {Q}^{t}+(\mathbf {S}^{t}P^T-\mathcal {R}^p_m\mathbf {F}_n^{t}+\mathbf {F}_m^{t}P^R+\mathcal {R}^g\mathbf {R}^{t})^H \mathbf {T}^t, \end{aligned}$$

(15)

where B is an upper bound on the term $\frac{1}{2}[(1+(\mathcal {R}^g)^2)(\mathbf {R}^{t})^{H}(\mathbf {R}^{t})+(\mathbf {F}_m^{t})^{H}(\mathbf {F}_m^{t})+(\mathcal {R}^p\mathbf {R}^{t})^{2}+\mathbf {S}^{2}(P^T)^2+(P^R\mathbf {F}_m^{t})^2]$, which holds under the fact that the data rates satisfy the properties of boundness. Adding $-V{\bar{\mathbf{E}}}[U^{t}({\alpha }^{t}, \mathbf {h}^{t})|{\varvec{\Theta }}^{t}]$ to both sides of Eq. (15) and taking a expectation, yields

$$\begin{aligned} \varDelta ({\varvec{\Theta }})^{t}&-V \mathbb {{E}}\left[ U^{t}\left( \alpha ^{t}, \mathbf {h}^{t}\right) |{\varvec{\Theta }}^{t}\right] \le B-V \mathbb {{E}}[ U^{t}(\alpha ^{t}, \mathbf {h}^{t})|{\varvec{\Theta }}^{t}]+\mathbb {{E}}\left[ (\mathbf {F}_m^{t}-\mathbf {R}^t)^{H}\mathbf {Q}^{t}|{\varvec{\Theta }}^{t}\right] ,\nonumber \\&+\left[ \left( \mathbf {S}^{t}P^T-\mathcal {R}^p_m\mathbf {F}_n^{t}+\mathbf {F}_m^{t}P^R+\mathcal {R}^g\mathbf {R}^{t}\right) ^{H}\mathbf {T}^t|{\varvec{\Theta }}^{t}\right] . \end{aligned}$$

(16)

This completes the proof of Lemma 1.

1.2 Appendix 2: Proof of Lemma 2

Since there at least exists a rate allocation policy that stabilizes the wireless network under arriving data session rate $\varvec{\lambda }+\varepsilon$. Then, we obtain that $\varvec{\lambda }$ is strictly interior to the capacity region $\varLambda$, and that $\varvec{\lambda }+\varepsilon$ is still in $\varLambda$ for a positive $\varepsilon$, there exist a $\mathbf {h}$-only policy that

$$\begin{aligned} \mathbb {E}\left[ \tilde{r}_{n,m}^{f_{m},t}\right]- & {} \mathbb {E}\left[ \tilde{r}_{s,n}^{f_{m},t}\right] \ge \varepsilon ,\\ \mathbb {E}\left[ \mathcal {R}_m^p\tilde{r}_{s,n}^{f_{n},t}\right]- & {} \mathbb {E}\left[ \tilde{s}_{n,m}^{t}P^T+\mathcal {R}^g\tilde{r}_{n,m}^{f_{m},t}+\left( \tilde{r}_{s,n}^{f_{m},t}\right) P^R\right] \ge \sigma , \end{aligned}$$

where $\tilde{r}_{s,n}^{f_{m},t}$, $\tilde{r}_{n,m}^{f_{m},t}$, $\tilde{s}_{n,m}^{t}$ and $\tilde{r}_{s,n}^{f_{n},t}$ are the resulting values at time-slot t under $\mathbf {h}$-only policy.

Due to the rate allocation in Eq. (12), then

$$\begin{aligned}&\mathbb {{E}}\left[ \sum _{n}\left( V \sum _{m}w_{n,m}\log \left( {r}_{s,n}^{f_{m},t}+e\right) +V w_{n}\log \left( {r}_{s,n}^{f_{n},t}+e\right) -\sum _{m}\left( Q_{n,m}^t{r}_{s,n}^{f_{m},t}-P^RT^t_{n,m}{r}_{s,n}^{f_{m},t}\right) +\mathcal {R}^p{r}_{s,n}^{f_{n},t}\sum _{m}T_{n,m}^t\right) \right] \\&\quad \ge \mathbb {{E}}\left[ \sum _{n}\left( V \sum _{m}w_{n,m}\log \left( \tilde{r}_{s,n}^{f_{m},t}+e\right) +V w_{n}\log \left( \tilde{r}_{s,n}^{f_{n},t}+e\right) -\sum _{m}(Q_{n,m}^t\tilde{r}_{s,n}^{f_{m},t}-T^t_{n,m}P^R\tilde{r}_{s,n}^{f_{m},t}) +\mathcal {R}^p\tilde{r}_{s,n}^{f_{n},t}\sum _{m}T_{n,m}^t\right) \right] . \end{aligned}$$

Similarly,

$$\begin{aligned}&\mathbb {{E}}\left[ \sum _{n,m}\left( Q_{n,m}^t-\mathcal {R}^gT_{n,m}^t\right) \log \left( 1+{s}_{n,m}^tP^T|h_{n,m}^t|^{2}\right) -\sum _{n,m}T_{n,m}^t{s}_{n,m}^tP^T\right] \\&\quad \ge \mathbb {{E}}\left[ \sum _{n,m}\left( Q_{n,m}^t-\mathcal {R}^gT_{n,m}^t\right) \log \left( 1+\tilde{s}_{n,m}^tP^T|h_{n,m}^t|^{2}\right) -\sum _{n,m}T_{n,m}^t\tilde{s}_{n,m}^tP^T\right] . \end{aligned}$$

The rate set ${\mathbf {F}}^{t}$ and relaying strategy ${\mathbf {S}}^{t}$ obtained by Algorithm 1 have better performance than the $\mathbf {h}$-only policy. There comes the equation at the top of the former page.

1.3 Appendix 3: Proof of Theorem 1

(a) Note that $Q_{n,m}^t\ge 0$ and $T_{n,m}^t\ge 0$ in this paper. Then, combining Eq. (10) in Lemma 1 and the conclusions in Lemma 2, we have

$$\begin{aligned} \varDelta ({\varvec{\Theta }})^{t}-V \mathbb {{E}}[U^{t}({\alpha }^{t}, \mathbf {h}^{t})|{\varvec{\Theta }}^{t}] \le B-V\tilde{U}^t(\tilde{\alpha }^{t}, \mathbf {h}^{t})-\sum _{n,m}\left( \varepsilon Q_{n,m}^t+\delta T_{n,m}^t\right) \le B-V \overline{U}^{opt}, \end{aligned}$$

(17)

where $\overline{U}^{opt}$ is the average optimal utility of $\mathbf {h}$-only policy. From Proposition 1, we have $\varDelta ({\varvec{\Theta }})^{t}\le \psi$ where $\psi$ is a constant. Taking telescoping sum over $t\in \{0,1,\ldots ,\varsigma -1\}$ in the above inequality and using Eq. (10), we get

$$\begin{aligned} \mathbb {E}\{T_n(\varsigma )^2\}\le 2\varsigma \psi +2\mathbb {E}\{L({\varvec{\Theta }})^{0})\}. \end{aligned}$$

(18)

From the variance formula: $D(\mid T_n(\varsigma )\mid )=\mathbb {E}(\mid T_n(\varsigma )\mid ^2)-[\mathbb {E}(\mid T_n(\varsigma )\mid )]^2$, we have $\mathbb {E}(\mid T_n(\varsigma )\mid ^2)\ge [\mathbb {E}(\mid T_n(\varsigma )\mid )]^2$. Thus

$$\begin{aligned} \mathbb {E}\{\mid T_n(\varsigma )\mid \}\le \sqrt{2\varsigma \psi +2\mathbb {E}\{L({\varvec{\Theta }})^{0})}. \end{aligned}$$

Dividing by $\varsigma$ and taking a limit as $\varsigma \rightarrow \infty$ prove $\lim _{\varsigma \rightarrow \infty }\frac{\mathbb {E}\{\mid T_n(\varsigma )\mid \}}{\varsigma }=0$.

Hence, queues $T_{n,m}^t$ are mean rate stable from Definition 1. Similarly, we can prove that queues $Q_{n,m}^t$ are mean rate stable as well.

(b) After having summed over $t\in \{0,1,\cdots ,\varsigma -1\}$ for Eq. (17), we get

$$\begin{aligned} \mathbb {{E}}[L({\varvec{\Theta }}^{\varsigma -1})|{\varvec{\Theta }}^{t}]-\mathbb {{E}}[L({\varvec{\Theta }}^{0})]-V\sum _{t=0}^{\varsigma -1} \mathbb {{E}}[U^{t}|{\varvec{\Theta }}^{t}] \le B\varsigma -V \varsigma \overline{U}^{opt}. \end{aligned}$$

Then, following from the general derivation in [8], we get

$$\begin{aligned} -\frac{\mathbb {{E}}[L({\varvec{\Theta }}^{0})]}{\varsigma V}+\overline{U}^{opt} \le \frac{B}{V}+\frac{1}{\varsigma }\sum _{t=0}^{\varsigma -1}\mathbb {{E}}[U^{t}|{\varvec{\Theta }}^{t}]. \end{aligned}$$

Taking limit as $\varsigma \rightarrow \infty$, we comes the solution that $\overline{U} \ge \overline{U}^{opt}- B/V$.

(c). Summing Eq. (17) over $t\in \{0,1,\cdots ,\varsigma -1\}$, we get

$$\begin{aligned} \mathbb {{E}}[L({\varvec{\Theta }}^{\varsigma -1})|{\varvec{\Theta }}^{t}]-\mathbb {{E}}[L({\varvec{\Theta }}^{0})] -V\sum _{t=0}^{\varsigma -1} \mathbb {{E}}[U^{t}|\mathbf {Q}^{t}] \le B\tau -V\varsigma \overline{U}^{opt}-\varepsilon \sum _{t=0}^{\varsigma -1}\sum _{n,m}\mathbb {{E}}\left[ Q_{n,m}^t\right] . \end{aligned}$$

Similarly, employing the derivation method, we get

$$\begin{aligned} \frac{\mathbb {{E}}[L(\mathbf {Q}^{0})]}{\varsigma }-V\frac{1}{\varsigma }\sum _{t=0}^{\varsigma -1} \mathbb {{E}}[U^{t}|\mathbf {Q}^{t}]\le B-V \overline{U}^{opt}-\varepsilon \frac{1}{\varsigma }\sum _{t=0}^{\varsigma -1}\sum _{n,m}\mathbb {{E}}\left[ Q_{n,m}^t\right] , \end{aligned}$$

Taking the limit as $\varsigma \rightarrow \infty$ and dividing by $\varepsilon$, we have

$$\begin{aligned} \lim _{\varsigma \rightarrow \infty } \frac{1}{\varsigma }\sum _{t=0}^{\varsigma -1}\sum _{n,m}\mathbb {{E}}\left[ Q_{n,m}^t\right] \le \frac{B+V\left( U_{\max }-\overline{U}^{opt} \right) }{\varepsilon }, \end{aligned}$$

based on the Proposition 1. Thus, the theorem for time averaged queues is proved.