Distributed Scheduling in Wireless Multiple Decode-and-Forward Relay Networks

Abstract

In this paper, we study the distributed DOS problem for wireless multiple relay networks. Formulating the problem as an extended three-level optimal stopping problem, an optimal strategy is proposed guiding distributed channel access for multiple source-to-destination communications under the help of multiple relays. The optimality of the strategy is rigorously proved, and abides by a tri-level structure of pure threshold. For network operation, easy implementation is presented of low complexity. The close-form expression of the maximal expected system throughput is also derived. Furthermore, numerical results are provided to demonstrate the correctness of our analytical expressions, and the effectiveness is verified.

A Proof of Lemma 1

For relay number $k\in \{1,2,...,L\}$, we define $r_n':=r_n\!-\!P_s|h(n)|^2$. For function $Z_k$, the first-order derivative is calculated as that

$$\begin{aligned}&\frac{\partial {Z}_k}{\partial r_n'}=\tau _d\Big (\frac{\log _2e}{1\!+\!r_n'\!+\!P_s|h(n)|^2}\big (1\!-\!(1-e^{-\frac{r_n'}{\sigma _g^2}})^k\big ) -\frac{1}{\sigma _g^2}e^{-\frac{r_n'}{\sigma _g^2}} \\ {}&~~~~ (1\!-\!e^{-\frac{r_n'}{\sigma _g^2}})^{k-1}k \big (\log _2(1\!+\!r_n'\!+\!P_s|h(n)|^2)-\lambda \big )\Big ). \end{aligned}$$

By replacing $1\!-\!e^{-\frac{r_n'}{\sigma _g^2}}$ with $y(r_n')$, the derivative is rewritten in Eq. (15).

$$\begin{aligned} \begin{aligned}&\frac{\partial {Z}_k}{\partial r_n'} = (1\!-\!y(r_n')) ky^{1\!-\!k}(r_n')\Big (\!-\frac{\tau _d}{\sigma _g^2} \left( \log _2(1\!+\!r_n'\!+\!P_s|h(n)|^2) -\!\lambda \big ) \right. \\&\left. \quad \quad \quad +\tau _d\log _2e \frac{1}{k}\sum \limits _{i=0}^{k-1}y^{-i}(\bar{R}_n)\frac{1}{1+r_n'\!+\!P_s|h(n)|^2} \right) . \end{aligned} \end{aligned}$$

(15)

In the region $r_n'\!\ge \!0$, the factor in Eq. (15) satisfies $(1-y(r_n'))y^{1-k}(r_n')>0$. Therefore, it suffices to compare $\frac{\tau _d\log _2e}{1+r_n'+P_s|h(n)|^2} \sum \limits _{i=0}^{k-1}y^{-i}(r_n')$ and $\frac{1}{\sigma _g^2} k \big (\log _2(1\!+\!r_n'\!+\!P_s|h(n)|^2)\tau _d\!-\!\lambda \tau _d\big )$ to determine the derivative.

It can be proved that $\sum \limits _{i=0}^{k-1}y^{-i}(r_n')\frac{1}{1+r_n'+P_s|h(n)|^2}$ is decreasing and $\log _2(1+r_n'+P_s|h(n)|^2)-\lambda$ is increasing in $r_n'\!\ge \!0$, respectively. Meanwhile, as $r_n'$ is large, $\sum \limits _{i=0}^{k-1}y^{-i}(r_n')$ will approach to k and $\frac{1}{1+r_n'+P_s|h(n)|^2}$ is small, while $\{\log _2(1+r_n'+P_s|h(n)|^2)\tau _d-\lambda \tau _d'\}$ is large. Thus, the existence of stationary point, denoted by $\zeta _k\!-\!P_s|h(n)|^2$ is guaranteed, which are unique such that $\frac{\partial {Z}_k}{\partial r_n'}\!=\!0$.

As a result, function $Z_k$ increases in $r_n\!<\!\zeta _k$, and decreases in $r_n\!\ge \!\zeta _k$.

Then, relation of these points $\{\zeta _k\},k=1,2,...,L$ is further investigated.

For $k\!>\!1$, by valuing $r_n'\!=\!0$, we have the derivative satisfy

$$\begin{aligned} \frac{\partial {Z}_k}{\partial r_n'}\big |_{r_n'=0}=\frac{\log _2e\cdot \tau _d }{1\!+\!P_s|h(n)|^2}> 0.\end{aligned}$$

(16)

which means ${\zeta }_k\!>\!P_s|h(n)|^2$.

For $k\!=\!1$, we have the derivative satisfy that

$$\begin{aligned} \frac{\partial {Z}_k}{\partial r_n'}\!\big |_{r_n'\!=\!0} = \frac{\tau _d\log _2e}{1\!+\!P_s|h(n)|^2} \!-\!\frac{\log _2(1\!+\!P_s|h(n)|^2)\tau _d\!-\!\lambda \tau _d}{\sigma _g^2}. \end{aligned}$$

(17)

Suppose $\zeta _1$ satisfies $\frac{\partial {Z}_k}{\partial r_n'}=0$, we compare the points $\{\zeta _1,\zeta _2,...,\zeta _L\}$. The situation where $\zeta _1\!=\!P_s|h_{s(n)}(n)|^2$ is similar as ${\zeta }_k\!>\!P_s|h_{s(n)}(n)|^2$ for $\forall k\ge 2$.

Since $\zeta _k$ such that $\frac{\partial {Z}_k}{\partial \bar{R}_n}\!=\!0$, by induction it suffices from (15) to compare $\frac{1}{k}\sum \limits _{i=0}^{k-1}y^{-i}(r_n')$.

Based the relation shown in the following that

$$\begin{aligned}&\frac{1}{k}\sum \limits _{i=0}^{k-1}y^{-i}(r_n')-\frac{1}{k+1} \sum \limits _{i=0}^{k}y^{-i}(r_n')\\&~~~~~~~~ =\frac{1}{k(k+1)}\big (\sum \limits _{i=0}^{k-1}y^{-i}(r_n') -k y^{-k}(r_n')\big )<0 \end{aligned}$$

it proves that $\zeta _k<\zeta _{k+1}$ for $\forall k=1,2,...,L\!-\!1$. In other words, $\zeta _1<\zeta _2<...<\zeta _L$.