Interference-aware relay assignment scheme for multi-hop wireless networks

Abstract

This paper proposes an interference-limited relay assignment scheme for multi-hop wireless networks that exploits cooperative diversity to cope with problems of wireless channels and to enhance data transmission reliability. By combining the selection and incremental relaying schemes and by taking into account the channel status information and queue length at each node, the cooperative scheme improves the packet dropped ratio and end-to-end delay due to retransmissions. The proposed method is based on local channel measurement and requires no topology information. In addition, this paper also investigates the interference problem produced by the relay nodes and the failure probability of the best relay selection. Extensive simulations conducted to evaluate the performance of the proposed scheme indicate that it effectively enhances the network performance in terms of the packet delivery ratio, energy consumption, and overall packet delay.

Appendices

Appendix 1: Products of a continuous and a discrete random variables

Let X be a continuous random variable with known PDF $f_{X}(x)$. Let also Y be a discrete random variable with PMF P(Y = y) which is defined on the interval $[0,L]\,(L>0)$. The CDF of U = XY is given by

$$F_{U}(u)=P(XY \le u)=\sum \limits _{y=0}^{L}P(Y=y)P(yX \le u|Y=y),$$

(18)

where $P(yX \le u|Y=y)=P(X \le \frac{u}{y}|Y=y)=F_{X}(\frac{u}{y})$ with $y>0$. On the other hand, $P(0X \le u|Y=0)=1$ for $u \ge 0$ or $P(0X \le u|Y=0)=0$ for $u<0$. With $u \ge 0$, Eq. (18) reduces to

$$F_{U}(u)=P(Y=0)+\sum \limits _{y=1}^{L}P(Y=y)F_{X}\left( \frac{u}{y}\right).$$

(19)

However, a continuous and a discrete random variables don’t have a joint PDF because their joint distribution is not absolutely continuous in 2-dimensional plane. We make the following estimation

$$f_{U}(u) \approx \frac{d}{du}F_{U}(u) = \sum \limits _{y=1}^{L}\frac{1}{y}P(Y=y)f_{X}\left( \frac{u}{y}\right).$$

(20)

Appendix 2: Approximation for the M/G/1 queue length distribution

Consider an M/G/1 queue with average service time $\overline{x}$, Poisson arrival rate $\lambda$, and a squared coefficient of variation of service time $c_{x}^{2}$. The probability that there are at least k + 1 packets in the queue is given by [17]

$$P[N>k]\approx U_{0} \alpha ^{k}, \quad k=0,1,2,\ldots,L-1.$$

(21)

where $U_{0}$ is the server utilization $(P[N>0]=U_{0},\,P[N=0]=1-U_{0})$ and $\alpha =\frac{U_{0}(c_{x}^{2}+1)}{2+U_{0}(c_{x}^{2}-1)}\,(\alpha \le 1)$. Then, the CDF of queue distribution can be written as follows

$$P[N\le k]=1-P[N>k] \approx 1 - U_{0} \alpha ^{k}.$$

(22)

for $k=0,1,2,\ldots,L-1$, and $P[N\le L]=1$. The PMF of queue distribution is given by

$$\begin{aligned} P[N=k]&= P[N>k-1] - P[N>k] \\&\approx U_{0} \alpha ^{k-1}(1-\alpha ), \quad k=1,2,3,\ldots,L. \end{aligned}$$

(23)

From queue length distribution, the PMF and CDF of the remaining queue length distribution can be given by

$$\begin{aligned} P[\overline{N}=k]&= P[N=L-k] \\&\approx U_{0} \alpha ^{L-k-1}(1-\alpha ), \end{aligned}$$

(24)

for $k=0,1,2,\ldots,L-1$ and $P[\overline{N}=L]=1-U_{0}$.

$$\begin{aligned} P[\overline{N}\le k]&= P[N\ge L-k] \\&= P[N>L-k] + P[N=L-k] \\&\approx U_{0} \alpha ^{L-k} + U_{0} \alpha ^{L-k-1}(1-\alpha ), \end{aligned}$$

(25)

for $k=1,2,\ldots,L-1$ and $P[\overline{N}\le 0]=P[N=L]=U_{0} \alpha ^{L-1}(1-\alpha ), P[\overline{N}\le L]=1$.

The CDF calculated by Eqs. (24) and (25) conduct different results because of the estimation in Eq. (21). For instance, k = 2, then the probabilities $P[\overline{N}\le 2]$ calculated by these equations are respectively

$$\begin{aligned} P[\overline{N}\le 2]&= P[\overline{N}=0] + P[\overline{N}=1] + P[\overline{N}=2] \\&\approx U_{0} (1-\alpha )(\alpha ^{L-1} + \alpha ^{L-2}) + U_{0} \alpha ^{L-3}(1-\alpha ), \\ P[\overline{N}\le 2]&= P[N>L-2] + P[N=L-2] \\&\approx U_{0} \alpha ^{L-2} + U_{0} \alpha ^{L-3}(1-\alpha ). \end{aligned}$$

The difference is that $(1-\alpha )(\alpha ^{L-1} + \alpha ^{L-2}) = \alpha ^{L-2} - \alpha ^{L} \ne \alpha ^{L-2}$. However, when the server utilization $U_{0}<1$, then $\alpha < 1$. For a large value of queue capacity $L, \alpha ^{L}$ converges to 0. Therefore, the difference is negligible.

Figure 10 plots the approximation for queue distribution with $U_{0}=0.85, c_{x}^{2}=5$, and maximum queue length of 100 incoming packets.

Fig. 10

Approximation CDF for current occupied queue length distribution (line A) and remaining queue length distribution (line B)