Energy-Aware Cooperative MAC with Uncoordinated Group Relays

Abstract

In this chapter, we extend the single S-D pair cooperation scenario considered in Chap. 3 to a new framework where multiple S-D pairs share a group of relays with energy constraint. To support the fast-growing multimedia services in a green manner, we introduce an energy-aware distributed cooperation scheme based on the backoff timer. The theoretical performance bounds of the proposed strategy are derived with respect to the collision probability and the transmission success probability. The numerical and simulation results show that the proposed strategy outperforms a probability-based uncoordinated strategy in terms of average packet delay, delay outage probability, and energy consumption. Further, we investigate the scalability of our proposed strategy and find it can be deployed in a large-scale network.

Appendices

Appendix A: Proof of (4.17)

According to (4.14), when θ > c, we have

$$\displaystyle{ I_{c}\ =\ N(N - 1)(I_{c_{1}} + I_{c_{2}} + I_{c_{3}}) }$$

(4.27)

where $I_{c_{1}}$, $I_{c_{2}}$ and $I_{c_{3}}$ are given by

$$\displaystyle\begin{array}{rcl} I_{c_{1}}& =& \int _{c}^{\theta } \frac{t} {\theta (1-\theta )}\bigg[1 - \frac{t^{2}/2} {\theta (1-\theta )}\bigg]^{N-2}\ \frac{(t - c)^{2}/2} {\theta (1-\theta )} \mathrm{d}t{}\end{array}$$

(4.28)

$$\displaystyle\begin{array}{rcl} I_{c_{2}} =\int _{ \theta }^{1-\theta } \frac{1} {1-\theta }\bigg[1 -\frac{t -\theta /2} {1-\theta } \bigg]^{N-2}\ \frac{t - c -\theta /2} {1-\theta } \mathrm{d}t& &{}\end{array}$$

(4.29)

$$\displaystyle\begin{array}{rcl} I_{c_{3}} =\int _{ 1-\theta }^{1}\frac{1 - t} {\theta (1-\theta )}\bigg[\frac{(1 - t)^{2}} {2\theta (1-\theta )} \bigg]^{N-2}\bigg[1 -\frac{(1 - t + c)^{2}} {2\theta (1-\theta )} \bigg]\mathrm{d}t.& &{}\end{array}$$

(4.30)

As a closed-form expression is not tractable for $I_{c_{1}}$, we take t ≤ θ and have

$$\displaystyle{ \begin{array}{rl} I_{c_{1}} & \ \geq \ \int _{c}^{\theta } \frac{t} {\theta (1-\theta )}\ \bigg[1 - \frac{\theta ^{2}/2} {\theta (1-\theta )}\bigg]^{N-2}\ \frac{(t-c)^{2}/2} {\theta (1-\theta )} \mathrm{d}t \\ &\ =\ \Big( \frac{1} {1-\theta }\Big)^{N}\bigg(1 -\frac{3} {2}\theta \Big)^{N-2}\Big(\theta -c\Big)^{3}\Big(\frac{1} {8\theta } + \frac{c} {24\theta ^{2}} \Big). \end{array} }$$

(4.31)

The closed-form expressions of $I_{c_{2}}$ and $I_{c_{3}}$ can be obtained as

$$\displaystyle\begin{array}{rcl} I_{c_{2}}& =& \Big( \frac{1} {1-\theta }\Big)^{N}\Bigg\{\frac{1 -\theta -c} {N - 1} \bigg[\Big(1 -\frac{3} {2}\theta \Big)^{N-1} -\Big ( \frac{\theta } {2}\Big)^{N-1}\bigg] \\ & & \quad - \frac{1} {N}\bigg[\Big(1 -\frac{3} {2}\theta \Big)^{N} -\Big ( \frac{\theta } {2}\Big)^{N}\bigg]\Bigg\} {}\end{array}$$

(4.32)

$$\displaystyle\begin{array}{rcl} I_{c_{3}}& =& \Big( \frac{\theta /2} {1-\theta }\Big)^{N}\Big(\frac{2} {\theta } \Big)\bigg[\frac{2(1-\theta ) - c^{2}/\theta } {2N - 2} - \frac{2c} {2N - 1} - \frac{\theta } {2N}\bigg].{}\end{array}$$

(4.33)

The last three equations conclude the proof to (4.17).

Appendix B: Proof of (4.25)

According to (4.2), we obtain the transmission success probability over the relay-to-destination channel with α = 2 as

$$\displaystyle{ P_{RD}\ =\ e^{-\phi d^{2} } }$$

(4.34)

where ϕ = T ₀ N ₀∕P ₀. Given the PDF of d in (4.10), we derive the CDF of P _RD by

$$\displaystyle{ \begin{array}{rl} \Pr \{P_{RD} \leq p\}& =\Pr \{ e^{-\phi d^{2} } \leq p\} = 1 -\Pr \bigg\{ d \leq \sqrt{-\frac{\ln p} {\phi }} \ \bigg\} \\ & = 1 -\int _{0}^{\sqrt{-\frac{1} {\phi } \ln p}}f(x)\mathrm{d}x = 1 - \frac{d^{2}} {L^{2}} \Bigg\vert _{0}^{\sqrt{-\frac{1} {\phi } \ln p}} \\ & = 1 + \frac{P_{0}} {N_{0}T_{0}L^{2}} \ln p. \end{array} }$$

Thus, it is easy to show that the probability that a relay has the maximum transmission success probability over the relay-to-destination channel among N candidates is given by (4.25).