Energy-Efficient Uncoordinated Cooperative MAC with Uncertain Relay Distribution Intensity

Abstract

As a promising technique, cooperative communications make use of the broadcasting nature of wireless medium to facilitate data transmission, and thereby reduce energy consumption. However, in many studies on wireless cooperative diversity, it is often assumed that the number of relays or the relay distribution intensity is known a priori. In this chapter, we relax such assumption and propose an algorithm to estimate the relay intensity for a backoff-based cooperative scheme, where the relays are distributed as a homogeneous Poisson point process (PPP). It is proved that the algorithm can converge to an optimal solution with the minimum estimation error. Based on the estimated relay intensity, we further investigate a distributed energy saving strategy, which selectively turns off some relays to reduce energy consumption while maintaining the required transmission success probability. The performance of the proposed cooperative scheme is analytically evaluated with respect to the collision probability. The numerical and simulation results demonstrate the high accuracy and efficiency of the intensity estimation algorithm and also validate the theoretical analysis. Moreover, the proposed cooperative scheme exhibits significant energy saving and satisfactory transmission performance, which offers a good match to accommodate green communications in wireless networks.

Appendices

Appendix A: Extended Proof of Lemma 3.2 with N Relays

Let p ₁, p ₂, …, p _N be the probabilities that N relays correctly receive a packet from S, and ζ ₁, ζ ₂, …, ζ _N be the active probabilities of the N relays, respectively. Any unknown ζ _n (2 ≤ n ≤ N) can be determined according to known ζ ₁, …, ζ _n−1. Thus, ζ ₁, ζ ₂, …, ζ _N can be obtained sequentially so as to minimize the overall energy consumption of these N relays.

The average energy consumption of a packet transmission is given by

$$\displaystyle{E = \mathcal{P}_{1} \cdot E_{t} + \mathcal{P}_{2} \cdot E_{r} + \mathcal{P}_{3} \cdot E}$$

where $\mathcal{P}_{1}$ is the probability that at least one active relay correctly overhears the packet from the source, $\mathcal{P}_{2}$ is the average number of relays that are active during the packet transmission, and $\mathcal{P}_{3}$ is the probability that none of the active relays successfully overhears the packet from the source. For N relays, we have

$$\displaystyle{\mathcal{P}_{1} = 1 -\prod _{i=1}^{N}(1 - p_{ i}\zeta _{i}),\quad \mathcal{P}_{2} =\sum _{ i=1}^{N}\zeta _{ i},\quad \mathcal{P}_{3} = 1 -\mathcal{P}_{1}.}$$

Thus, we have

$$\displaystyle{ E = E_{t} + \frac{\zeta _{1} +\zeta _{2} + \cdots +\zeta _{N}} {1 - (1 - p_{1}\zeta _{1})(1 - p_{2}\zeta _{2})\cdots (1 - p_{N}\zeta _{N})} \cdot E_{r}. }$$

Similar to (3.21), (3.22), and (3.23), it can be easily inferred that, given known ζ ₁, …, ζ _N−1, the active probability ζ _N should be either 0 or 1, so as to minimize the average energy consumption E. The setting of 0 or 1 to ζ _N depends on the successful receiving probability p _N and the status of the other relays.

Appendix B: Proof of (3.34) and (3.35)

The conditional collision probability P _c is given by

$$\displaystyle{ \begin{array}{rl} P_{c}& =\mathrm{ P}[r_{2} \leq r_{1} + w] = 1 -\mathrm{ P}[r_{2} > r_{1} + w] = 1 -\int _{0}^{L_{0}-w}\int _{r_{ 1}+w}^{L_{0}}g(r_{ 1},r_{2})\mathrm{d}r_{2}\mathrm{d}r_{1} \\ & = 1 - \frac{1} {P_{2^{+}}} \int _{0}^{L_{0}-w}\Bigg\{\Big[2\lambda r_{1}e^{-K(L^{2}+r_{ 1}^{2}) }\int _{0}^{\arccos (\frac{r_{1}} {L} )}e^{2KLr_{1}\cos \theta }\mathrm{d}\theta \Big] \\ &\ \ \qquad \qquad \int _{r_{1}+w}^{L_{0}}\Big[e^{-\varLambda _{r_{2}}}\Big(2\lambda r_{2}e^{-K(L^{2}+r_{ 2}^{2}) }\int _{0}^{\arccos (\frac{r_{2}} {L} )}e^{2KLr_{2}\cos \theta }\mathrm{d}\theta \Big)\Big]\mathrm{d}r_{2}\Bigg\}\mathrm{d}r_{1} \\ & = 1 - \frac{1} {P_{2^{+}}} \int _{0}^{L_{0}-w}\Bigg\{\Big[2\lambda r_{1}e^{-K(L^{2}+r_{ 1}^{2}) }\int _{0}^{\arccos (\frac{r_{1}} {L} )}e^{2KLr_{1}\cos \theta }\mathrm{d}\theta \Big] \\ &\ \ \ \ \qquad \qquad \qquad \qquad \cdot \Big (e^{-\varLambda _{r_{1}}} \cdot e^{-\varLambda _{\varDelta w}} - e^{-\varLambda _{L_{0}}}\Big)\Bigg\}\mathrm{d}r_{1} \end{array} }$$

$$\displaystyle\begin{array}{rcl} \phantom{P_{c}}& =& 1 - \frac{1} {P_{2^{+}}} \int _{0}^{\varLambda _{L_{0}-w} }\Big(e^{-\varLambda _{r_{1}} } \cdot e^{-\varLambda _{\varDelta w} } - e^{-\varLambda _{L_{0}} }\Big)\mathrm{d}\varLambda _{r_{1}} \\ & =& 1 + \frac{\varLambda _{L_{0}-w} \cdot e^{-\varLambda _{L_{0}}}} {P_{2^{+}}} - \frac{1} {P_{2^{+}}} \int _{0}^{\varLambda _{L_{0}-w} }e^{-\varLambda _{r_{1}} } \cdot e^{-\varLambda _{\varDelta w} }\mathrm{d}\varLambda _{r_{1}}.{}\end{array}$$

(3.37)

Therefore, (3.34) is proved.

Since Λ _{Δ w}  ≤ Ψ, according to (3.6), we have

$$\displaystyle{ \begin{array}{rl} P_{c}& \leq 1 + \frac{\varLambda _{L_{0}-w}\cdot e^{-\varLambda _{L_{0}} }} {P_{2^{+}}} - \frac{1} {P_{2^{+}}} \int _{0}^{\varLambda _{L_{0}-w}}e^{-\varLambda _{r_{1}}} \cdot e^{-\varPsi }\mathrm{d}\varLambda _{r_{ 1}} \\ & = 1 + \frac{\varLambda _{L_{0}-w}\cdot e^{-\varLambda _{L_{0}} }} {P_{2^{+}}} - \frac{e^{-\varPsi }} {P_{2^{+}}} \int _{0}^{\varLambda _{L_{0}-w}}e^{-\varLambda _{r_{1}}}\mathrm{d}\varLambda _{r_{ 1}} \\ & = 1 + \frac{\varLambda _{L_{0}-w}\cdot e^{-\varLambda _{L_{0}} }} {P_{2^{+}}} - \frac{e^{-\varPsi }} {P_{2^{+}}} \Big[\big(-e^{-\varLambda _{r_{1}}}\big)\big\vert _{0}^{\varLambda _{L_{0}-w}}\Big] \\ & = 1 + \frac{\varLambda _{L_{0}-w}\cdot e^{-\varLambda _{L_{0}} }} {P_{2^{+}}} - \frac{e^{-\varPsi }} {P_{2^{+}}} \big(1 - e^{-\varLambda _{L_{0}-w}}\big) \end{array} }$$

(3.38)

which gives the result in (3.35).