Min–Max Energy-Efficiency Analysis of Green Multiuser Wireless Systems

Abstract

In this paper, we propose an optimal power allocation scheme that minimizes the energy per bit of worst user (i.e., the user with highest energy per bit) in a multiuser wireless communication network. The problem of determining energy efficient power allocation to improve the worst user is a constrained non-convex nonlinear fractional programming problem. We propose an iterative energy efficient power allocation algorithm that guarantees optimal solution. We use parametric equivalent formulation to get the optimal solution. Numerical solutions obtained using simulations are presented and compared with equal power allocation scheme.

Appendix: Proof of Theorem 1

Proof

Before the proof of theorem 1, we first present some observations and remarks. $\square$

Remark 1

• (a). It is easy to observe that both $\mathcal {S}$ and $\mathcal {K}$ are closed and bounded, which means $\mathcal {S}$ and $\mathcal {K}$ are compact.

• (b). Since $p_k \ge 0, \forall k\in \mathcal {K}$, $\theta$ will always be non-negative.

• (c). Due to the positivity of $f_d\left( p_k\right)$, the function $F(\theta )$ is decreasing with $\theta$.

Proposition 1

If both $\mathcal {S}$ and $\mathcal {K}$ are compact, then $F(\theta ) < 0$ if and only if $\theta > \theta ^*$.

Proof

If $F(\theta ) < 0$ then there exists $\varvec{\hat{p}} \in \mathcal {S}$ such that $f_n\left( \hat{p}_k\right) -\theta f_d\left( \hat{p}_k\right) <0, \forall k\in \mathcal {K}$. Hence, $\theta > \underset{k\in \mathcal {K}}{\text {max}} \; \frac{f_n\left( \hat{p}_k\right) }{f_d\left( \hat{p}_k\right) } \ge \theta ^*$. Conversely, if $\theta > \theta ^*$, then $\theta ^* = \underset{k\in \mathcal {K}}{\text {max}} \; \frac{f_n\left( p^*_k\right) }{f_d\left( p^*_k\right) } < \theta$.This implies that $f_n\left( p^*_k\right) - \theta f_d\left( p^*_k\right) < 0, \forall k\in \mathcal {K}$, indicating that $F(\theta ) < 0$. $\square$

Remark 2

For all $t\ge 1$ and $\varvec{p}\in \mathcal {S}$, $\theta ^t=\underset{k\in \mathcal {K}}{\text {max}} \, \frac{f_n(p_k^t)}{f_d(p_k^t)} \ge \theta *$. Hence, from Lemma 1, $F(\theta )\le 0$.

Lemma 2

The sequence $\{\theta ^t\}$ is monotonically decreasing.

Proof

Let $\xi$ be the index used to specify $\theta ^{t+1}$, that is $\theta ^{t+1}=\underset{k\in \mathcal {K}}{\text {max}} \, \frac{f_n(p_k^t)}{f_d(p_k^t)}=\frac{f_n(p_{\xi }^t)}{f_n(p_{\xi }^t)}$. Then

$$\begin{aligned} F(\theta ^t)&= \underset{k\in \mathcal {K}}{\text {max}} f_n\left( p_k^t\right) -\theta ^t f_d\left( p_k^t\right) \nonumber \\&\ge f_n\left( p_{\xi }^t\right) -\theta ^t f_d\left( p_{\xi }^t\right) \\&= \theta ^{t+1}f_d\left( p_{\xi }^t\right) -\theta ^t f_d\left( p_{\xi }^t\right) \nonumber \\&= (\theta ^{t+1}-\theta ^t) f_d\left( p_{\xi }^t\right) \nonumber \end{aligned}$$

(6)

Hence, due to the positivity of $f_d\left( p_{\xi }^t\right)$, $(\theta ^{t+1}-\theta ^t) \le \frac{F(\theta ^t)}{f_d\left( p_{\xi }^t\right) } < 0$. $\square$

Let us define $f_d^{min}= \underset{k\in \mathcal {K}}{\text {min}}\, f_d(p_k^t)$ and $f_d^{max}=\underset{k\in \mathcal {K}}{\text {max}} \, f_d(p_k^t)$.

Lemma 3

If $\theta ^t \in \mathcal {R}$ and $\varvec{p}$ is the optimal solution of (4) then

$$\begin{aligned}&F(\theta ) \le F(\theta ^t)- f_d^{min}(\theta -\theta ^t) \quad {\textit{if}} \ (\theta > \theta ^t)\nonumber \\&F(\theta ) \le F(\theta ^t)- f_d^{max}(\theta -\theta ^t) \quad {\textit{if}} \ (\theta < \theta ^t) \end{aligned}$$

(7)

Proof

For any $\theta ^t \in \mathcal {R}$

$$\begin{aligned} F(\theta ^t)&= \underset{\varvec{p}\in \mathcal {S},\theta }{\text {min}} \, \underset{k\in \mathcal {K}}{\text {max}} \, f_n\left( p_k^t\right) -\theta f_d\left( p_k^t\right) \nonumber \\&\le \underset{k\in \mathcal {K}}{\text {max}} \, f_n\left( p_k^t\right) -\theta f_d\left( p_k^t\right) \end{aligned}$$

(8)

Let the maximum of (8) be attained at index $\xi \in \mathcal {K}$, then

$$\begin{aligned} F(\theta )&\le f_n\left( p_{\xi }^t\right) -\theta f_d\left( p_{\xi }^t\right) \nonumber \\&= f_n\left( p_{\xi }^t\right) -\theta f_d\left( p_{\xi }^t\right) +\theta ^t f_d\left( p_{\xi }^t\right) -\theta ^t f_d\left( p_{\xi }^t\right) \\&= F(\theta ^t)-(\theta -\theta ^t)f_d\left( p_{\xi }^t\right) \nonumber \end{aligned}$$

(9)

It is easy to observe (7) from the definition of $f_d^{min}$ and $f_d^{max}$. $\square$

Remark 3

If $|\mathcal {S}|=1$, we have $f_d^{min}=f_d^{max}=\hat{f}$. Then (7) implies that $-\hat{f}$ is a sub-gradient of $F$ at $\theta$.

Let, $\varvec{p}^*$ be the optimal solution of (1). We can rewrite min–max problem as:

$$\begin{aligned} \theta ^* = \underset{\varvec{p}\in \mathcal {S}}{\text {min}} \, \underset{k\in \mathcal {K}}{\text {max}} \qquad \frac{f_n\left( p_k\right) /f_d\left( p^*_k\right) }{f_d\left( p_k\right) /f_d\left( p^*_k\right) } \end{aligned}$$

(10)

the associated parametric problem is,

$$\begin{aligned} \bar{F}(\theta ^*) = \underset{\varvec{p}\in \mathcal {S}}{\text {min}} \, \underset{k\in \mathcal {K}}{\text {max}} \qquad \frac{f_n\left( p_k\right) -\theta f_d\left( p_k\right) }{f_d\left( p^*_k\right) } \end{aligned}$$

(11)

From Lemma 3, we have

$$\begin{aligned}&\bar{F}(\theta ^*) \le \bar{F}(\theta ^t)- \Gamma _d^{min}(\theta -\theta ^t) \quad {\textit{if}} \ (\theta > \theta ^t)\nonumber \\&\bar{F}(\theta ^*) \le \bar{F}(\theta ^t)- \Gamma _d^{max}(\theta -\theta ^t) \quad {\textit{if}} \ (\theta < \theta ^t) \end{aligned}$$

(12)

where $\Gamma _d^{min}= \underset{k\in \mathcal {K}}{\text {min}}\, f_d(p^*_k)/f_d(p^*_k)=1=\Gamma _d^{max}=\underset{k\in \mathcal {K}}{\text {max}} \, f_d(p^*_k)/f_d(p^*_k)$. From Lemma 3 and Remark 3, it is easy to observe that the Algorithm 1 reduces to a Newton method in the neighborhood of $\theta ^*$ and rate of convergence is at least linear as noticed in [16].