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

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.

Authors and Affiliations

1. Department of Electrical and Computer Engineering, Ryerson University, Toronto, ON, Canada

   Muhammad Naeem, Alagan Anpalagan & Muhammad Jaseemuddin

2. COMSATS Institute of IT, Wah Campus, Islamabad, Pakistan

   Muhammad Naeem

Corresponding author

Correspondence to Alagan Anpalagan.

This work was supported in part by NSERC Discovery Grants.

Appendix: Proof of Theorem 1

Proof

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

Remark 1

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.

2. (b).

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

3. (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

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

Proof

For any \(\theta ^t \in \mathcal {R}\)

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

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:

the associated parametric problem is,

From Lemma 3, we have

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].

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