## 3 Energy-Efficient Optimal Power Allocation
### Energy-Efficient Optimal Power Allocation for Non-Real-Time Applications
#### 3.1.1 Average Transmit Power Constraint
: _The energy-efficient optimal transmit power for non-real-time applications with ATP constraint is given as_  
\[P_{t}^{*}=\left[\frac{1}{\left(\eta_{\xi}^{2}+\tau+\mu G_{t}h_{I}\right)\ln 2 }-\frac{N_{s}}{G_{S}h_{S}}\right]^{+}, \tag{14}\]  
_where \(\left[x\right]^{+}=\max\left(0,x\right)\), which means the maximum between \(x\) and \(0\)._  
We can see that (9) can be efficiently solved via (14) for a given \(\eta\). To solve (8) and find the maximum EE \(\eta^{*}\), we resort to the [PERSON]'s method [26]. Then, we propose the iterative power allocation algorithm to solve (8), which is denoted by Algorithm 1. It has been proved that [PERSON]'s method can converge to the optimal solution with a superlinear convergence rate [39, 40]. The proof of the convergence is shown as below. Before the proof of convergence, two Lemmas are given as follows. ```
Set parameters: \(\xi_{0}>0\), \(\xi_{1}>0\), \(\xi_{2}>0\) : Error tolerances; \(t_{1}>0\), \(t_{2}>0\) : Step sizes; \(N_{i}\) : Iteration number. Initialization: \(\eta=\eta_{0}\), \(\tau=\tau_{0}\), \(\mu=\mu_{0}\), \(\delta=\delta_{0}\);  Calculate \(P_{t}^{0}\) using (14); \(\delta_{1}=\left|\tau^{k}\left(P_{av}-\mathrm{E}\left(P_{t}^{0}\right)\right) \right|\), \(\delta_{2}=\left|\mu^{k}\left(I_{th}-\mathrm{E}\left(P_{t}^{0}\right)\right) \right|\). Search optimal values: \(n=0\), \(\eta_{0}=0\); While\(\delta>\zeta_{0}\) \(k=0\);  Update \(\tau\) and \(\mu\) by subgradient method as follows: While\(\delta_{1}>\xi_{1}\) or \(\delta_{2}>\xi_{2}\) \(\tau^{k+1}=\left[\tau^{k}-t_{1}\left(P_{av}-\mathrm{E}\left(P_{t}^{k}\right) \right)\right]^{+}\); \(\mu^{k+1}=\left[\mu^{k}-t_{2}\left(I_{th}-\mathrm{E}\left(P_{t}^{k}\right) \right)\right]^{+}\); \(k=k+1\);  Calculate \(P_{t}^{k}\) using (14); \(\delta_{1}=\left|\tau^{k}\left(P_{av}-\mathrm{E}\left(P_{t}^{k}\right)\right) \right|\); \(\delta_{2}=\left|\mu^{k}\left(I_{th}-\mathrm{E}\left(P_{t}^{k}\right)\right) \right|\); End; \(\eta_{n+1}=\frac{\mathrm{E}\left[\log_{2}\left(1+\frac{P_{t}^{k}}{N_{n}} \right)\right]}{\mathrm{E}\left(\zeta P_{t}^{k}+P_{c}\right)}\); \(\delta=\eta_{n+1}-\eta_{n}\); End; \(P_{t}^{*}=P_{t}^{k}\); \(\eta^{*}=\eta_{n+1}\). ```  
**Algorithm 1**Iterative Power Allocation Algorithm for (8). Lemma 1: \(T(\eta)\) _defined in (9) is strictly monotonic decreasing, i.e., \(T(\eta^{\diamond})<T(\eta^{\heartsuit})\) if \(\eta^{\diamond}>\eta^{\heartsuit}\)._  
Proof: Let \(P_{t}^{\diamond}\) maximize \(T(\eta^{\diamond})\), then  
\[T(\eta^{\diamond}) =\max_{P_{t}}\{E[\log_{2}(1+\frac{P_{r}}{N_{s}})]-\eta^{\diamond} E(\xi P_{t}+P_{c})\} \tag{15}\] \[=E[\log_{2}(1+\frac{P_{r}^{\heartsuit}}{N_{s}})]-\eta^{\diamond}E( \xi P_{t}^{\heartsuit}+P_{c})\] (16) \[<E[\log_{2}(1+\frac{P_{r}^{\heartsuit}}{N_{s}})]-\eta^{\heartsuit} E(\xi P_{t}^{\heartsuit}+P_{c})\] (17) \[\leq\max_{P_{t}}\{E[\log_{2}(1+\frac{P_{r}}{N_{s}})]-\eta^{ \heartsuit}E(\xi P_{t}+P_{c})\}=T(\eta^{\heartsuit}), \tag{18}\]  
where the first inequality is based on \(E(\xi P_{t}+P_{c})>0\). Lemma 2: _Given \(P_{t}^{\clubsuit}\) satisfying (a1) and (b) in (8) and \(\eta^{\clubsuit}=\frac{E[\log_{2}(1+\frac{P_{r}^{\clubsuit}}{N_{s}})]}{E(\xi P _{t}^{\clubsuit}+P_{c})}\), we have \(T(\eta^{\clubsuit})\geq 0\)._  
**Proof. **  
\[L\left(P_{t},\tau,\mu\right)=\mathrm{E}\left[\log_{2}\left(1+\frac{P_{r}}{N_{ s}}\right)\right]-\eta\mathrm{E}\left(\xi P_{t}+P_{c}\right)-\tau\left[ \mathrm{E}\left(P_{t}\right)-P_{uv}\right]-\mu\left[\mathrm{E}\left(P_{i} \right)-I_{th}\right], \tag{10}\]0  
\(\square\)  
**Theorem 3**: _The iterative variable \(\eta_{n+1}=\frac{E[\log_{2}(1+\frac{P_{r}^{k}}{N_{\rm s}})]}{E(\xi P_{t}^{k}+P_ {c})}\) produces an increasing sequence of \(\eta\) values, which converges to the optimal value \(\eta^{*}\)._  
**Proof. ** First, we prove \(\eta_{n+1}>\eta_{n}\) for all \(n\) with \(T(\eta_{n})>0\). Lemma 2 makes \(T(\eta_{n})\geq 0\).