# User-Perceived Capacity: Theory, Computation, and Achievable Policies
## 4 The Cross-Layer Approach
### The Power Metric and the Continuous Reading Constraint  
We next express the long-term average power consumption in terms of the steady-state distribution \(\pi_{\mathbf{F}}\) and the scheduling policy \(\mathbf{F}\) as follows. Theorem 1: _The long-term average power consumption \(P_{ave}=\lim\limits_{N\to+\infty}\frac{1}{N}\sum\limits_{t=1}^{N}P[t]\) is calculated as_  
\[P_{ave}=\sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\sum_{h_{l}\in\mathcal{H}} \sum_{s\in\mathcal{S}}P(s,l)\eta_{l}\pi_{\mathbf{F}}(q,m)f^{s}_{q,m,l}. \tag{21}\]  
Proof: We calculate the average power consumption from a probability perspective. The probability of the queue state \(q[t]=q\), time index \(m[t]=m\), channel state \(h[t]=h_{l}\), and transmission rate \(s[t]=s\) is given by \(\eta_{l}\pi_{\mathbf{F}}(q,m)f^{s}_{q,m,l}\), with a power consumption of \(P(s,l)\). Subsequently, proving Theorem 1 involves computing the mathematical expectation. The continuous reading constraint in Equation (10) can be transformed into  
\[\sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\pi_{\mathbf{F}}(q,m)\mathds{1}\{qT +mB>TB\}=0. \tag{22}\]  
Therefore, we formulate an optimization problem to minimize the average power while satisfying the continuous reading constraint,  
\[\min_{\{\pi_{\mathbf{F}}(q,m),f^{s}_{q,m,l}\}} \sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\sum_{h_{l}\in\mathcal{ H}}\sum_{s\in\mathcal{S}}P(s,l)\eta_{l}\pi_{\mathbf{F}}(q,m)f^{s}_{q,m,l} \tag{23a}\] \[\text{s.t.} \sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\pi_{\mathbf{F}}(q,m) \mathds{1}\{qT+mB>TB\}=0,\] (23b) \[\sum_{s\in\mathcal{S}}f^{s}_{q,m,l}=1,\forall q\in\mathcal{Q},m \in\mathcal{M},h_{l}\in\mathcal{H},\] (23c) \[\sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\pi_{\mathbf{F}}(q,m) =1,\] (23d) \[\sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\lambda_{(q,m),(q,\bar {m})}\pi_{\mathbf{F}}(q,m)=\pi_{\mathbf{F}}(\bar{q},\bar{m}),\forall\bar{q} \in\mathcal{Q},\bar{m}\in\mathcal{M},\] (23e) \[\pi_{\mathbf{F}}(q,m)\geq 0,f^{s}_{q,m,l}\geq 0. \tag{23f}\]  
Note that the derived problem (23) is non-convex, making it challenging to find the optimal solution. We next convert this problem into a linear programming problem by introducing a series of variables, which are defined as  
\[y^{s}_{q,m,l}=\pi_{\mathbf{F}}(q,m)\eta_{l}f^{s}_{q,m,l}. \tag{24}\]  
For the new variable, there is an intuitive understanding: the original variable \(f^{s}_{q,m,l}\) is the probability of taking action \(s[t]=s\) when the queue state \(q[t]=q\), time index \(m[t]=m\), and channel state \(h[t]=h_{l}\), whereas the new variable \(y^{s}_{q,m,l}\) represents the probability of being in the queue state \(q[t]=q\), time index \(m[t]=m\), channel state \(h[t]=h_{l}\), and taking action \(s[t]=s\). Subsequently, we propose Theorem 2. Theorem 2: _The optimization problem (23) is equivalent to the following linear programming problem,_  
\[\min_{\{\pi_{\mathbf{F}}(q,m),f^{s}_{q,m,l}\}}\sum_{q\in\mathcal{Q}}\sum_{m\in \mathcal{M}}\sum_{h_{l}\in\mathcal{H}}\sum_{s\in\mathcal{S}}P(s,l)y^{s}_{q,m,l} \tag{25a}\]  
The solutions \(\pi_{\text{F}}(q,m)\) and \(f^{\text{s}}_{q,m,l}\) of optimization problem (23) can generate the solutions \(y^{\text{s}}_{q,m,l}\) of optimization problem (25) according to Equation (24). Similarly, \(y^{\text{s}}_{q,m,l}\) in problem (25) generates \(\pi_{\text{F}}(q,m)\) by  
\[\sum_{q\in\mathbb{Q}}\sum_{m\in\mathcal{M}}\pi_{\mathbf{F}}(q,m)\mathds{1}\{qT +mB>TB\}=0. \tag{22}\]0  
Likewise, we have  
\[f^{\text{s}}_{q,m,l}= \begin{cases}\frac{y^{\text{s}}_{q,m,l}}{\pi_{\text{F}}(q, m)}&\text{if }\pi_{\text{F}}(q,m)>0,\\ \mathds{1}\{s=0\}&\text{otherwise}.\end{cases}  \tag{32}\]  
We have demonstrated that if \(\left(\pi_{\text{F}}(q,m),f^{\text{s}}_{q,m,l}\right)\) constitutes an optimal solution to problem (23), then the corresponding \(y^{\text{s}}_{q,m,l}\) is an optimal solution to problem (25), and vice versa. The attentive reader should have noticed that we are optimizing power consumption given \(B\) and \(T\).