Energy efficiency resource allocation based on spectrum-power tradeoff in distributed satellite cluster network

Abstract

Distributed satellite cluster network is a novel structure in spatial information network. This paper investigates the spectrum-power tradeoff algorithm for distributed satellite cluster (DSC), which enables the master satellite (MS) to offload data traffic to the slave satellite (SS). The SS provides power resource to the master satellite users (MSUs) with bandwidth as compensation. The purpose of the proposed algorithm is to maximize the energy efficiency (EE) of the SS while ensuring the QoS of the DSC. Considering data rate and bandwidth constraints of the MS, the SS jointly allocates bandwidth and power resources based on the served MSU set, meanwhile, data rate and power constraints of the SS should be also satisfied. We first prove that compensating bandwidth of a MSU can only be used by one slave satellite user. Then objective function is simplified to the equivalent subtractive form that can be solved by convex optimization method. Besides, we discuss the MSU selection conditions and employ them to the proposed spectrum-power tradeoff algorithm. The numerical results validate the performance improvement of the proposed algorithm and demonstrate the impact of power, bandwidth and data rate on the EE.

Appendices

Appendix A: Proof of Theorem 1

The most effective tradeoff strategy is that only one SSU with the biggest channel power gain shares bandwidth of MSU_k, and then we prove this with proof by contradiction as follows. It is supposed that $\tilde{A} = \{ \tilde{p}_{n} ,\tilde{p}_{k,n} ,\tilde{v}_{k} ,\tilde{b}_{k,n} ,\tilde{t}_{k} ,\tilde{z}_{k} \}$ is optical solution for (9), and the bandwidth of the given MSU_e can be shared by two SSUs, SSU_m and SSU_m’, m ≠ m’. g_e,m is the biggest channel power gain between SS and SSUs on T_e, and since the channel fading is continuous and random, the channel power gain of the two SSUs cannot be equal. Let $\hat{A} = \{ \hat{p}_{n} ,\hat{p}_{k,n} ,\hat{v}_{k} ,\hat{b}_{k,n} ,\hat{t}_{k} ,\hat{z}_{k} \}$ be an another different solution for (9), where $\tilde{p}_{n} = \hat{p}_{n}$, $\tilde{v}_{k} = \hat{v}_{k}$, $\tilde{t}_{k} = \hat{t}_{k}$, $\tilde{z}_{k} = \hat{z}_{k}$,

$$ \hat{p}_{k,n} = \left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\tilde{p}_{{{\text{e}},m}} + \tilde{p}_{{e,m^{\prime}}} ,} \hfill \\ {0,} \hfill \\ {\tilde{p}_{k,n} ,} \hfill \\ \end{array} } \hfill & {\begin{array}{*{20}l} {k = e,n = m} \hfill \\ {k = e,n \ne m} \hfill \\ {k \ne e} \hfill \\ \end{array} } \hfill \\ \end{array} } \right.\quad {\text{and}}\quad \hat{b}_{k,n} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}l} {\tilde{b}_{{{\text{e}},m}} + \tilde{b}_{{e,m^{\prime}}} ,} \hfill \\ {0,} \hfill \\ {\tilde{b}_{k,n} ,} \hfill \\ \end{array} } & {\begin{array}{*{20}l} {k = e,n = m} \hfill \\ {k = e,n \ne m} \hfill \\ {k \ne e} \hfill \\ \end{array} } \\ \end{array} } \right.. $$

The solution $\hat{A}$ is feasible due to meeting all constraints in (9). And then, the data rate of SS on MSU_e’s bandwidth is denoted as follows

$$ \begin{aligned} & \hat{b}_{e,m} \log_{2} \left( {1 + \frac{{\lambda G_{s} G_{u} g_{e,m} \hat{p}_{e,m} }}{{(4\pi d)^{2} N_{0} \hat{b}_{e,m} }}} \right) \\ & = (\tilde{b}_{{{\text{e}},m}} + \tilde{b}_{{e,m^{\prime}}} )\log_{2} \left( {1 + \frac{{\lambda G_{s} G_{u} g_{e,m} (\tilde{p}_{e,m} + \tilde{p}_{{e,m^{\prime}}} )}}{{(4\pi d)^{2} N_{0} (\tilde{b}_{{{\text{e}},m}} + \tilde{b}_{{e,m^{\prime}}} )}}} \right) \\ & \mathop \ge \limits^{(\alpha )} \tilde{b}_{{{\text{e}},m}} \log_{2} \left( {1 + \frac{{\lambda G_{s} G_{u} g_{e,m} \tilde{p}_{e,m} }}{{(4\pi d)^{2} N_{0} \tilde{b}_{{{\text{e}},m}} }}} \right) + \tilde{b}_{{e,m^{\prime}}} \log_{2} \left( {1 + \frac{{\lambda G_{s} G_{u} g_{e,m} \tilde{p}_{{e,m^{\prime}}} }}{{(4\pi d)^{2} N_{0} \tilde{b}_{{e,m^{\prime}}} }}} \right) \\ & \mathop > \limits^{(\beta )} \tilde{b}_{{{\text{e}},m}} \log_{2} \left( {1 + \frac{{\lambda G_{s} G_{u} g_{e,m} \tilde{p}_{e,m} }}{{(4\pi d)^{2} N_{0} \tilde{b}_{{{\text{e}},m}} }}} \right) + \tilde{b}_{{e,m^{\prime}}} \log_{2} \left( {1 + \frac{{\lambda G_{s} G_{u} g_{{e,m^{\prime}}} \tilde{p}_{{e,m^{\prime}}} }}{{(4\pi d)^{2} N_{0} \tilde{b}_{{e,m^{\prime}}} }}} \right) \\ \end{aligned} . $$

(28)

The inequality (α) holds because the function $x\log_{2} (1 + {y \mathord{\left/ {\vphantom {y x}} \right. \kern-0pt} x})$ is concave, and strict inequality (β) holds because of g_e,m > g_e,m’. It is concluded that the solution $\hat{A}$ obtains higher total data rate when the power consumption is the same. Hence, $\hat{A}$ achieves higher EE compared with $\tilde{A}$, which is in contradiction with the original hypothesis that $\tilde{A}$ is the optimal solution.

Then we prove that C2 and C3 satisfy the equality at the resource allocation solution. When z_k = 0, the constraint C2 in (9) transform to $t_{k} + b_{k,n} \le 0$. Considering t_k and b_k,n ≥ 0, so t_k = b_k,n = 0, and thus, C2 and C3 meet the equality. When z_k = 1, if strict inequality $t_{k} + b_{k,n} < z_{k} T_{k}$ and $t_{k} \log_{2} (1 + \frac{{\lambda G_{s} G_{u} v_{k} h_{k} }}{{(4\pi d)^{2} t_{k} N_{0} }}) > z_{k} R_{k}^{\text{MSU}}$ hold under the optimal solution, another different solution could be acquired via raising b_k,n and reducing v_k, which makes C2 and C3 tend to inequality while getting higher EE. This result contradicts the hypothesis that the optimal solution is obtained when strict inequalities of C2 and C3 hold.

Appendix B: Proof of Theorem 2

At first, introduce the following lemma [45]:

Lemma

If x₁, x₂, x₃, and x₄are arbitrary positive numbers, then:

$$ \hbox{min} \left\{ {\frac{{x_{1} }}{{x_{2} }},\frac{{x_{3} }}{{x_{4} }}} \right\} \le \frac{{x_{1} + x_{3} }}{{x_{2} + x_{4} }} \le \hbox{max} \left\{ {\frac{{x_{1} }}{{x_{2} }},\frac{{x_{3} }}{{x_{4} }}} \right\}. $$

(29)

The equality holds If and only if ${{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {x_{2} }}} \right. \kern-0pt} {x_{2} }} = {{x_{3} } \mathord{\left/ {\vphantom {{x_{3} } {x_{4} }}} \right. \kern-0pt} {x_{4} }}$.

Then (1) of Theorem 2 is proved as follows. $\bar{A} = \{ \bar{p}_{n} ,\bar{p}_{{k,n^{\prime}}} ,\bar{v}_{k} ,\bar{b}_{{k,n^{\prime}}} ,\bar{t}_{k} \}$ is defined as the optimum solution of problem (26), and $\tilde{A} = \{ \tilde{p}_{n} ,\tilde{p}_{{k,n^{\prime}}} ,\tilde{v}_{k} ,\tilde{b}_{{k,n^{\prime}}} ,\tilde{t}_{k} \}$ and $\hat{A} = \{ \hat{p}_{n} ,\hat{p}_{{k,n^{\prime}}} ,\hat{v}_{k} ,\hat{b}_{{k,n^{\prime}}} ,\hat{t}_{k} \}$ are defined as the optimum solutions of problem (10) with $k \in \varPhi$ and $k \in \varPhi \cup \{ j\}$. Besides, the corresponding maximum EEs are defined as $EE_{k}^{*}$, $EE_{\varPhi }^{*}$ and $EE_{{\varPhi \cup \{ j\} }}^{*}$, respectively. Therefore, we can obtain

$$ \begin{aligned} EE_{{\varPhi \cup \{ j\} }}^{*} & = \frac{{\sum\nolimits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\hat{p}_{n} )} + \sum\nolimits_{k \ne j} {r_{{k,n^{\prime } }} (\hat{p}_{{k,n^{\prime } }} ,\hat{b}_{{k,n^{\prime } }} )} + r_{j,e} (\hat{p}_{j,e} ,\hat{b}_{j,e} )}}{{\sum\nolimits_{n = 1}^{N} {\hat{p}_{n} } + \sum\nolimits_{k \ne j} {\hat{p}_{{k,n^{\prime } }} } + \sum\nolimits_{k \ne j} {\hat{v}_{k} } + P_{s} + \hat{p}_{j,e} + \hat{v}_{j} }} \\ & \mathop \ge \limits^{\alpha } \frac{{\sum\nolimits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\tilde{p}_{n} )} + \sum\nolimits_{k \ne j} {r_{{k,n^{\prime } }} (\tilde{p}_{{k,n^{\prime } }} ,\tilde{b}_{{k,n^{\prime } }} )} + r_{j,e} (\bar{p}_{j,e} ,\bar{b}_{j,e} )}}{{\sum\nolimits_{n = 1}^{N} {\tilde{p}_{n} } + \sum\nolimits_{k \ne j} {\tilde{p}_{{k,n^{\prime } }} } + \sum\nolimits_{k \ne j} {\tilde{v}_{k} } + P_{s} + \bar{p}_{j,e} + \bar{v}_{j} }} \\ & \mathop \ge \limits^{\beta } \hbox{min} \left\{ {\frac{{\sum\nolimits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\tilde{p}_{n} )} + \sum\nolimits_{k \ne j} {r_{{k,n^{\prime } }} (\tilde{p}_{{k,n^{\prime } }} ,\tilde{b}_{{k,n^{\prime } }} )} }}{{\sum\nolimits_{n = 1}^{N} {\tilde{p}_{n} } + \sum\nolimits_{k \ne j} {\tilde{p}_{{k,n^{\prime } }} } + \sum\nolimits_{k \ne j} {\tilde{v}_{k} } + P_{s} }},\frac{{r_{j,e} (\bar{p}_{j,e} ,\bar{b}_{j,e} )}}{{\bar{p}_{j,e} + \bar{v}_{j} }}} \right\} \\ & = \hbox{min} \left\{ {EE_{\varPhi }^{*} ,EE_{j}^{*} } \right\}, \\ \end{aligned} $$

(30)

where $e = \arg \hbox{max} g_{j.n}$. The inequality (α) holds because $\hat{A}$ is the optimum solution of problem (10) with $k \in \varPhi \cup \{ j\}$, and the inequality (β) holds because of the Lemma. Therefore, it is concluded that $EE_{j}^{*} > EE_{\varPhi }^{*}$ is the sufficient condition for $EE_{{\varPhi \cup \{ j\} }}^{*} > EE_{\varPhi }^{*}$. Then we prove that $EE_{j}^{*} > EE_{\varPhi }^{*}$ is the necessary condition for $EE_{{\varPhi \cup \{ j\} }}^{*} > EE_{\varPhi }^{*}$, we can obtain:

$$ \begin{aligned} EE_{{\varPhi \cup \{ j\} }}^{*} & = \frac{{\sum\nolimits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\hat{p}_{n} )} + \sum\nolimits_{k \ne j} {r_{{k,n^{\prime } }} (\hat{p}_{{k,n^{\prime } }} ,\hat{b}_{{k,n^{\prime } }} )} + r_{j,e} (\hat{p}_{j,e} ,\hat{b}_{j,e} )}}{{\sum\nolimits_{n = 1}^{N} {\hat{p}_{n} } + \sum\nolimits_{k \ne j} {\hat{p}_{{k,n^{\prime } }} } + \sum\nolimits_{k \ne j} {\hat{v}_{k} } + P_{s} + \hat{p}_{j,e} + \hat{v}_{j} }} \\ & \mathop \le \limits^{\chi } \hbox{max} \left\{ {\frac{{\sum\nolimits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\hat{p}_{n} )} + \sum\nolimits_{k \ne j} {r_{{k,n^{\prime } }} (\hat{p}_{{k,n^{\prime } }} ,\hat{b}_{{k,n^{\prime } }} )} }}{{\sum\nolimits_{n = 1}^{N} {\hat{p}_{n} } + \sum\nolimits_{k \ne j} {\hat{p}_{{k,n^{\prime } }} } + \sum\nolimits_{k \ne j} {\hat{v}_{k} } + P_{s} }},\frac{{r_{j,e} (\bar{p}_{j,e} ,\bar{b}_{j,e} )}}{{\bar{p}_{j,e} + \bar{v}_{j} }}} \right\} \\ & \mathop \le \limits^{\delta } \hbox{max} \left\{ {\frac{{\sum\nolimits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\tilde{p}_{n} )} + \sum\nolimits_{k \ne j} {r_{{k,n^{\prime } }} (\tilde{p}_{{k,n^{\prime } }} ,\tilde{b}_{{k,n^{\prime } }} )} }}{{\sum\nolimits_{n = 1}^{N} {\tilde{p}_{n} } + \sum\nolimits_{k \ne j} {\tilde{p}_{{k,n^{\prime } }} } + \sum\nolimits_{k \ne j} {\tilde{v}_{k} } + P_{s} }},\frac{{r_{j,e} (\bar{p}_{j,e} ,\bar{b}_{j,e} )}}{{\bar{p}_{j,e} + \bar{v}_{j} }}} \right\} \\ & = \hbox{max} \left\{ {EE_{\varPhi }^{*} ,EE_{j}^{*} } \right\}, \\ \end{aligned} $$

(31)

The inequality (χ) holds because of the Lemma, and inequality (δ) holds because $\tilde{A}$ and $\bar{A}$ are the optimum solutions of problem (10) with $k \in \varPhi$ and problem (26), respectively. Therefore, it is concluded that $EE_{j}^{*} \le EE_{\varPhi }^{*}$$\Rightarrow$$EE_{\varPhi }^{*} \le EE_{{\varPhi \cup \{ j\} }}^{*}$, which is the contrapositive proposition of the $EE_{\varPhi }^{*} > EE_{{\varPhi \cup \{ j\} }}^{*}$$\Rightarrow$$EE_{j}^{*} > EE_{\varPhi }^{*}$. Thus, we prove that $EE_{j}^{*} > EE_{\varPhi }^{*}$ is the necessary and sufficient condition for $EE_{{\varPhi \cup \{ j\} }}^{*} > EE_{\varPhi }^{*}$.

Next we prove (2) of Theorem 2. When constraint C4 is removed from problem (10), based on (1) of Theorem 2, the inequality (α) in (30) would not hold due to the different power constraint on different solutions. $\tilde{A}$ and $\bar{A}$ are limited by

$$ \sum\limits_{n = 1}^{N} {\tilde{p}_{n} } + \sum\limits_{k \ne j} {\tilde{p}_{{k,n^{\prime}}} } + \sum\limits_{k \ne j} {\tilde{v}_{k} } \le P_{\hbox{max} }^{\text{SS}} , $$

(32)

$$ \bar{p}_{j,e} ,\bar{v}_{j} \ge 0, $$

(33)

but $\hat{A}$ is limited by

$$ \sum\limits_{n = 1}^{N} {\hat{p}_{n} } + \sum\limits_{k \ne j} {\hat{p}_{{k,n^{\prime}}} } + \sum\limits_{k \ne j} {\hat{v}_{k} } + \hat{p}_{j,e} + \hat{v}_{j} \le P_{\hbox{max} }^{\text{SS}} . $$

(34)

Therefore, from (32) to (34), it is concluded that the feasible domain of $\tilde{A}$ and $\bar{A}$ is greater than the feasible domain of $\hat{A}$, which results in the inequality (α) in (30) not holding. But this result demonstrates that the inequality (δ) in (31) holds.

Finally we prove (3) of Theorem 2. When constraint C1 is removed from problem (10), based on (1) of Theorem 2, the inequality (δ) in (31) would not hold due to the different data rate constraint on different solutions. $\hat{A}$ is limited by

$$ \sum\limits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\hat{p}_{n} )} + \sum\limits_{k \ne j} {r_{{k,n^{\prime}}} (\hat{p}_{{k,n^{\prime}}} ,\hat{b}_{{k,n^{\prime}}} )} + r_{j,e} (\hat{p}_{j,e} ,\hat{b}_{j,e} ) \ge R^{\text{SS}} , $$

(35)

but $\tilde{A}$ is limited by

$$ \sum\limits_{n = 1}^{N} {r_{n}^{\text{SSU}} (\tilde{p}_{n} )} + \sum\limits_{k \ne j} {r_{{k,n^{\prime}}} (\tilde{p}_{{k,n^{\prime}}} ,\tilde{b}_{{k,n^{\prime}}} )} \ge R^{\text{SS}} . $$

(36)

Therefore, from (35) to (36), it is concluded that the feasible domain of $\hat{A}$ is greater than the feasible domain of $\tilde{A}$ and $\bar{A}$, which results in the inequality (δ) in (31) not holding. But this result demonstrates that the inequality (α) in (30) holds. Thus the proof of Theorem 2 is completed.