Dynamic energy-efficient resource allocation in wireless powered communication network

Abstract

In this paper, we investigate the dynamic energy-efficient resource allocation and analyze delay in newly emerging wireless powered communication network (WPCN). Considering the time-varying channel and stochastic data arrivals, we formulate the resource allocation (i.e., time allocation and power control) problem as a dynamic stochastic optimization model, which maximizes the system energy efficiency (EE) subject to both the data queue stability and the harvested energy availability, and simultaneously satisfies a certain quality of service (QoS) in terms of delay. With the aid of fractional programming, Lyapunov optimization theory and Lagrange method, we solve the problem and propose an dynamic energy-efficient resource allocation algorithm (DEERAA), which does not require any prior distribution knowledge of the channel state information (CSI) or stochastic data arrivals. We find that the performance of EE and delay can be adjusted by a system control parameter V. The effectiveness of the proposed algorithm is demonstrated by the mathematical analysis and simulation results.

Appendix

1.1 Proof of Theorem 1

Theorem 1 is proved by separately proving the necessary condition and sufficient condition of it. First, we prove the necessary condition. Let $\pmb {\tau ^*}$ and $\pmb {P^{tr*}}$ be the solution of (9) and according to the formulation (10), we arrive in

$$\begin{aligned} \rho _{ee}^* = \frac{\overline{R}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})}{\overline{W}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})} \ge \frac{\overline{R}_{A}(\pmb {\tau }, \pmb {P^{tr}})}{\overline{W}_{A}(\pmb {\tau }, \pmb {P^{tr}})}, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S}. \end{aligned}$$

(37)

Rewrite the formulation above, i.e. (37), then we obtain

$$\begin{aligned}&\overline{R}_{A}(\pmb {\tau }, \pmb {P^{tr}}) - \rho _{ee}^*\overline{W}_{A}(\pmb {\tau }, \pmb {P^{tr}}) \le 0, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S} \end{aligned}$$

(38)

$$\begin{aligned}&\overline{R}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}}) - \rho _{ee}^* \overline{W}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})= 0, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S}. \end{aligned}$$

(39)

According to (38), we have $max\{ \overline{R}_{A}(\pmb {\tau }, \pmb {P^{tr}}) - \rho _{ee}^*\overline{W}_{A}(\pmb {\tau }, \pmb {P^{tr}}), \forall \pmb {\tau }| \pmb {P^{tr}} \in {\mathfrak {S}} = 0$. Combining with (38), we have the maximum value is taken when $\{\pmb {\tau }, \pmb {P^{tr}}\}=\{\pmb {\tau ^*},\pmb {P^{tr*}}\}$. Thus finished the proof of necessary condition. To prove the sufficient conditon, let $\{\{\pmb {\tau ^*},\pmb {P^{tr*}}\}$ be a solution of (11), i.e. $\overline{R}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}}) - \rho _{ee}^* \overline{W}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})= 0$, and then

$$\begin{aligned} \overline{R}_{A}(\pmb {\tau }, \pmb {P^{tr}}) - \rho _{ee}^*\overline{W}_{A}(\pmb {\tau }, \pmb {P^{tr}}) \le \overline{R}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}}) - \rho _{ee}^* \overline{W}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})= 0, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S}. \end{aligned}$$

(40)

Hence

$$\begin{aligned}&\overline{R}_{A}(\pmb {\tau }, \pmb {P^{tr}}) - \rho _{ee}^*\overline{W}_{A}(\pmb {\tau }, \pmb {P^{tr}}) \le 0, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S}, \end{aligned}$$

(41)

$$\begin{aligned}&\overline{R}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}}) - \rho _{ee}^* \overline{W}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})= 0, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S}. \end{aligned}$$

(42)

From (41)we have $\rho _{ee}^* \ge \frac{\overline{R}_{A}(\pmb {\tau }, \pmb {P^{tr}})}{\overline{W}_{A}(\pmb {\tau }, \pmb {P^{tr}})}, \forall \pmb {\tau }, \pmb {P^{tr}} \in \mathfrak {S}$, that is $\rho _{ee}^*$ is the maximum of (10). From (42) we have $\frac{\overline{R}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})}{\overline{W}_{A}(\pmb {\tau ^*},\pmb {P^{tr*}})}=\rho _{ee}^*$, that is $\{\{\pmb {\tau ^*},\pmb {P^{tr*}}\}$ is a solution of (10). Thus finished the proof of Theorem 1. $\square$

1.2 Proof of Theorem 2

Theorem 2 with is proved with the aid of contradiction. We propose two solutions to $\mathcal {P}3$. The first one is supposing the optimal solution to problem $\mathcal {P}3$ in time-slot t is $\varPhi ^* =\{P_0^{tr*}(t), P_k^{tr*}(t), \tau _0^*(t), \tau _k^*(t)\}, \forall k \ne 0$, and supposing the PS transmit power satisfies $P_0^{tr*}(t)< P_0^{max}(t)$. Under such a assumption we attain the optimal solution $\varPhi ^*$ is denoted as $\rho ^*$. The other one solution is $\varPhi ^\prime =\{P_0^{tr\prime }(t), P_k^{tr\prime }(t), \tau _0^\prime (t), \tau _k^\prime (t)\},\forall k \ne 0$, and suppose the PS transmit power satisfies $P_0^{tr\prime }(t)= P_0^{max}(t)$. The corresponding optimal value is $\rho ^\prime$. Furthermore, we assume the difference only exists in the WET stage, i.e., the condition in WIT stage between two solutions is same. Then we have $\forall k \ne 0$,$P_0^{tr*}(t)\tau _0^*(t)=P_0^{tr\prime }(t)\tau _0^\prime (t), \tau _k^*(t)=\tau _k^\prime (t), P_k^{tr*}(t)=P_k^{tr\prime }(t), W_k^*(v)=W_k^\prime (v),R_k^*(v)=R_k^\prime (v)$. Now, we can compare the value $\rho ^*$ and $\rho ^\prime$, which are given by Eqs. (43) and (44), at the top of next page to find the relationship between them. According to the assumption that $P_0^{tr\prime }(t)= P_0^{max}(t)>P_0^{tr*}(t)$, $P_0^{tr*}(t)\tau _0^*(t)=P_0^{tr\prime }(t)\tau _0^\prime (t)$, then we obtain $\tau _0^\prime (t)<\tau _0^*(t)$. Comparing the two equations, i.e.,Eqs. (43) and (44), we can obtain $\rho ^*<\rho ^\prime$ which contradicts the assumption that $\rho ^*$ is the optimal solution. Thus Theorem 2 is proved. $\square$

$$\begin{aligned} \rho ^\prime= & {} \frac{\sum _{v=1}^t\left( \sum _{k=1}^KR_k^\prime (v)\right) }{\sum _{v=1}^{t-1}\left( P_0^{tr\prime }(v)\tau _0^\prime (v)+P_0^{ci}\tau _0^\prime (v)-\sum _{k=1}^KW_k^\prime (v)+\sum _{k=1}^K\left( P_k^{tr\prime }(v)+P_k^{ci\prime }(v)\right) \tau _k^\prime (v)\right) }. \end{aligned}$$

(43)

$$\begin{aligned} \rho ^*= & {} \frac{\sum _{v=1}^t\left( \sum _{k=1}^KR_k^*(v)\right) }{\sum _{v=1}^{t-1}\left( P_0^{tr*}(v)\tau _0^*(v)+P_0^{ci}\tau _0^*(v)-\sum _{k=1}^KW_k^*(v)+\sum _{k=1}^K\left( P_k^{tr*}(v)+P_k^{ci*}(v)\right) \tau _k^*(v)\right) }. \end{aligned}$$

(44)

1.3 Proof of (17)

Squaring both sides of Eq. (7) results in

$$\begin{aligned} \left( Q_k^D(t+1)\right) ^2\le&(Q_k^D(t))^2+(R_k(t))^2+(D_k(t))^2-2Q_k^D(t)(R_k(t)-D_k(t)). \end{aligned}$$

(45)

Sum overall $k(k=1, ...,K)$, take the conditional expectation $\mathbb {E}\{\cdot |Z(t)\}$, and recall $R_k(t)\le R_k^{max}$, $D_k(t)\le D_k^{max}$, then

$$\begin{aligned} \varDelta \{Z(t)\}\le & {} \frac{1}{2}\sum _{k=1}^K\mathbb {E}\{(D_k(t))^2+(R_k(t))^2|Z(t)\} -\sum _{k=1}^K\mathbb {E}\{Q_k^D(t)\left( (R_k(t)-D_k(t))|Z(t)\right) \}\nonumber \\\le & {} c^{M}-\sum _{k=1}^K\mathbb {E}\{Q_k^D(t)\left( R_k(t)|Z(t)\right) \} +\sum _{k=1}^K\mathbb {E}\{Q_k^D(t)\left( D_k(t)|Z(t)\right) \}, \end{aligned}$$

(46)

where $c^{M}=\frac{1}{2}\sum _{k=1}^K\left( (D_k^{max}(t))^2+(R_k^{max}(t))^2\right)$. Define $C^{M}=c^{M}+\sum _{k=1}^K\mathbb {E}\{Q_k^D(t)\left( D_k(t)|Z(t)\right) \}$, and then we obtain (17). This finished the proof. $\square$

1.4 Proof of (35) and (36)

Adopt an i.i.d. algorithm and we can transform (18) into (47)

$$\begin{aligned}&\triangle \{Z(t)\}-VE\{{R_{A}(t )-\rho _{ee}(t ){W_{A}}(t)}\left| \right. Z(t)\}\nonumber \\&\quad \le C^{M} -E\left\{ V\left\{ R_{A}^\omega (t )-\rho _{ee}(t ){W_{A}^\omega }(t)\right\} \left| \right. Z(t)\right\} -E\left\{ \sum _{k=1}^KQ_k^D(t)R_k^\omega (t)\left| \right. Z(t)\right\} \end{aligned}$$

(47)

Plugging (33) and (34) into (47) and taking a limit as $\varepsilon \rightarrow 0$ yield

$$\begin{aligned}&\triangle \{Z(t)\}-VE\{{R_{A}}(t) - \rho _{ee}(t ){W_{A}}(t)|Z(t)\} \le c^{M}-V\rho _{ee}^{opt}E\{{W_{A}^\omega }(t)\}+VE\{\rho _{ee}(t){W_{A}^\omega }(t)\}. \end{aligned}$$

(48)

By taking iterated expectation and using telescoping sums over $t \in \{1, \ldots , T\}$ in (48), and then we obtain

$$\begin{aligned} E\{L(Z(T))\}-E\{L(Z(1))\}\le&T(c^{M}-V\rho _{ee}^{opt}E\{{W_{A}^\omega }(t)\})\nonumber \\&+V\sum _{t=1}^TE\{\rho _{ee}(t){R_{A}^\omega }(t)\} +V\sum _{t=1}^TE\{{R_{A}}(t) - \rho _{ee}(t ){W_{A}}(t)|Z(t)\}. \end{aligned}$$

(49)

Dividing (49) by VT and using the fact that $E\{L(Z(T))\}\ge 0$, we acquire

$$\begin{aligned} \frac{1}{T}\sum _{t=1}^TE\{{R_{A}}(t) - \rho _{ee}(t ){W_{A}}(t)\} \ge -\frac{E\{L(Z(1))\}}{VT}-\frac{c^{M}}{V}+\rho _{ee}^{opt}E\{{W_{A}^\omega }(t)\} -E\{{W_{A}^\omega }(t)\}\frac{1}{T}\sum _{t=1}^TE\{\rho _{ee}(t)\}. \end{aligned}$$

(50)

Taking the limit as $T\rightarrow \infty$ , and after some manipulations, then we obtain

$$\begin{aligned} \rho _{ee}\ge \rho _{ee}^{opt} - \frac{c^{M}/E\{{W_{A}^\omega }(t)\}}{V} \ge \rho _{ee}^{opt} - \frac{c^{M}/W_{A_{min}}}{V}. \end{aligned}$$

(51)

Similarly, plug (33) and (34) into (47), take iterated expectation and use telescoping sums over $t \in \{1, \cdots , T\}$, then

$$\begin{aligned}&E\{L(Z(T))\}-E\{L(Z(1))\}+\varepsilon \sum _{t=1}^T\sum _{k=1}^KQ_k^D(t)-V\sum _{t=1}^TE\{\rho _{ee}(t){W_{A}^\omega }(t)\}\nonumber \\&\quad \le T(c^{M}-V\rho _{ee}^{opt}E\{{W_{A}^\omega }(t)\}) +V\sum _{t=1}^TE\{{R_{A}}(t) - \rho _{ee}(t ){W_{A}}(t)\}. \end{aligned}$$

(52)

Divide (52) by $\varepsilon T$ and take limit as $T\rightarrow \infty$

$$\begin{aligned} \overline{Q}=\lim _{T \rightarrow \infty } \frac{1}{T} \sum _{t=1}^T \sum _{k=1}^K Q_k^D(t)\le \frac{c^{M}+V\left\{ R_{A_{max}}+\rho _{ee}^{opt}(W_{A_{max}}-W_{A_{min}})\right\} }{\varepsilon }. \end{aligned}$$

(53)

This finished the proof of Eqs. (35) and (36). $\square$