Optimization of energy-constrained wireless powered communication networks with heterogeneous nodes

Abstract

In this paper, we generalize conventional time division multiple access (TDMA) wireless networks to a new type of wireless networks coined generalized wireless powered communication networks (g-WPCNs). Our prime objective is to optimize the design of g-WPCNs where nodes are equipped with radio frequency (RF) energy harvesting circuitries along with constant energy supplies. This constitutes an important step towards a generalized optimization framework for more realistic systems, beyond prior studies where nodes are solely powered by the inherently limited RF energy harvesting. Towards this objective, we formulate two optimization problems with different objective functions, namely, maximizing the sum throughput and maximizing the minimum throughput (maxmin) to address fairness. First, we study the sum throughput maximization problem, investigate its complexity and solve it efficiently using an algorithm based on alternating optimization approach. Afterwards, we shift our attention to the maxmin optimization problem to improve the fairness limitations associated with the sum throughput maximization problem. The proposed problem is generalized, compared to prior work, as it seemlessly lends itself to prior formulations in the literature as special cases representing extreme scenarios, namely, conventional TDMA wireless networks (no RF energy harvesting) and standard WPCNs, with only RF energy harvesting nodes. In addition, the generalized formulation encompasses a scenario of practical interest we introduce, namely, WPCNs with two types of nodes (with and without RF energy harvesting capability) where legacy nodes without RF energy harvesting can be utilized to enhance the system sum throughput, even beyond WPCNs with all RF energy harvesting nodes studied earlier in the literature. We establish the convexity of all formulated problems which opens room for efficient solution using standard techniques. Our numerical results show that conventional TDMA wireless networks and WPCNs with only RF energy harvesting nodes are considered as lower bounds on the performance of the generalized problem setting in terms of the maximum sum throughput and maxmin throughput. Moreover, the results reveal valuable insights and throughput-fairness trade-offs unique to our new problem setting.

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Notes

  1. 1.

    Note that slot time allocations are assumed to take continuous values. This, in turn, requires accurate synchronization methods to implement such scheme in realistic systems.

  2. 2.

    The assumption that CSI is perfectly pre-estimated at the BS in the beginning of each slot is an idealization of actual practical systems. This calls for the necessity of using estimators with high accuracy to sufficiently reduce the potential estimation errors.

  3. 3.

    Note that this paper falls within the context of WPCNs where the efficiency of energy harvesting circuitries is assumed to be linear [25,26,27,28,29,30,31]. Incorporating the assumption of non-linear energy harvesting efficiency to our model is a challenging direction of future work.

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Correspondence to Mohamed A. Abd-Elmagid.

Appendices

Appendix A

Thanks to the fact that the perspective function of a concave function is also a concave function [37]. \(\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right)\) is the perspective function of the concave function \(\log _{2} \left( 1 + \alpha _{i} E_{i}\right)\) which preserves the concavity of \(R_{i}\) with respect to \((E_{i},\tau _{i})\). Since the non-negative weighted sum of concave functions is also concave [37], then the objective function of \(\mathbf P1\), which is the non-negative weighted summation of concave functions, i.e., \(R_{i}\) for \(i=1, \ldots ,K\), is a concave function in \((\mathbf {E},\pmb {\tau })\). In addition, all constraints of \(\mathbf P1\) are affine in \((\mathbf {E},\pmb {\tau })\). This establishes the proof.

Appendix B

For a given \(\mathbf {E}\) that satisfies (5) and \(0 \le E_{i} < E^b_{i} + \eta _{i} P_{B} h_{i}, i=1, \ldots ,K\), P1 reduces as follows.

$$\begin{aligned} \nonumber \mathbf P1 ^{\prime }: \qquad&\nonumber \underset{\pmb {\tau }}{\text {max}}\;\; \sum _{i=1}^{K}{\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right) }\nonumber \\ \quad {\text {s.t.}} \qquad&\sum _{i=1}^{K}{\tau _{i}} \le 1 - \tau _{0}, \end{aligned}$$
(28)
$$\begin{aligned}&\pmb {\tau } \succeq \mathbf {0},\end{aligned}$$
(29)
$$\begin{aligned}&\tau _{0} \ge \dfrac{E_{i} - E^b_{i}}{\eta _{i} P_{B} h_{i}},\qquad i=1, \ldots ,K. \end{aligned}$$
(30)

It can be easily shown that \(R_{i} = \tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right)\) is a monotonically increasing function in \((E_{i}, \tau _{i})\) [25, Lemma 3.2], \(i=1,\ldots ,K\). Therefore, the constraint in (28) should hold with equality at the optimality (otherwise, the objective function can be further increased by increasing some \(\tau _{i}\)’s). Hence, from (30), the optimal harvesting time duration is given by

$$\begin{aligned} \tau _{0}^{*} = {\text {min}} \left[ \left( \underset{i}{\text {max}}\left\{ \dfrac{E_{i} - E^b_{i}}{\eta _{i} P_{B} h_{i}}\right\} \right) ^{+},\; 1 \right] . \end{aligned}$$
(31)

Hence, \(\mathbf P1 ^{\prime }\) reduces to

$$\begin{aligned} \nonumber \mathbf P1 ^{\prime \prime }: \qquad&\nonumber \underset{\pmb {\tau ^{\prime }}}{\text {max}}\;\; \sum _{i=1}^{K}{\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right) }\nonumber \\ \quad {\text {s.t.}} \qquad&\sum _{i=1}^{K}{\tau _{i}} = 1 - \tau _{0}^{*}, \end{aligned}$$
(32)
$$\begin{aligned}&\pmb {\tau ^{\prime }} \succeq \mathbf {0}. \end{aligned}$$
(33)

Recall that \(\pmb {\tau ^{\prime }}=[\tau _{1}, \ldots ,\tau _{K}]\). Based on Theorem 1, \(\mathbf P1 ^{\prime \prime }\) is a convex optimization problem and its Lagrangian is given by

$$\begin{aligned} \mathcal {L}\left( \pmb {\tau ^{\prime }},\mu \right) = R_{sum}\left( \pmb {\tau ^{\prime }}\right) + \mu \left( \sum _{i=1}^{K}{\tau _{i}}- \left( 1 - \tau _{0}^{*}\right) \right) , \end{aligned}$$
(34)

where \(R_{sum}\left( \pmb {\tau ^{\prime }}\right) = \sum _{i=1}^{K}{\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right) }\) and \(\mu\) is the Lagrangian dual variable associated with the total slot duration constraint (32). It can be easily shown that there exists a \(\pmb {\tau ^{\prime }}\) that strictly satisfies all constraints of \(\mathbf P1 ^{\prime \prime }\). Hence, according to Slater’s condition [37], strong duality holds for this problem; therefore, the KKT conditions are necessary and sufficient for the global optimality of \(\mathbf P1 ^{\prime \prime }\), which are given by

$$\begin{aligned} \dfrac{\partial }{\partial \tau _{i}^{*}} \mathcal {L}\left( \pmb {\tau ^{\prime *}},\mu ^{*}\right)= & {} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}^{*}}\right) - \dfrac{\alpha _{i} \dfrac{E_{i}}{ \tau _{i}^{*}}}{\ln (2)\left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}^{*}}\right) } + \mu ^{*}= 0,\; i=1, \ldots ,K, \end{aligned}$$
(35)
$$\begin{aligned}&\sum _{i=1}^{K}{\tau _{i}^{*}} = 1 - \tau _{0}^{*}, \end{aligned}$$
(36)

where \(\pmb {\tau ^{\prime *}}\) and \(\mu ^{*}\) denote, respectively, the optimal primal and dual solutions of \(\mathbf P1 ^{\prime \prime }\). Therefore, from (35) and (36), we have

$$\begin{aligned} \alpha _{1} \dfrac{E_{1}}{\tau _{1}^{*}} = \alpha _{2} \dfrac{E_{2}}{\tau _{2}^{*}} \cdots \alpha _{K} \dfrac{E_{K}}{\tau _{K}^{*}} = \dfrac{\sum _{j=1}^{K}{\alpha _{i} E_{i}}}{1 - \tau _{0}^{*}}. \end{aligned}$$
(37)

Thus from (37), the optimal time allocations are given by

$$\begin{aligned} \tau _{i}^{*}=\dfrac{\alpha _{i} E_{i} \left( 1 - \tau _{0}^{*}\right) }{\sum _{j=1}^{K}{\alpha _{j} E_{j}}},\; i= 1, \ldots ,K. \end{aligned}$$
(38)

This establishes the proof.

Appendix C

For a given \(\pmb {\tau }\) that satisfies (6)–(8), P1 reduces as follows.

$$\begin{aligned} \nonumber \mathbf P1 ^{\dagger }: \qquad&\nonumber \underset{\mathbf {E}}{\text {max}}\; \; \sum _{i=1}^{K}{\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right) }\nonumber \\ \quad {\text {s.t.}} \qquad&\sum _{i=1}^{K}{E_{i}} \le E_{max}, \end{aligned}$$
(39)
$$\begin{aligned}&0 \le E_{i} \le E^b_{i} + \eta _{i} P_{B} h_{i} \tau _{0},\qquad i=1, \ldots ,K. \end{aligned}$$
(40)

Recall that \(R_{i} = \tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right)\) is a monotonically increasing function in \((E_{i}, \tau _{i})\), \(i=1,\ldots ,K\). Therefore, when \(E_{max} \ge \sum _{j = 1}^{K}{\left( E^b_{j} + \eta _{j} P_{B} h_{j} \tau _{0}\right) }\), \(\mathbf P1 ^{\dagger }\) has a trivial solution that \(E_{i}^{*} = E^b_{i} + \eta _{i} P_{B} h_{i} \tau _{0}\), \(i=1, \ldots ,K\). On the other hand, when \(E_{max} < \sum _{j = 1}^{K}{\left( E^b_{j} + \eta _{j} P_{B} h_{j} \tau _{0}\right) }\), the optimal solution of \(\mathbf P1 ^{\dagger }\) can be characterized as follows. First, the constraint in (39) should hold with equality at the optimality (otherwise, the objective function can be further increased by increasing some \(E_{i}\)’s). Based on Theorem 1, \(\mathbf P1 ^{\dagger }\) is a convex optimization problem and its Lagrangian is given by

$$\begin{aligned} \mathcal {L}\left( \mathbf {E},\lambda \right) = R_{sum}\left( \mathbf {E}\right) + \lambda \left( \sum _{i=1}^{K}{E_{i}} - E_{max}\right) , \end{aligned}$$
(41)

where \(\lambda\) is the Lagrangian dual variable associated with the total allowable consumed energy per slot constraint (39). The strong duality holds for \(\mathbf P1 ^{\dagger }\); therefore, the KKT conditions are necessary and sufficient for the global optimality of \(\mathbf P1 ^{\dagger }\), which are given by

$$\begin{aligned} \dfrac{\partial }{\partial E_{i}^{*}} \mathcal {L}\left( \mathbf {E}^{*},\lambda ^{*}\right) = \dfrac{\alpha _{i}}{\ln (2)\left( 1 + \dfrac{\alpha _{i} E_{i}^{*}}{\tau _{i}}\right) } + \lambda ^{*}= 0,\; i=1, \ldots ,K, \end{aligned}$$
(42)
$$\begin{aligned} \sum _{i=1}^{K}{E_{i}^{*}} = E_{max}, \end{aligned}$$
(43)

where \(\mathbf {E}^{*}\) and \(\lambda ^{*}\) denote, respectively, the optimal primal and dual solutions of \(\mathbf P1 ^{\dagger }\). Therefore, from (42), we have

$$\begin{aligned} E_{i}^{*} = - \dfrac{\tau _{i}}{\alpha _{i}}\left( \dfrac{\alpha _{i}}{\lambda ^{*} \ln (2)} + 1\right) ,\; i=1, \ldots ,K. \end{aligned}$$
(44)

Taking into account the constraints in (40), the optimal energy allocations are given by

$$\begin{aligned} E_{i}^{*} = {\text {min}}\left[ \left( - \dfrac{\tau _{i}}{\alpha _{i}}\left( \dfrac{\alpha _{i}}{\lambda ^{*} \ln (2)} + 1\right) \right) ^{+},\;E^{b}_{i} + \eta _{i} P_{B} h_{i} \tau _{0} \right] ,\; i=1,\ldots ,K, \end{aligned}$$
(45)

where \(\lambda ^{*}\) satisfies the equality constraint \(\sum _{i=1}^{K}{E_{i}^{*}} = E_{max}\). This establishes the proof.

Appendix D

P4 is a convex optimization problem and its Lagrangian is given by

$$\begin{aligned} \mathcal {L}\left( \pmb {\tau ^{\prime \prime }},\bar{E},\lambda ,\mu \right) = R_{sum}\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right) - \mu \left( \tau _{0}+\sum _{i=1}^{M}{\tau _{1,i}}+\sum _{j=1}^{N}{\tau _{2,j}} - 1\right) - \lambda \left( a\tau _{0} + N \bar{E} - E_{max}\right) , \end{aligned}$$
(46)

where \(\mu\) and \(\lambda\) are the Lagrangian dual variables associated with the slot duration and the total allowable consumed energy per slot constraints, respectively, and \(R_{sum}\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right) = \sum _{i=1}^{M}{R_{1,i}\left( \tau _{0},\tau _{1,i}\right) } + \sum _{j=1}^{N}{R_{2,j}\left( \bar{E},\tau _{2,j}\right) }\). Hence, the dual function can be expressed as

$$\begin{aligned} G\left( \lambda ,\mu \right) = \underset{\pmb {\tau ^{\prime \prime }},\bar{E}\in \mathcal {S}}{\max } \; \mathcal {L}\left( \pmb {\tau ^{\prime \prime }},\bar{E},\lambda ,\mu \right) , \end{aligned}$$
(47)

where \(\mathcal {S}\) is the feasible set specified by \(\pmb {\tau ^{\prime \prime }} \succeq \mathbf {0}\) and \(\bar{E} \ge 0\). It can be easily shown that there exists a \((\pmb {\tau ^{\prime \prime }},\bar{E})\) that strictly satisfies all constraints of P4. Hence, according to Slater’s condition [37], strong duality holds for this problem; therefore, the KKT conditions are necessary and sufficient for the global optimality of P4, which are given by

$$\begin{aligned}&\tau _{0}^{*}+\sum _{i=1}^{M}{\tau _{1,i}^{*}}+\sum _{j=1}^{N}{\tau _{2,j}^{*}} \le 1, \end{aligned}$$
(48)
$$\begin{aligned}&a\tau _{0}^{*} + N \bar{E}^{*} \le E_{max}, \end{aligned}$$
(49)
$$\begin{aligned}&\mu ^{*} \left( \tau _{0}^{*}+\sum _{i=1}^{M}{\tau _{1,i}^{*}} +\sum _{j=1}^{N}{\tau _{2,j}^{*}} - 1\right) = 0, \end{aligned}$$
(50)
$$\begin{aligned}&\lambda ^{*} \left( a\tau _{0}^{*} + N \bar{E}^{*} - E_{max}\right) = 0, \end{aligned}$$
(51)
$$\begin{aligned}&\dfrac{\partial }{\partial \tau _{0}}R_{sum}\left( \pmb {\tau ^{\prime \prime *}},\bar{E}^{*}\right) - \left( a \lambda ^{*} + \mu ^{*}\right) = 0, \end{aligned}$$
(52)
$$\begin{aligned}&\dfrac{\partial }{\partial \tau _{1,i}}R_{sum}\left( \pmb {\tau ^{\prime \prime *}},\bar{E}^{*}\right) - \mu ^{*} = 0, \; i=1, \ldots ,M, \end{aligned}$$
(53)
$$\begin{aligned}&\dfrac{\partial }{\partial \tau _{2,j}}R_{sum}\left( \pmb {\tau ^{\prime \prime *}},\bar{E}^{*}\right) - \mu ^{*} = 0,\; j=1, \ldots ,N, \end{aligned}$$
(54)
$$\begin{aligned}&\dfrac{\partial }{\partial \bar{E}^{*}}R_{sum}\left( \pmb {\tau ^{\prime \prime *}},\bar{E}^{*}\right) - N \lambda ^{*} = 0, \end{aligned}$$
(55)

where \(\left( \pmb {\tau ^{\prime \prime *}},\bar{E}^{*}\right)\) and \(\left( \lambda ^{*},\mu ^{*}\right)\) denote, respectively, the optimal primal and dual solutions of P4. Since \(R_{sum}\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right)\) is a monotonic increasing function in \(\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right)\), therefore \(\tau _{0}^{*}+\sum _{i=1}^{M}{\tau _{1,i}^{*}}+\sum _{j=1}^{N}{\tau _{2,j}^{*}} = 1\) and \(a\tau _{0}^{*} + N \bar{E}^{*} = E_{max}\) must hold. From (52)–(55), we have

$$\begin{aligned}&\sum _{i=1}^{M}{\dfrac{\gamma _{i}}{1 + \gamma _{i} \dfrac{\tau _{0}^{*}}{\tau _{1,i}^{*}}}} = \left( a \lambda ^{*} + \mu ^{*}\right) \ln (2), \end{aligned}$$
(56)
$$\begin{aligned}&\ln \left( 1 + \gamma _{i} \dfrac{\tau _{0}^{*}}{\tau _{1,i}^{*}}\right) - \dfrac{\gamma _{i}\dfrac{\tau _{0}^{*}}{\tau _{1,i}^{*}}}{1+\gamma _{i}\dfrac{\tau _{0}^{*}}{\tau _{1,i}^{*}}} = \mu ^{*} \ln (2),\; i=1, \ldots ,M, \end{aligned}$$
(57)
$$\begin{aligned}&\ln \left( 1+\dfrac{\bar{E}^{*} \theta _{j}}{\tau _{2,j}^{*}}\right) - \dfrac{\dfrac{\bar{E}^{*} \theta _{j}}{\tau _{2,j}^{*}}}{1 + \dfrac{\bar{E}^{*} \theta _{j}}{\tau _{2,j}^{*}}} = \mu ^{*} \ln (2), \; j=1, \ldots , N. \end{aligned}$$
(58)
$$\begin{aligned}&\sum _{j=1}^{N}{\dfrac{\theta _{j}}{1 + \theta _{j} \dfrac{\bar{E}^{*}}{\tau _{2,j}^{*}}}} = N\lambda ^{*}\ln (2), \end{aligned}$$
(59)

Therefore, from (57) and (58), we have

$$\begin{aligned} \dfrac{\gamma _{1} \tau _{0}^{*}}{\tau _{1,1}^{*}} = \dfrac{\gamma _{2} \tau _{0}^{*}}{\tau _{1,2}^{*}} = \cdots \dfrac{\gamma _{M} \tau _{0}^{*}}{\tau _{1,M}^{*}} = \dfrac{\bar{E}^{*}\theta _{1}}{\tau _{2,1}^{*}} = \dfrac{\bar{E}^{*}\theta _{2}}{\tau _{2,2}^{*}} = \cdots \dfrac{\bar{E}^{*}\theta _{N}}{\tau _{2,N}^{*}} = x_{1}. \end{aligned}$$
(60)

From \(\tau _{0}^{*}+\sum _{i=1}^{M}{\tau _{1,i}^{*}}+\sum _{j=1}^{N}{\tau _{2,j}^{*}} = 1\) and (60), \(\tau _{1,i}^{*}\) and \(\tau _{2,j}^{*}\) can be expressed, respectively, by

$$\begin{aligned} \tau _{1,i}^{*}= & {} \dfrac{\gamma _{i} \left( N\left( x_{1}^{*}- 1\right) - E_{max} A_{2} \right) }{\left( x_{1}^{*} - 1\right) \left( N\left( x_{1}^{*} - 1 + A_{1}\right) - a A_{2} \right) }, \; i=1, \ldots , M, \end{aligned}$$
(61)
$$\begin{aligned} \tau _{2,j}^{*}= & {} \dfrac{\theta _{j} \left( E_{max}\left( x_{1}^{*} - 1 + A_{1} \right) - a \left( x_{1}^{*} - 1\right) \right) }{\left( x_{1}^{*} - 1\right) \left( N\left( x_{1}^{*} - 1 + A_{1}\right) - a A_{2} \right) }, \; j=1, \ldots , N, \end{aligned}$$
(62)

where \(A_{1} = \sum _{i=1}^{M}{\gamma _{i}}\) and \(A_{2} = \sum _{j=1}^{N}{\theta _{j}}\). From (56) and (59), it follows that

$$\begin{aligned} \lambda ^{*}= & {} \dfrac{A_{2}}{ N x_{1} \ln (2)}, \end{aligned}$$
(63)
$$\begin{aligned} \mu ^{*}= & {} \dfrac{A_{1} - \dfrac{a}{N}A_{2}}{x_{1} \ln (2)}. \end{aligned}$$
(64)

By substituting with \(\mu ^{*}\) into (57), we have

$$\begin{aligned} x_{1}\ln (x_{1}) - x_{1} + 1 = A_{1} - \dfrac{a}{N} A_{2}. \end{aligned}$$
(65)

From (61) and (62), it is clear that \(x_{1} > 1\) if \(A_{1} > 0\), \(A_{2} > 0\) and \(0< \tau _{0}^{*} < 1\). According to [25, Lemma 3.2], there exists a unique solution \(x_{1}^{*} > 1\) for (65) if \(A_{1} \ge \dfrac{a}{N} A_{2}\), otherwise the total slot time and the total allowable consumed energy per slot will be assigned to the Type II nodes for uplink information transmissions. Thus from (60)–(65), the optimal time and energy allocations are given by

$$\begin{aligned} \tau _{0}^{*}= & {} \dfrac{N\left( x_{1}^{*} - 1 \right) - E_{max} A_{2} }{N\left( x_{1}^{*} - 1 + A_{1}\right) - a A_{2}}, \end{aligned}$$
(66)
$$\begin{aligned} \tau _{1,i}^{*}= & {} \dfrac{\gamma _{i} \left( x_{1}^{*} - E_{max} A_{2} - 1\right) }{\left( x_{1}^{*} - 1\right) \left( x_{1}^{*} + A_{1} - a A_{2} -1\right) }, \; i=1, \ldots ,M, \end{aligned}$$
(67)
$$\begin{aligned} \tau _{2,j}^{*}= & {} \dfrac{\theta _{j} \left( E_{max}\left( x_{1}^{*} + A_{1} - 1\right) - a \left( x_{1}^{*} - 1\right) \right) }{K\left( x_{1}^{*} - 1\right) \left( x_{1}^{*} + A_{1} - a A_{2} -1\right) }, \; j=1, \ldots ,N. \end{aligned}$$
(68)
$$\begin{aligned} \bar{E}^{*}= & {} \dfrac{E_{max}\left( x_{1}^{*} - 1 + A_{1} \right) - a \left( x_{1}^{*} - 1\right) }{N\left( x_{1}^{*} - 1 + A_{1}\right) - a A_{2}}. \end{aligned}$$
(69)

From (66)–(69), for \([\tau _{0}^{*}, \tau _{1,1}^{*}, \ldots ,\tau _{1,M}^{*}, \tau _{2,1}^{*}, \ldots ,\tau _{2,N}^{*}, \bar{E}^{*}] \succeq \mathbf {0}\), we must have \(\dfrac{a(x_{1}^{*} - 1)}{A_{1} + x_{1}^{*} - 1} \le E_{max} \le \dfrac{N}{A_{2}}(x_{1}^{*} - 1)\). If \(E_{max} > \dfrac{N}{A_{2}}(x_{1}^{*} - 1)\), then we have \([\tau _{0}^{*}, \tau _{1,1}^{*}, \ldots ,\tau _{1,M}^{*}] \prec \mathbf {0}\). Hence, the total slot time and the total allowable consumed energy per slot will be assigned to the Type II nodes for uplink information transmissions. Therefore, from (50) and (58), the optimal time and energy allocations are given by (21)–(24).

On the other hand, if \(E_{max} < \dfrac{a(x_{1}^{*} - 1)}{A_{1} + x_{1}^{*} - 1}\), then we have \([\tau _{2,1}^{*}, \ldots ,\tau _{2,N}^{*}, \bar{E}^{*}] \prec \mathbf {0}\). Hence, the total slot time and the total allowable consumed energy per slot will be assigned to the Type I nodes for uplink information transmissions. Therefore, from (50), (56) and (57), the optimal time and energy allocations are given by (21)–(24). This establishes the proof.

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Abd-Elmagid, M.A., ElBatt, T. & Seddik, K.G. Optimization of energy-constrained wireless powered communication networks with heterogeneous nodes. Wireless Netw 25, 713–730 (2019). https://doi.org/10.1007/s11276-017-1587-x

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Keywords

  • Cellular networks
  • Green communications
  • RF energy harvesting
  • Convex optimization
  • Numerical results