Resource allocation in two-tier small-cell networks with energy consumption constraints

Abstract

Small-cell networks (SCNs) technology is being considered as a promising solution to improve the coverage and capacity for small-cell wireless equipment (SWE). However, the deployment of SCNs is challenging due to wireless channel interference, stochastic tasks arrival, and more prominently, the long-term energy consumption constraint of SWEs. In this paper, we provide a novel distributed dynamic resource management approach for energy-aware applications in two-tier SCNs. The joint admission control (AC) at the transport layer and resource allocation (RA) at the physical layer in SWEs is proposed to solve these challenges. Specifically, the AC and RA problem under two-tier SCNs is formulated as a stochastic optimization model which aims at maximizing the long-term average throughput of SWEs in SCN subject to time-average energy consumption limitation of each SWE and network stability constraint. By adopting Lyapunov optimization theory and Lagrangian dual decomposition technique, we propose a distributed online energy-constraint throughput optimal algorithm (ETOA) to obtain optimal AC and RA decisions. Furthermore, we derive the analytical bounds for the time-average system throughput and the time-average queue backlog achieved by our proposed approach under the constraints of long-term average energy consumption and network stability. The evaluation confirms theoretical analysis on the performance of ETOA and also shows that our approach outperforms other resource allocation methods in satisfying the time-average energy consumption requirement of SWEs.

Appendices

Appendix: A

Lemma 2

For any nonnegative real numbers x, yandz, there holds [max(x − y, 0) + z]² ≤ x² + y² + z² + 2x (z − y).

Squaring both sides of Eq. 6 and applying Lemma 2 produces

$$\begin{array}{@{}rcl@{}} {Q_{ij}^{2}}{({t + 1})} - {Q_{ij}^{2}}{(t)} &\le& {R_{ij}^{2}}{(t)} + {D_{ij}^{2}}{(t)}\\ &&+ 2{Q_{ij}}(t)\left( {{D_{ij}}(t) - {R_{ij}}(t)}\right) \end{array} $$

(38)

Summing over all queue backlog of SWEs at both sides of Eq. 38 and rearranging terms yield

$$\begin{array}{@{}rcl@{}} \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} \frac{{Q_{ij}^{2}}(t + 1)+{Q_{ij}^{2}}(t)}{2} &\le&\ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} \frac{{R_{ij}^{2}}{(t)} + {D_{ij}^{2}}{(t)}}{2}\\ &&+\sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J}{Q_{ij}}(t)\\ &&\times\left( {{D_{ij}}(t) - {R_{ij}}(t)}\right) \end{array} $$

(39)

For the update equation of E_ij(t), i.e., (9), we similarly have

$$\begin{array}{@{}rcl@{}} \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J}\frac{{E_{ij}^{2}}{({t + 1})} - {E_{ij}^{2}}{(t)}}{2} &\le& \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} \frac{\text{TP}_{ij}^{2}(t)+(P_{ij}^{\text{av}})^{2}}{2} \\ &&+ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} E_{ij}(t)\\ &&\times\left( \text{TP}_{ij}(t)-P_{ij}^{\text{av}}\right) \end{array} $$

(40)

Rombining (39) and (40) and exploiting (11) yield

$$\begin{array}{@{}rcl@{}} L\left( {\boldsymbol{\Psi}({t + 1})}\right) - L\left( {\boldsymbol{\Psi}(t)}\right) &\le&\ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} {\frac{{{R_{ij}^{2}}{(t)} + {D_{ij}^{2}}{(t)}}}{2}} \\ &&+ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} {\frac{\text{TP}_{ij}^{2}(t)+(P_{ij}^{\text{av}})^{2}}{2}} \\ &&+ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} {E_{ij}}(t)\\ &&\left( \text{TP}_{ij}(t)-P_{ij}^{\text{av}}\right)\\ &&+ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J}\\ &&\times {{Q_{ij}}(t)\left( {{D_{ij}}(t) - {R_{ij}}(t)}\right)} \end{array} $$

(41)

Taking conditional expectations on Eq. 41, exploiting (12), and subtracting $\nu \mathbb {E}\left \{D(t)\left | \boldsymbol {\Psi }(t)\right .\right \}$ at both sides, we can prove (14).

Appendix: B

We then review the basic result in Theorem 1, for the proposed ETOA approach, which is given as follows

$$\begin{array}{@{}rcl@{}} &&{\Delta} \left( {\boldsymbol{\Psi} (t)}\right) - \nu\mathbb{E}\left\{D(t) \left|{\boldsymbol{\Psi}(t)}\right.\right\}\\ &\le& C + \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} Q_{ij}(t)\mathbb{E}\left\{D_{ij}(t)-R_{ij}(t)\left|{\boldsymbol{\Psi}(t)}\right.\right\} \\ &&+ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} {{E_{ij}}}(t) \mathbb{E}\left\{\text{TP}_{ij}(t)-P_{ij}^{\text{av}}\left|{\boldsymbol{\Psi}(t)}\right.\right\} \\ &&- \nu\mathbb{E}\left\{D(t)\left| {\boldsymbol{\Psi}(t)}\right.\right\} \\ &\le& C + \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} Q_{ij}(t)\mathbb{E}\left\{D_{ij}^{*}(t)-R_{ij}^{*}(t)\left|{\boldsymbol{\Psi}(t)}\right.\right\} \\ &&+ \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} {{E_{ij}}}(t) \mathbb{E}\left\{\text{TP}_{ij}^{*}(t)-P_{ij}^{\text{av}}\left|{\boldsymbol{\Psi}(t)}\right.\right\} \\ &&- \nu\mathbb{E}\left\{D^{*}(t)\left|{\boldsymbol{\Psi}(t)}\right.\right\} \end{array} $$

(42)

where the second inequality sign of Eq. 42 holds owing to the fact that the proposed ETOA approach minimizes the RHS of Eq. 14 over all feasible control policy Φ(t) which includes the optimal randomized stationary policy $\mathbb {\alpha }^{*}(t)$.

Substituting (29) and (30) into (42) and using telescoping sums over all time slots in the above inequality, we obtain the following inequation as δ → 0

$$\begin{array}{@{}rcl@{}} &&\mathbb{E}\left\{L\left( \boldsymbol{\Psi}(t)\right)\right\} - \mathbb{E}\left\{L\left( \boldsymbol{\Psi}(0)\right)\right\} - \nu\sum\limits_{t = 0}^{T-1}\mathbb{E}\left\{D(t)\right\} \\ &\le& TC - \sum\limits_{t = 0}^{T-1}\sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} \vartheta Q_{ij}(t) \\ &&- \sum\limits_{t = 0}^{T-1}\nu\mathbb{E}\left\{D^{*}(t)\right\} \end{array} $$

(43)

(a) Applying the definition of L (Ψ(t)), we can simply the inequality as follows

$$ \frac{1}{2} \sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J}\mathbb{E} \left\{ E_{ij}^{2}(t)\right\} \le TC + T\nu IJA_{\max} + \mathbb{E}\left\{L\left( \boldsymbol{\Psi}(0)\right)\right\} $$

(44)

According to the variance formula $D\left \{E_{ij}(t)\right \}=\mathbb {E}\left \{E_{ij}^{2}(t)\right \}-\mathbb {E}^{2}\left \{E_{ij}(t)\right \}$ and D {E_ij(t)} > 0, we obtain

$$ \mathbb{E}\left\{E_{ij}(t)\right\} \le \sqrt{2TC+ 2T{\nu}IJA_{\max}+ \mathbb{E}\left\{L\left( \boldsymbol{\Psi}(0)\right)\right\}} $$

(45)

Dividing both sides by T and taking a lim inf as T →∞, we can obtain

$$ \lim_{T \rightarrow \infty} \frac{\mathbb{E}\left\{E_{ij}(T)\right\}}{T}= 0 $$

(46)

Hence, virtual energy queues E_ij(t) are mean rate stable based on Definition 1 and thus, the time-average energy consumption constraint C1 is satisfied according to Lemma 1. Similarly, we can prove that data queues Q_ij(t) are mean rate stable as well.

(b) Since the conclusion Q_ij(t) < A_max + ν can be proven using similar techniques as [13], we omit the detailed proof here for brevity. Considering the fact that $\mathbb {E}\left \{L\left ({\Psi }(t)\right )\right \} \ge 0$ and Q_ij(t) ≥ 0, rearranging (43), we obtain the following inequality

$$ \nu\sum\limits_{t = 0}^{T-1} \mathbb{E} \left\{D(t)\right\} \ge T{\nu}D^{*} - TC - \mathbb{E}\left\{L\left( {\Psi}(0)\right)\right\} $$

(47)

Dividing both sides of the above inequality by νT and then taking a limit as T →∞, we can obtain

$$ \lim_{T\to\infty} \frac{1}{T}\sum\limits_{t = 0}^{T-1} \mathbb{E} \left\{D(t)\right\} \ge D^{*} - \frac{C}{\nu} $$

(48)

(c) Rearranging (43), we obtain

$$\begin{array}{@{}rcl@{}} \sum\limits_{t = 0}^{T-1}\sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J}\vartheta Q_{ij}(t) &\le& TC + \mathbb{E}\left\{L\left( {\Psi}(0)\right)\right\} \\ &&+ \nu\sum\limits_{t = 0}^{T-1}\mathbb{E}\left\{D^{*}(t)\right\} \\ &\le& TC + \mathbb{E}\left\{L\left( {\Psi}(0)\right)\right\} \\ &&+ {\nu}TIJA_{\max} \end{array} $$

(49)

Similarly, dividing (49) by 𝜗T and then taking a limit as T →∞, we prove

$$ \lim_{T \to \infty} \frac{1}{T} \sum\limits_{t = 0}^{T-1}\sum\limits_{i = 1}^{I}\sum\limits_{j = 1}^{J} Q_{ij}(t) \le \frac{C+{\nu}IJA_{\max}}{\vartheta} $$

(50)