Dynamic power and subcarrier allocation for downlink OFDMA systems under imperfect CSI

Abstract

In this paper, we investigate the joint power and subcarrier allocation for the downlink of orthogonal frequency division multiplexing access systems, with various practical considerations including imperfect estimation of channel state information, a stochastic packet arrival and a time-varying channel. To this end, we formulate the stochastic optimization problem to minimize the time-averaged power consumption, whilst keeping all queues at the base-station stable. The data transmission rate is defined as a function of the transmit power, the assigned subcarrier and the estimation error. With the aid of Lyapunov optimization method, the original problem is transformed into a series of mixed-integer programming problems, which are then solved via the dual decomposition technique. We determine analytical bounds for the time-averaged power consumption and queue length achieved by our proposed algorithm, which depend on the channel estimation error. Moreover, the theoretical analysis and simulation results show that the proposed algorithm reduces the energy consumption at the expense of queue backlog (i.e., achieves a energy-queue tradeoff), and quantitatively strike the energy-queue tradeoff by simply tuning an introduced control parameter V.

Appendix

1.1 Proof of Theorem 3

According to Lemma 1, for the proposed DPSAI algorithm, we obtain

$$\begin{aligned}&\varDelta ({\mathbf {Q}}(\tau ))+V{\mathbb {E}}\left\{ \sum _{i=1}^M\sum _{j=1}^Na^o_{ij}(\tau )P^o_{ij}(\tau )\mid {\mathbf {Q}}(\tau )\right\} \nonumber \\&\quad \le C_1+V{\mathbb {E}}\left\{ \sum _{i=1}^M\sum _{j=1}^Na^o_{ij}(\tau )P^o_{ij}(\tau )\mid {\mathbf {Q}}(\tau )\right\} \nonumber \\&\qquad +\,{\mathbb {E}}\left\{ \sum _{i=1}^M\sum _{j=1}^N Q_i(\tau )[A_i(\tau )-a^o_{ij}(\tau )R_{ij}(P^o_{ij}(\tau ))]\left| \right. {\mathbf {Q}}(\tau )\right\} \nonumber \\&\quad \le C_1+V{\mathbb {E}}\left\{ \sum _{i=1}^M\sum _{j=1}^Na'_{ij}(\tau )P'_{ij}(\tau )\mid {\mathbf {Q}}(\tau )\right\} \nonumber \\&\qquad +\,{\mathbb {E}}\left\{ \sum _{i=1}^M\sum _{j=1}^N Q_i(\tau )[A_i(\tau )-a'_{ij}(\tau )R_{ij}(P'_{ij}(\tau ))]\left| \right. {\mathbf {Q}}(\tau )\right\} ,\nonumber \\ \end{aligned}$$

(42)

where the resource allocation decisions $a'_{ij}(\tau )$ and $P'_{ij}(\tau )$ are implemented with any stationary randomized strategy. The second inequality sign of (42) holds due to the fact that the proposed resource allocation scheme is optimal to minimize the RHS of the bounds in (26) compared with any other strategies.

Suppose that $\pmb {\lambda }$ is strictly interior to the capacity region $\pmb {\varGamma }$, that $\pmb {\lambda }+\vartheta$ is still in $\pmb {\varGamma }$ for a positive $\vartheta$. According to the stochastic network optimization theory [20, 25], if the constraints (9)–(14) are feasible, then for any $\xi >0$ there exists a stationary randomized policy satisfying

$$\begin{aligned} {\mathbb {E}}\left[ \sum ^N_{j=1}a'_{ij}(\tau )R_{ij}(P'_{ij}(\tau ))\left| \right. {\mathbf {Q}}(\tau )\right]\ge & {} \lambda _i+\vartheta , \end{aligned}$$

(43)

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^M\sum _{j=1}^Na'_{ij}(\tau )P'_{ij}(\tau )\left| \right. {\mathbf {Q}}(\tau )\right]\le & {} \overline{P}^{opt}_{tot}+\xi , \end{aligned}$$

(44)

where $\overline{P}^{opt}_{tot}$ is the minimum time-averaged power expenditure over all feasible resource policies. Substituting (43) and (44) into (42), we get the following inequation as $\xi \rightarrow 0$.

$$\begin{aligned}&\varDelta ({\mathbf {Q}}(\tau ))+V{\mathbb {E}}\left\{ P_{tot}(\tau )\mid {\mathbf {Q}}(\tau )\right\} \nonumber \\&\quad \le C_1+V\overline{P}^{opt}_{tot}-\vartheta \sum _{i=1}^M{\mathbb {E}}\{Q_i(\tau )\}. \end{aligned}$$

(45)

Plugging (16) into (45) and using telescoping sums over $\tau \in \{0,1,\ldots ,T-1\}$ yield

$$\begin{aligned}&{\mathbb {E}}\{L({\mathbf {Q}}(T))\}-{\mathbb {E}}\{L({\mathbf {Q}}(0))\}+V\sum _{\tau =1}^{T-1}{\mathbb {E}}\left\{ P_{tot}(\tau )\right\} \nonumber \\&\quad \le TC_1+TV\overline{P}^{opt}_{tot}-\sum _{\tau =1}^{T-1}\sum _{i=1}^M\vartheta {\mathbb {E}}\{Q_i(\tau )\}. \end{aligned}$$

(46)

• (a) According to the definition of $L({\mathbf {Q}}(\tau ))$ (15), the inequality (46) is further simplified as

  $$\begin{aligned}&\frac{1}{2}\sum _{i=1}^M{\mathbb {E}}\left\{ Q_i(T)^2\right\} \le TC_1+TV\overline{P}^{opt}_{tot}+{\mathbb {E}}\left\{ L\left( {\mathbf {Q}}(0)\right) \right\} . \end{aligned}$$

  (47)

  From the variance formula: $D\{Q_i(T)\}={\mathbb {E}}\{Q_i(T)^2\}-{\mathbb {E}}^2\{Q_i(T)\}$, we have ${\mathbb {E}}\{Q^2_i(T)\}\ge {\mathbb {E}}^2\{Q_i(T)\}$ due to the fact that $D\{Q_i(T)\}>0$. Thus

  $$\begin{aligned}&{\mathbb {E}}\{Q_i(T)\}\le \sqrt{2TC_1+2TV\overline{P}^{opt}_{tot}+2{\mathbb {E}}\left\{ L\left( {\mathbf {Q}}(0)\right) \right\} }. \end{aligned}$$

  (48)

  Dividing the above inequality by T and taking a limit as $T\rightarrow \infty$, we proves

  $$\begin{aligned} \lim _{T\rightarrow \infty }\frac{{\mathbb {E}}\left\{ Q_i(T)\right\} }{T}=0. \end{aligned}$$

• (b) Using the fact that ${\mathbb {E}}\{L({\mathbf {Q}}(\tau ))\}>0$ and $Q_i(\tau )>0$, rearranging (46), we obtain that

  $$\begin{aligned}&V\sum _{\tau =1}^{T-1}{\mathbb {E}}\left\{ P_{tot}(\tau )\right\} \le TC_1+TV\overline{P}^{opt}_{tot} +{\mathbb {E}}\left\{ L\left( {\mathbf {Q}}(0)\right) \right\} . \end{aligned}$$

  (49)

  Dividing inequality (49) by VT and taking a limit as $T\rightarrow \infty$ yields

  $$\begin{aligned}&\lim _{T\rightarrow \infty }\frac{1}{T}\sum _{\tau =1}^{T-1}{\mathbb {E}}\left\{ P_{tot}(\tau )\right\} \le \frac{C_1}{V}+\overline{P}^{opt}_{tot}. \end{aligned}$$

  (50)

• (c) Similarly, we rewritten (46) as

  $$\begin{aligned}&\frac{1}{T}\sum _{\tau =1}^{T-1}\sum _{i=1}^M{\mathbb {E}}\{Q_i(\tau )\} \le \frac{C_1+V\overline{P}^{opt}_{tot}}{\vartheta }+\frac{{\mathbb {E}}\{L({\mathbf {Q}}(0))\}}{T\vartheta }. \end{aligned}$$

  (51)

  Taking a limit as $T\rightarrow \infty$, we prove

  $$\begin{aligned}&\lim _{T\rightarrow \infty }\frac{1}{T}\sum _{\tau =1}^{T-1}\sum _{i=1}^M{\mathbb {E}}\{Q_i(\tau )\} \le \frac{C_1+V\overline{P}^{opt}_{tot}}{\vartheta }. \end{aligned}$$

  (52)