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.

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Notes

  1. 1.

    Define the network capacity region \(\pmb {\varGamma }\) as the set of data arrival rate that can be stably supported by the network, considering all possible power and subcarrier allocation decisions, namely, there at lest exists a policy that stabilizes the network under this arrival rate [13].

References

  1. 1.

    Gortzen, S., & Schmeink, A. (2012). Optimality of dual methods for discrete multiuser multicarrier resource allocation problems. IEEE Transactions on Wireless Communications, 11(10), 3810–3817.

    Article  Google Scholar 

  2. 2.

    Lopez-Perez, D., Chu, X., Vasilakos, A. V., & Claussen, H. (2014). Power minimization based resource allocation for interference mitigation in OFDMA femtocell networks. IEEE Journal on Selected Areas in Communication, 32(2), 333–344.

    Article  Google Scholar 

  3. 3.

    Chang, T., Feng, K., Lin, J., & Wang, L. (2013). Green resource allocation schemes for relay-enhanced MIMO-OFDM networks. IEEE Transactions on Vehicular Technology, 62(9), 4539–4554.

    Article  Google Scholar 

  4. 4.

    Luo, S., Zhang, R., & Lim, T. J. (2014). Joint transmitter and receiver energy minimization in multiuser OFDM systems. IEEE Transactions on Communications, 62(10), 3504–3516.

    Article  Google Scholar 

  5. 5.

    Pao, W., Chen, Y., & Tsai, M. (2014). An adaptive allocation scheme in multiuser OFDM systems with time-varying channels. IEEE Transactions on Wireless Communications, 13(2), 669–679.

    Article  Google Scholar 

  6. 6.

    Kim, S., Lee, B. G., & Park, D. (2013). Radio resource allocation for energy consumption minimization in multi-homed wireless networks. In IEEE international conference on communications, pp. 5589–8894

  7. 7.

    Joung, J., Ho, C. K., Tan, P., & Sun, S. (2012). Energy minimization in OFDMA downlink systems: A sequential linear assignment algorithm for resource allocation. IEEE Wireless Communications Letters, 1(4), 300–303.

    Article  Google Scholar 

  8. 8.

    Gursoy, M. C. (2009). On the capacity and energy efficiency of the training-base transmissions over fading channels. IEEE Transactions on Information Theroy, 55(10), 4543–4567.

    Article  MATH  Google Scholar 

  9. 9.

    Adireddy, S., Tong, L., & Viswanathan, H. (2002). Optimal placement of training for frequency-selective block-fading channels. IEEE Transactions on Information Theory, 48(8), 2338–2353.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Wong, I. C., & Evans, B. L. (2009). Optimal resource allocation in OFDMA downlink with imperfect channle knowledge. IEEE Transactions on Communications, 57(1), 232–241.

    Article  Google Scholar 

  11. 11.

    Awad, M. K., Mahinthan, V., Mehrjoo, M., Shen, X., & Mark, J. W. (2010). A dual-decomposition-based resource allocation for OFDMA networks with imperfect CSI. IEEE Transactions on Vehicular Technology, 59(5), 2394–2403.

    Article  Google Scholar 

  12. 12.

    Wang, J., Su, Q., Wang, J., Feng, M., Chen, M., Jiang, B., et al. (2014). Imperfect CSI-based joint resource allocaiton in multirelay OFDMA networks. IEEE Transactions on Vehicular Technology, 63(8), 3806–3817.

    Article  Google Scholar 

  13. 13.

    Neely, M. J. (2006). Energy optimal control for time-varying wireless networks. IEEE Transactions on Information Theory, 52(7), 2915–2934.

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Urgaonkar, R., & Neely, M. J. (2009). Opportunistic scheduling with reliability guarantees in cognitive radio networks. IEEE Transactions on Mobile Computing, 8(6), 766–777.

    Article  Google Scholar 

  15. 15.

    Ju, H., Liang, B., Li, J., & Yang, X. (2013). Dynamic joint resource optimization for LET-advanced relay networks. IEEE Transactions on Wireless Communications, 12(11), 5668–5678.

    Article  Google Scholar 

  16. 16.

    Sheng, M., Li, Y., Wang, X., Li, J., & Shi, Y. (2016). Energy-efficiency and delay tradeoff in device-to-device communications underlaying cellular networks. IEEE Journal on Selected Areas in Communications, 34(1), 92–106.

    Article  Google Scholar 

  17. 17.

    Ahmed, H., Jagannathan, K., & Bhashyam, S. (2015). Queue-aware optimal resource allocation for the LET downlink with best M subband feedback. IEEE Transactions on Wireless Communications, 14(9), 4923–4933.

    Article  Google Scholar 

  18. 18.

    Wu, Y., Louie, R. H. Y., & McKay, M. R. (2013). Analysis and design of wireless ad hoc networks with estimation errors. IEEE Transactions on Signal Communications, 61(6), 1447–1459.

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Georgiadis, L., Neely, M. J., & Tassiulas, L. (2006). Resource allocation and cross-layer control in wireless networks. Foundations and Trends in Networking, 1(1), 1–144.

    Article  MATH  Google Scholar 

  20. 20.

    Neely, M. J. (2010). Stochastic network optimization with application to communication and queueing systems. San Rafael, CA: Morgan & Claypool.

    Google Scholar 

  21. 21.

    Huang, J., Subramanian, V., Agrawal, R., & Berry, R. (2009). Downlink scheduling and resource allocation for OFDM systems. IEEE Transactions on Wireless Communicatons, 8(1), 288–296.

    Article  Google Scholar 

  22. 22.

    Huang, J., Subramanian, V., Agrawal, R., & Berry, R. (2009). Joint scheduling and resource allocation in uplink OFDM systems for broadband wireless access networks. IEEE Journal on Selected Areas in Communications, 27(2), 226–234.

    Article  Google Scholar 

  23. 23.

    Ju, H., Liang, B., Li, J., & Yang, X. (2013). Dynamic joint resource optimization for LTE-advanced relay networks. IEEE Transactions on Wireless Communications, 12(11), 5668–5678.

    Article  Google Scholar 

  24. 24.

    Boyd, S., & vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.

    Google Scholar 

  25. 25.

    Neely, M. J. (2013). Dynamic optimization and learning for renewal systems. IEEE Transactions on Automatic Control, 58(1), 32–46.

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Bertsekas, D., & Gallager, R. (1987). Data networks. Upper Saddle River: Prentice-Hall.

    Google Scholar 

Download references

Acknowledgements

This research was supported in part by NSF China (61471287), 111 Project (B08038) and MSIT, Korea, under ITRC Program (IITP-2017-2014-0-00729) supervised by IITP.

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Correspondence to Qinghai Yang.

Appendix

Appendix

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)
  1. (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

    $$\begin{aligned} \lim _{T\rightarrow \infty }\frac{{\mathbb {E}}\left\{ Q_i(T)\right\} }{T}=0. \end{aligned}$$
  2. (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)
  3. (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)

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Liu, F., Yang, Q., He, Q. et al. Dynamic power and subcarrier allocation for downlink OFDMA systems under imperfect CSI. Wireless Netw 25, 545–558 (2019). https://doi.org/10.1007/s11276-017-1574-2

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Keywords

  • Imperfect CSI
  • Channel estimation
  • Queue stability
  • Dynamic power and subcarrier allocation