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 timevarying channel. To this end, we formulate the stochastic optimization problem to minimize the timeaveraged power consumption, whilst keeping all queues at the basestation 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 mixedinteger programming problems, which are then solved via the dual decomposition technique. We determine analytical bounds for the timeaveraged 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 energyqueue tradeoff), and quantitatively strike the energyqueue tradeoff by simply tuning an introduced control parameter V.
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Notes
 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.
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.
 2.
LopezPerez, 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.
 3.
Chang, T., Feng, K., Lin, J., & Wang, L. (2013). Green resource allocation schemes for relayenhanced MIMOOFDM networks. IEEE Transactions on Vehicular Technology, 62(9), 4539–4554.
 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.
 5.
Pao, W., Chen, Y., & Tsai, M. (2014). An adaptive allocation scheme in multiuser OFDM systems with timevarying channels. IEEE Transactions on Wireless Communications, 13(2), 669–679.
 6.
Kim, S., Lee, B. G., & Park, D. (2013). Radio resource allocation for energy consumption minimization in multihomed wireless networks. In IEEE international conference on communications, pp. 5589–8894
 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.
 8.
Gursoy, M. C. (2009). On the capacity and energy efficiency of the trainingbase transmissions over fading channels. IEEE Transactions on Information Theroy, 55(10), 4543–4567.
 9.
Adireddy, S., Tong, L., & Viswanathan, H. (2002). Optimal placement of training for frequencyselective blockfading channels. IEEE Transactions on Information Theory, 48(8), 2338–2353.
 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.
 11.
Awad, M. K., Mahinthan, V., Mehrjoo, M., Shen, X., & Mark, J. W. (2010). A dualdecompositionbased resource allocation for OFDMA networks with imperfect CSI. IEEE Transactions on Vehicular Technology, 59(5), 2394–2403.
 12.
Wang, J., Su, Q., Wang, J., Feng, M., Chen, M., Jiang, B., et al. (2014). Imperfect CSIbased joint resource allocaiton in multirelay OFDMA networks. IEEE Transactions on Vehicular Technology, 63(8), 3806–3817.
 13.
Neely, M. J. (2006). Energy optimal control for timevarying wireless networks. IEEE Transactions on Information Theory, 52(7), 2915–2934.
 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.
 15.
Ju, H., Liang, B., Li, J., & Yang, X. (2013). Dynamic joint resource optimization for LETadvanced relay networks. IEEE Transactions on Wireless Communications, 12(11), 5668–5678.
 16.
Sheng, M., Li, Y., Wang, X., Li, J., & Shi, Y. (2016). Energyefficiency and delay tradeoff in devicetodevice communications underlaying cellular networks. IEEE Journal on Selected Areas in Communications, 34(1), 92–106.
 17.
Ahmed, H., Jagannathan, K., & Bhashyam, S. (2015). Queueaware optimal resource allocation for the LET downlink with best M subband feedback. IEEE Transactions on Wireless Communications, 14(9), 4923–4933.
 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.
 19.
Georgiadis, L., Neely, M. J., & Tassiulas, L. (2006). Resource allocation and crosslayer control in wireless networks. Foundations and Trends in Networking, 1(1), 1–144.
 20.
Neely, M. J. (2010). Stochastic network optimization with application to communication and queueing systems. San Rafael, CA: Morgan & Claypool.
 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.
 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.
 23.
Ju, H., Liang, B., Li, J., & Yang, X. (2013). Dynamic joint resource optimization for LTEadvanced relay networks. IEEE Transactions on Wireless Communications, 12(11), 5668–5678.
 24.
Boyd, S., & vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.
 25.
Neely, M. J. (2013). Dynamic optimization and learning for renewal systems. IEEE Transactions on Automatic Control, 58(1), 32–46.
 26.
Bertsekas, D., & Gallager, R. (1987). Data networks. Upper Saddle River: PrenticeHall.
Acknowledgements
This research was supported in part by NSF China (61471287), 111 Project (B08038) and MSIT, Korea, under ITRC Program (IITP20172014000729) supervised by IITP.
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Appendix
Appendix
Proof of Theorem 3
According to Lemma 1, for the proposed DPSAI algorithm, we obtain
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
where \(\overline{P}^{opt}_{tot}\) is the minimum timeaveraged power expenditure over all feasible resource policies. Substituting (43) and (44) into (42), we get the following inequation as \(\xi \rightarrow 0\).
Plugging (16) into (45) and using telescoping sums over \(\tau \in \{0,1,\ldots ,T1\}\) yield

(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
$$\begin{aligned}&V\sum _{\tau =1}^{T1}{\mathbb {E}}\left\{ P_{tot}(\tau )\right\} \le TC_1+TV\overline{P}^{opt}_{tot} +{\mathbb {E}}\left\{ L\left( {\mathbf {Q}}(0)\right) \right\} . \end{aligned}$$(49)Dividing inequality (49) by VT and taking a limit as \(T\rightarrow \infty \) yields
$$\begin{aligned}&\lim _{T\rightarrow \infty }\frac{1}{T}\sum _{\tau =1}^{T1}{\mathbb {E}}\left\{ P_{tot}(\tau )\right\} \le \frac{C_1}{V}+\overline{P}^{opt}_{tot}. \end{aligned}$$(50) 
(c)
Similarly, we rewritten (46) as
$$\begin{aligned}&\frac{1}{T}\sum _{\tau =1}^{T1}\sum _{i=1}^M{\mathbb {E}}\{Q_i(\tau )\} \le \frac{C_1+V\overline{P}^{opt}_{tot}}{\vartheta }+\frac{{\mathbb {E}}\{L({\mathbf {Q}}(0))\}}{T\vartheta }. \end{aligned}$$(51)Taking a limit as \(T\rightarrow \infty \), we prove
$$\begin{aligned}&\lim _{T\rightarrow \infty }\frac{1}{T}\sum _{\tau =1}^{T1}\sum _{i=1}^M{\mathbb {E}}\{Q_i(\tau )\} \le \frac{C_1+V\overline{P}^{opt}_{tot}}{\vartheta }. \end{aligned}$$(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/s1127601715742
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Keywords
 Imperfect CSI
 Channel estimation
 Queue stability
 Dynamic power and subcarrier allocation