Abstract
In this paper, an innovative technique is given to reduce the implementation time taken for doing the seamless communication in heterogeneous networks. When a new user arrives into the locality of one of the heterogeneous networks, resource allocation is to be done fastly by that heterogeneous network so as not to interrupt the data transfer to the newly arrived user. To do this, heterogeneous network allots optimal powers to S channels for maximizing the capacity of the newly arrived user while confirming to the interference bounds of the other users of the heterogeneous network and at the same time preserving the power (to save the energy) in heterogeneous network. To resolve this power allocation problem, the proposed algorithm finds out the tight lower bound and tight upper bound to the number of positive powers (\(=N\)) first and then finds out the indices of positive power allotted channels thereby discerning N itself (so as to calculate N powers exclusively). This is the unprecedented method as existing algorithms are iterative algorithms that deal with all of the assigned S channels. Hence, the proposed algorithm reduces the worst-case computational complexity of the erstwhile algorithms by the factor of \(O(S^2)\).
Similar content being viewed by others
Notes
|Y| is cardinality of Y.
\(\gamma _v\), v \(\le\) V are initialized to [0.01,0.011, \(\ldots\), 0.01 + (\(V-1\)) 0.001] and step size in the sub-gradient = \(\frac{1}{S}\). Number of iterations is given in brackets.
References
Helou, M. E., Ibrahim, M., Lahoud, S., et al. (2015). A network-assisted approach for rat selection in heterogeneous cellular networks. IEEE Journal on Selected Areas in Communications, 33(6), 1055–1067.
Yu, G., et al. (2015). Multi-objective energy-efficient resource allocation for multi-rat heterogeneous networks. IEEE Journal on Selected Areas in Communications, 33(10), 2118–2127.
Wang, C. X., Haider, F., et al. (2014). Cellular architecture and key technologies for 5 g wireless communication networks. IEEE Communications Magazine, 52(2), 122–130.
Chiu, K.-L. et al. (2011). Seamless session mobility scheme in heterogeneous wireless networks. International Journal of Communication Systems, 24(6), 789–809.
Wang, C. X., et al. (2014). Cellular architecture and key technologies for 5 g wireless communication networks. IEEE Communications Magazine, 52(2), 122–130.
Naidu, K., Kumar, R., & Vikas, K. (Jan 2017). The fastest possible solution to the weighted water-filling problems. In Proceedings of 7th IEEE IACC-2017, Hyderabad, India
Naidu, K., & Khan, M. Z. A. (2015). Fast computation of generalized water-filling problems. The IEEE Signal Processing Letters, 22(11), 1884–1887.
Naidu, K., & Battula, R. B. (Oct 2017). Quicker solution to reduce interference in wireless networks. In Communicated to IEEE transactions on vehicular technology, submission id : VT-2017-00275.
Naidu, K., Khan, M. Z. A., & Hanzo, L. (2016). An efficient direct solution of cave-filling problems. IEEE Transactions on Communications, 64(7), 3064–3077.
Xu, W., & Zhang, H. (2014). Uplink interference mitigation for heterogeneous networks with user-specific resource allocation and power control. EURASIP Journal on Wireless Communications and Networking, 2014(1), 55.
Zhang, X., Cheng, W., & Zhang, H. (2014). Heterogeneous statistical QOS provisioning over 5 g mobile wireless networks. IEEE Network, 28(6), 46–53.
Yu, G., Jiang, Y., Xu, L., & Li, G . Y. (2015). Multi-objective energy-efficient resource allocation for multi-rat heterogeneous networks. IEEE Journal on Selected Areas in Communications, 33(10), 2118–2127.
Khawam, K., Lahoud, S., Ibrahim, M., et al. (2016). Radio access technology selection in heterogeneous networks. Physical Communication, 18(P2), 125–139.
Naidu, K. (2016). Fast computation of waterfilling algorithms (online). http://raiith.iith.ac.in/2604/.
Zhang, H., et al. (2015). Resource allocation for cognitive small cell networks: A cooperative bargaining game theoretic approach. IEEE Transactions on Wireless Communications, 14(6), 3481–3493.
Vu, M. (2011). Miso capacity with per-antenna power constraint. IEEE Transactions on Communications, 59(5), 1268–1274.
Zhang, R. (2010). Cooperative multi-cell block diagonalization with per-base-station power constraints. IEEE Journal on Selected Areas in Communications, 28(9), 1435–1445.
Zhang, H., et al. (2017). Sensing time optimization and power control for energy efficient cognitive small cell with imperfect hybrid spectrum sensing. IEEE Transactions on Wireless Communications, 16(2), 730–743.
Tej, G., Nadkar, T., Thumar, V., Desai, U., & Merchant, S. (March 2011). Power allocation in cognitive radio: Single and multiple secondary users. In Wireless Communications and Networking Conference (WCNC), 2011 IEEE (pp. 1420–1425).
Hoshyar, R., Shariat, M., & Tafazolli, R. (2010). Subcarrier and power allocation with multiple power constraints in ofdma systems. IEEE Communications Letters, 14(7), 644–646.
Zhang, H., et al. (2016). Interference-limited resource optimization in cognitive femtocells with fairness and imperfect spectrum sensing. IEEE Transactions on Vehicular Technology, 65(3), 1761–1771.
Naidu, K., & Khan, M. Z. A. (Sep 2016). A fast algorithm for solving cave-filling problems. In Proceedings of IEEE 84th vehicular technology conference: VTC 2016-Fall, Montreal, Canada (pp. 18–21).
Kalpana, N., Khan, M. Z. A., & Desai, U. B. (Dec. 2011). Optimal power allocation for secondary users in CR networks. In Proceedings of IEEE ANTS 2011, Bangalore, India.
Zhang, H., et al. (2014). Resource allocation in spectrum-sharing ofdma femtocells with heterogeneous services. IEEE Transactions on Communications, 62(7), 2366–2377.
Vinh, N. V., Shouyi, Y., & Tran, L. C. (2014). Power allocation algorithm in ofdm-based cognitive radio systems. In Proceedings of ComManTel (pp. 13–18).
Bansal, G., Hossain, M. J., & Bhargava, V. K. (2008). Optimal and suboptimal power allocation schemes for OFDM-based cognitive radio systems. IEEE Transactions on Wireless Communications, 7(11), 4710–4718.
Zhang, L., Liang, Y.-C., & Xin, Y. (2008). Joint beamforming and power allocation for multiple access channels in cognitive radio networks. IEEE Journal on Selected Areas in Communications, 26(1), 38–51.
Ismail, M., & Zhuang, W. (2014). Green radio communications in a heterogeneous wireless medium. IEEE Wireless Communications, 21(3), 128–135.
Shaat, M., & Bader, F. (2010). Computationally efficient power allocation algorithm in multi carrier-based cognitive radio networks: Ofdm and fbmc systems. EURASIP Journal on Advances in Signal Processing, 2010, 528378.
Al-Imari, M., Xiao, P., Imran, M. A., & Tafazolli, R. (2013). Low complexity subcarrier and power allocation algorithm for uplink ofdma systems. EURASIP Journal on Wireless Communications and Networking, 2013(1), 98.
Naidu, K., Khan, M. Z. A, Desai, U. B. et al. (2014). A study on white and gray spaces in india. In A. K. Mishra & D. F. Johnson (Eds.), White space communication: Advances, developments and engineering challenges (pp. 49–73). doi:10.1007/978-3-319-08747-4_3.
Minimum mean square error. (online). http://en.wikipedia.org/wiki/Minimum_mean_square_error.
Estimation. (online). http://www.stanford.edu/class/ee363/lectures/estim.pdf.
Lee, D. Q. (2012). Numerically efficient methods for solving least squares problems (online). http://math.uchicago.edu/~may/REU2012/REUPapers/Lee.pdf.
Singular value decomposition. (online). http://en.wikipedia.org/wiki/Singular_value_decomposition.
Kwatra, V., & Han, M. Fast covariance computation and dimensionality reduction for sub-window features in images (Online). https://pdfs.semanticscholar.org/9615/6d48c668648fe783673460112d8ed5e3cdf3.pdf.
Zheng, L., & Tse, D. N. C. (2003). Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels. IEEE Transactions on Information Theory, 49(5), 1073–1096.
Want, R. (2009). When cell phones become computers. IEEE Pervasive Computing, 8(2), 2–5.
Wikipedia. (2015). Covariance (Online). http://en.wikipedia.org/wiki/Covariance.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
1.1 Proof for Proposition 5:
Proof
The covariance relation between two random variables A and B can be described as [39]
wherein \(E\left( A\right) = \frac{\sum _{v=1}^{V} a_v }{V}\) is mean of A random variable and \(Cov\left( A , B \right)\) is covariance of A and B random variables. (52) is used in the following proof.
The below given proof is comprised of three segments.
Part 4: Finding bound for the water level \({\frac{1}{ E\left( \gamma _v\right) }}\)
Figure 8 gives the water level for vth user in OPAN. In Fig. 8, the noise levels (\({R}^{*}_{mv}\), m = 1, 2, \(\ldots\)) are arranged as bottom steps of a container. When water is poured above the bottom steps (or noise levels) upto the water level (\(= \frac{1}{\gamma _{v}}\)), the amount of water that is there above the mth noise level (or power allotted to the mth channel) is \({P}^{*}_{mv} = \frac{1}{\gamma _{v}}-{R}^{*}_{mv}\). The area of occupied water (or slashed lines part in Fig. 8) is \(I_v\).
From Fig. 8, it can be grasped that the water level \(\frac{1}{\gamma _v}\) \(\propto\) \(I_v\) because the height of water level is directly dependent on the total amount (or area) of occupied water. Hence, the covariance between \(\gamma _v\) and \(I_v\) is negative. Applying this in (52) for \(\gamma _v\) and \(I_v\) variables, we get
In (53),
Likewise, the water level (\(\frac{1}{\gamma _v}\)) > the noise levels \(\left( {R}^{*}_{mv}, m \in X_N \right)\) since power (\(P^{*}_{mv} = \frac{1}{\gamma _v}\) - \({R}^{*}_{mv}\)) > 0, m \(\in X_N\) [7]. In Fig. 8, \(X_N\) is given as {1,2, \(\ldots\), N} for better understanding of the concepts. This further implies that the height of water level (\(\frac{1}{\gamma _v}\)) \(\propto {R}^{*}_{mv}, m \le N\). Consecutively, the covariance between \(\gamma _v\) and \({R}^{*}_{mv}\) is negative. Applying this concept in (52) for \(\gamma _v\) and \({R}^{*}_{mv}\) variables, we get
(9)–(10) is again given here for convenience.
Substituting (59) in (58); and then simplifying it further, we obtain
After using the definition of (55) in (60), (60) becomes
Applying (54) and (57) in (61) , we procure
Part 5: Procuring an additional bound for water level \({\frac{1}{ E\left( \gamma _v\right) }}\)
This part of the proof is given by contradiction. That is, the relation for water level for \(m \notin X_N\) is derived, and then, contradictory relation is applied for the water level for \(m \in X_N\).
(17) identifies that the channels that are assigned with zero powers have
Applying the definition of (55) in (64) to get
Figure 8 points out that \({R}^{*}_{(N+1)v}>\) the water level (\(\frac{1}{\gamma _v}\)) [7]. Then, it is evident that as water level (\(\frac{1}{\gamma _v}\)) increases, \({R}^{*}_{(N+1)v}\) also increases. In other words, \({R}^{*}_{(N+1)v} \propto \frac{1}{\gamma _v}\). But, \((N+1)\,\notin X_N\). Therefore, (57) can be applied as well to \(m \notin X_N\). That is,
Substituting (66) in (65), we attain
Contrary to the above, the opposite to (68) occurs for \(m \in X_N\). That is,
Along with (15), one more similarity can be observed in between WFP and OPAN now. For vth exclusive WFP, \(\frac{1}{\gamma _{_{v,WFP}}} > {R}^{*}_{Nv}\); whereas for OPAN, \(\frac{1}{ E\left( \gamma _v \right) } > \,\sum _{v=1}^{V} {R}^{*}_{Nv}\) with \(N = |X_N|\).
Part 6: Merging the above two proof parts to get the upper bound \({U_{N1}}\)
Water level for vth exclusive WFP is [7]
Likewise, the water level calculated for sum of noise levels \(\left( {R}^{*}_{m} = \sum _{v=1}^{V} {R}^{*}_{mv}\right)\) is :
The above is obtained from (70). Water level calculated for the sum of noise levels is shown in Fig. 9. Comparing both the water levels from (71) and (72), it can be perceived here that
-
1.
Water level for vth exclusive WFP (\(=\frac{1}{\gamma _{_{v, WFP}}}\)) is calculated for the individual noise levels \({R}^{*}_{mv}\) where as \(\frac{1}{\gamma _{_{sum}}}\) is calculated for the sum of noise levels \({R}^{*}_{m}\) = \(\sum _{v=1}^{V} {R}^{*}_{mv}\).
-
2.
The area of water occupied in vth exclusive WFP is \(I_{v}\); whereas the area of water occupied for the water level of \(\frac{1}{\gamma _{_{sum}}}\) is \(\sum _{v=1}^{V} I_{v}\).
-
3.
For vth exclusive WFP, \(\frac{1}{\gamma _{_{v, WFP}}}\) > \({R}^{*}_{mv}\), m \(\in X_N\) where as \(\frac{1}{\gamma _{_{sum}}} > \sum _{v=1}^{V} {R}^{*}_{mv}\), m \(\in X_N\) from (70).
(69) infers that ‘\(\frac{1}{ E\left( \gamma _v\right) } > \sum _{v=1}^{V} {R}^{*}_{mv}\)’ is followed for \(X_N\) values of m. Further, ‘\(\frac{1}{\gamma _{_{sum}}} > \frac{1}{ E\left( \gamma _v\right) }\)’ from (70). Then, it can be inferred that ‘\(\frac{1}{\gamma _{_{sum}}} > \sum _{v=1}^{V} {R}^{*}_{mv}\)’ is fulfilled either for \(X_N\) values of m (or) for more than \(X_N\) values of m (i.e. for \(U_{N1} > |X_N|\)).
Part 7: Obtaining another upper bound \({U_{N2}}\)
‘\(\frac{1}{ E\left( \gamma _v\right) }\)’ represents the water level for the mean of \(\gamma _v\)’s. Taking into account that OPAN includes vth exclusive WFP, upper bound for the number of positive powers can also be obtained from all the V exclusive WFPs. Subsequently, the upper bound \(U_{N2}\) becomes equivalent to the mean of \(X_{_{v,WFP}}\), \(v \le V\) wherein \(X_{_{v,WFP}}\) is attained as the ‘m’ values for which \(\{ \frac{1}{\gamma _{_{v, WFP}}} \ge {R}^{*}_{mv}, \hbox { m } \le S \}\) is fulfilled.
Part 8: Upper bound from \({U_{N1}}\) and \({U_{N2}}\)
Finally, the upper bound for the number of positive powers in OPAN is \(U_{N}\) = maximum of \(U_{N1}\) and \(U_{N2}\).
Hence, the Proposition is proved.\(\square\)
Rights and permissions
About this article
Cite this article
Naidu, K., Battula, R.B. Quick resource allocation in heterogeneous networks. Wireless Netw 24, 3171–3188 (2018). https://doi.org/10.1007/s11276-017-1527-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11276-017-1527-9