# American Institute of Mathematical Sciences

2011, 1(4): 675-689. doi: 10.3934/naco.2011.1.675

## Analysis of the statistical time-access fairness index of one-bit feedback fair scheduler

 1 Department of Systems Design and Engineering, Nanzan University, 27 Seirei, Seto, Aichi 489-0863, Japan

Received  June 2011 Revised  September 2011 Published  November 2011

Recently various schedulers exploiting multiuser diversity in wireless networks have been proposed and studied. Although the utilization of multiuser diversity can increase the information theoretic capacity, there exists a tradeoff between the capacity and fairness. Among schedulers exploiting multiuser diversity, the one-bit feedback fair scheduler is considered as an attractive choice for the reduction of feedback overheads and the ease of implementation. In this paper, we study the short term fairness of the one-bit feedback fair scheduler. Since the short term fairness has a strong impact on the quality-of-service of each mobile station, it is important to examine the short term fairness properties of the scheduler. As a short term fairness index, we consider the statistical time-access fairness index (STAFI). We then develop two numerical methods to estimate the STAFI of the scheduler. The first method calculates the exact value of the STAFI by using the inverse discrete FFT method. The second method estimates the asymptotic decay rate of the STAFI by using the theory of large deviations. Numerical results show that the threshold value of the one-bit feedback fair scheduler greatly affects its short term fairness properties.
Citation: Fumio Ishizaki. Analysis of the statistical time-access fairness index of one-bit feedback fair scheduler. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 675-689. doi: 10.3934/naco.2011.1.675
##### References:
 [1] J. A. Bucklew, "Large Deviation Techniques in Decision, Simulation and Estimation,", Wiley-Interscience, (1990). Google Scholar [2] C. S. Chang, "Performance Guarantees in Communication Networks,", Springer-Verlag, (2000). doi: 10.1007/978-1-4471-0459-9. Google Scholar [3] J. Diaz, O. Simeone and Y. Bar-Ness, Sum-rate of MIMO broadcast channels with one bit feedback,, in ''Proc. of IEEE International Symposium on Information Theory (ISIT '06), (2006), 1944. Google Scholar [4] F. Florén, O. Edfors and B. A. Molin, The effect of feedback quantization on the throughput of a multiuser diversity scheme,, in ''Proc. of IEEE GLOBECOM 2003, (2003), 497. Google Scholar [5] D. Gesbert and M. S. Alouini, How much feedback is multi-user diversity really worth?,, in ''Proc. of IEEE ICC '04, (2004), 234. Google Scholar [6] G. U. Hwang and F. Ishizaki, Design of a fair scheduling exploiting multiuser diversity with feedback reduction,, IEEE Communications Letters, 12 (2008), 124. doi: 10.1109/LCOMM.2008.071547. Google Scholar [7] G. U. Hwang and F. Ishizaki, Analysis of short term fairness and its impact on packet level performance,, Performance Evaluation, 67 (2010), 1340. doi: 10.1016/j.peva.2010.08.021. Google Scholar [8] F. Ishizaki and G. U. Hwang, Queuing delay analysis for packet schedulers with/without multiuser diversity over a fading channel,, IEEE Trans. Veh. Technol., 56 (2007), 3220. doi: 10.1109/TVT.2007.900523. Google Scholar [9] F. Ishizaki and G. U. Hwang, Throughput performance of quantized proportional fair scheduling with adaptive modulation and coding,, in ''Proc. of Wireless Telecommunications Symposium (WTS) 2009, (2009). Google Scholar [10] H. Kim and Y. Han, An opportunistic channel quality feedback scheme for proportional fair scheduling,, IEEE Communications Letters, 11 (2007), 501. doi: 10.1109/LCOMM.2007.070106. Google Scholar [11] R. Knopp and P. A. Humblet, Information capacity and power control in single-cell multiuser communications,, in ''Proc. of IEEE ICC '95, (1995), 331. doi: 10.1109/ICC.1995.525188. Google Scholar [12] Q. Liu, S. Zhou and G. B. Giannakis, Queuing with adaptive modulation and coding over wireless links: cross-layer analysis and design,, IEEE Trans. Wireless Commun., 4 (2005), 1142. doi: 10.1109/TWC.2005.847005. Google Scholar [13] Y. Liu, S. Gruhl and E. W. Knightly, WCFQ: an opportunistic wireless scheduler with statistical fairness bounds,, IEEE Trans. Wireless Commun., 2 (2003), 1017. doi: 10.1109/TWC.2003.816777. Google Scholar [14] X. Qin and R. Berry, Exploiting multiuser diversity for medium access control in wireless networks,, in ''Proc. of IEEE INFOCOM '03, (2003), 1084. Google Scholar [15] S. Sanayei and A. Nosratinia, Opportunistic downlink transmission with limited feedback,, IEEE Trans. Infor. Theory, 53 (2007), 4363. doi: 10.1109/TIT.2007.899529. Google Scholar [16] O. Somekh, A. M. Haimovich and Y. Bar-Ness, Sum-rate analysis of downlink channels with 1-bit feedback,, IEEE Communications Letters, 11 (2007), 137. doi: 10.1109/LCOMM.2007.061464. Google Scholar [17] H. C. Tijms, "A First Course in Stochastic Models,", John Wiley & Sons, (2003). doi: 10.1002/047001363X. Google Scholar [18] B. Tan, L. Ying and R. Srikant, Short-term fairness and long-term QoS,, in ''Proc. of Conference on Information Science and Systems (CISS), (2008), 1201. Google Scholar [19] D. Wu and R. Negi, Utilizing multiuser diversity for efficient support of quality of service over a fading channel,, IEEE Trans. Veh. Technol., 54 (2005), 1198. doi: 10.1109/TVT.2005.844671. Google Scholar [20] Y. Xue and T. Kaiser, Exploiting multiuser diversity with imperfect one-bit channel state feedback,, IEEE Trans. Veh. Technol., 56 (2007), 183. doi: 10.1109/TVT.2006.883784. Google Scholar [21] M. D. Yacoub, "Foundation of Mobile Radio Engineering,", CRC-Press, (1993). Google Scholar [22] L. Yang, M. Kang and M.-S. Alouini, On the capacity-fairness tradeoff in multiuser diversity systems,, IEEE Trans. Veh. Technol., 56 (2007), 1901. doi: 10.1109/TVT.2007.897229. Google Scholar [23] W. Zhang and K. B. Letaief, MIMO broadcast scheduling with limited feedback,, IEEE J. Sel. Areas Commun., 25 (2007), 1457. doi: 10.1109/JSAC.2007.070918. Google Scholar

show all references

##### References:
 [1] J. A. Bucklew, "Large Deviation Techniques in Decision, Simulation and Estimation,", Wiley-Interscience, (1990). Google Scholar [2] C. S. Chang, "Performance Guarantees in Communication Networks,", Springer-Verlag, (2000). doi: 10.1007/978-1-4471-0459-9. Google Scholar [3] J. Diaz, O. Simeone and Y. Bar-Ness, Sum-rate of MIMO broadcast channels with one bit feedback,, in ''Proc. of IEEE International Symposium on Information Theory (ISIT '06), (2006), 1944. Google Scholar [4] F. Florén, O. Edfors and B. A. Molin, The effect of feedback quantization on the throughput of a multiuser diversity scheme,, in ''Proc. of IEEE GLOBECOM 2003, (2003), 497. Google Scholar [5] D. Gesbert and M. S. Alouini, How much feedback is multi-user diversity really worth?,, in ''Proc. of IEEE ICC '04, (2004), 234. Google Scholar [6] G. U. Hwang and F. Ishizaki, Design of a fair scheduling exploiting multiuser diversity with feedback reduction,, IEEE Communications Letters, 12 (2008), 124. doi: 10.1109/LCOMM.2008.071547. Google Scholar [7] G. U. Hwang and F. Ishizaki, Analysis of short term fairness and its impact on packet level performance,, Performance Evaluation, 67 (2010), 1340. doi: 10.1016/j.peva.2010.08.021. Google Scholar [8] F. Ishizaki and G. U. Hwang, Queuing delay analysis for packet schedulers with/without multiuser diversity over a fading channel,, IEEE Trans. Veh. Technol., 56 (2007), 3220. doi: 10.1109/TVT.2007.900523. Google Scholar [9] F. Ishizaki and G. U. Hwang, Throughput performance of quantized proportional fair scheduling with adaptive modulation and coding,, in ''Proc. of Wireless Telecommunications Symposium (WTS) 2009, (2009). Google Scholar [10] H. Kim and Y. Han, An opportunistic channel quality feedback scheme for proportional fair scheduling,, IEEE Communications Letters, 11 (2007), 501. doi: 10.1109/LCOMM.2007.070106. Google Scholar [11] R. Knopp and P. A. Humblet, Information capacity and power control in single-cell multiuser communications,, in ''Proc. of IEEE ICC '95, (1995), 331. doi: 10.1109/ICC.1995.525188. Google Scholar [12] Q. Liu, S. Zhou and G. B. Giannakis, Queuing with adaptive modulation and coding over wireless links: cross-layer analysis and design,, IEEE Trans. Wireless Commun., 4 (2005), 1142. doi: 10.1109/TWC.2005.847005. Google Scholar [13] Y. Liu, S. Gruhl and E. W. Knightly, WCFQ: an opportunistic wireless scheduler with statistical fairness bounds,, IEEE Trans. Wireless Commun., 2 (2003), 1017. doi: 10.1109/TWC.2003.816777. Google Scholar [14] X. Qin and R. Berry, Exploiting multiuser diversity for medium access control in wireless networks,, in ''Proc. of IEEE INFOCOM '03, (2003), 1084. Google Scholar [15] S. Sanayei and A. Nosratinia, Opportunistic downlink transmission with limited feedback,, IEEE Trans. Infor. Theory, 53 (2007), 4363. doi: 10.1109/TIT.2007.899529. Google Scholar [16] O. Somekh, A. M. Haimovich and Y. Bar-Ness, Sum-rate analysis of downlink channels with 1-bit feedback,, IEEE Communications Letters, 11 (2007), 137. doi: 10.1109/LCOMM.2007.061464. Google Scholar [17] H. C. Tijms, "A First Course in Stochastic Models,", John Wiley & Sons, (2003). doi: 10.1002/047001363X. Google Scholar [18] B. Tan, L. Ying and R. Srikant, Short-term fairness and long-term QoS,, in ''Proc. of Conference on Information Science and Systems (CISS), (2008), 1201. Google Scholar [19] D. Wu and R. Negi, Utilizing multiuser diversity for efficient support of quality of service over a fading channel,, IEEE Trans. Veh. Technol., 54 (2005), 1198. doi: 10.1109/TVT.2005.844671. Google Scholar [20] Y. Xue and T. Kaiser, Exploiting multiuser diversity with imperfect one-bit channel state feedback,, IEEE Trans. Veh. Technol., 56 (2007), 183. doi: 10.1109/TVT.2006.883784. Google Scholar [21] M. D. Yacoub, "Foundation of Mobile Radio Engineering,", CRC-Press, (1993). Google Scholar [22] L. Yang, M. Kang and M.-S. Alouini, On the capacity-fairness tradeoff in multiuser diversity systems,, IEEE Trans. Veh. Technol., 56 (2007), 1901. doi: 10.1109/TVT.2007.897229. Google Scholar [23] W. Zhang and K. B. Letaief, MIMO broadcast scheduling with limited feedback,, IEEE J. Sel. Areas Commun., 25 (2007), 1457. doi: 10.1109/JSAC.2007.070918. Google Scholar
 [1] Gregory M. Zaverucha, Douglas R. Stinson. Short one-time signatures. Advances in Mathematics of Communications, 2011, 5 (3) : 473-488. doi: 10.3934/amc.2011.5.473 [2] Jiongmin Yong. Remarks on some short rate term structure models. Journal of Industrial & Management Optimization, 2006, 2 (2) : 119-134. doi: 10.3934/jimo.2006.2.119 [3] Yiju Wang, Wei Xing, Hengxia Gao. Optimal ordering policy for inventory mechanism with a stochastic short-term price discount. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018199 [4] Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011 [5] Francesca Faraci, Alexandru Kristály. One-dimensional scalar field equations involving an oscillatory nonlinear term. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 107-120. doi: 10.3934/dcds.2007.18.107 [6] Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034 [7] Qichun Wang, Chik How Tan, Pantelimon Stănică. Concatenations of the hidden weighted bit function and their cryptographic properties. Advances in Mathematics of Communications, 2014, 8 (2) : 153-165. doi: 10.3934/amc.2014.8.153 [8] Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671 [9] Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019 [10] José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149 [11] Dilek Günneç, Ezgi Demir. Fair-fixture: Minimizing carry-over effects in football leagues. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1565-1577. doi: 10.3934/jimo.2018110 [12] Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729 [13] B. Spagnolo, D. Valenti, A. Fiasconaro. Noise in ecosystems: A short review. Mathematical Biosciences & Engineering, 2004, 1 (1) : 185-211. doi: 10.3934/mbe.2004.1.185 [14] Shingo Iwami, Shinji Nakaoka, Yasuhiro Takeuchi. Mathematical analysis of a HIV model with frequency dependence and viral diversity. Mathematical Biosciences & Engineering, 2008, 5 (3) : 457-476. doi: 10.3934/mbe.2008.5.457 [15] Evariste Sanchez-Palencia, Jean-Pierre Françoise. Topological remarks and new examples of persistence of diversity in biological dynamics. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1775-1789. doi: 10.3934/dcdss.2019117 [16] Bin Li, Hai Huyen Dam, Antonio Cantoni. A low-complexity zero-forcing Beamformer design for multiuser MIMO systems via a dual gradient method. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012 [17] Chuan Ding, Kaihong Wang, Shaoyong Lai. Channel coordination mechanism with retailers having fairness preference ---An improved quantity discount mechanism. Journal of Industrial & Management Optimization, 2013, 9 (4) : 967-982. doi: 10.3934/jimo.2013.9.967 [18] Huimin Liu, Hui Yu. Fairness and retailer-led supply chain coordination under two different degrees of trust. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1347-1364. doi: 10.3934/jimo.2016076 [19] John D. Nagy. The Ecology and Evolutionary Biology of Cancer: A Review of Mathematical Models of Necrosis and Tumor Cell Diversity. Mathematical Biosciences & Engineering, 2005, 2 (2) : 381-418. doi: 10.3934/mbe.2005.2.381 [20] Xue Yan, Heap-Yih Chong, Jing Zhou, Zhaohan Sheng, Feng Xu. Fairness preference based decision-making model for concession period in PPP projects. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2018137

Impact Factor:

## Metrics

• PDF downloads (4)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]