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Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis

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Abstract

This paper considers max-min fair rate allocation and routing in energy harvesting networks where fairness is required among both the nodes and the time slots. Unlike most previous work on fairness, we focus on multihop topologies and consider different routing methods. We assume a predictable energy profile and focus on the design of efficient and optimal algorithms that can serve as benchmarks for distributed and approximate algorithms. We first develop an algorithm that obtains a max-min fair rate assignment for any routing that is specified at the input. We then turn to the problem of determining a “good” routing. For time-invariable unsplittable routing, we develop an algorithm that finds routes that maximize the minimum rate assigned to any node in any slot. For fractional routing, we derive a combinatorial algorithm for the time-invariable case with constant rates. We show that the time-variable case is at least as hard as the 2-commodity feasible flow problem and design an FPTAS to combat the high running time. Finally, we show that finding an unsplittable routing or a routing tree that provides lexicographically maximum rate assignment (i.e., the best in the max-min fairness terms) is NP-hard, even for a time horizon of a single slot. Our analysis provides insights into the problem structure and can be applied to other related fairness problems.

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Notes

  1. 1.

    Note that node i’s sensing rate (generated flow) can change over time, even though the routing does not change.

  2. 2.

    \(\widetilde{O}(.)\)-notation hides poly-log terms.

  3. 3.

    The notions of max-min fairness and lexicographical ordering of vectors are defined in Sect. 2.1.

  4. 4.

    Note that we treat Eq. (2) as a linear constraint, since the considered problems focus on maximizing \(\lambda _{i, t}\)’s (under the max-min fairness criterion), and (2) can be replaced by \(b_{i, t+1}\le B\) and \(b_{i, t+1}\le b_{i, t}+e_{i,t}- (c_{\text {rt}}f^{\Sigma }_{i, t} + c_{\text {st}}\lambda _{i, t})\) while leading to the same solution.

  5. 5.

    Notice that this is consistent with the definition of a descendant in a routing tree.

  6. 6.

    P is determined by linear equalities and inequalities, which implies that it is convex.

References

  1. 1.

    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Inc, Upper Saddle River, NJ (1993)

  2. 2.

    Bacinoglu, B., Uysal-Biyikoglu, E.: Finite-horizon online transmission rate and power adaptation on a communication link with Markovian energy harvesting. (2013). ArXiv preprint, http://arxiv.org/abs/1305.4558

  3. 3.

    Bertsekas, D., Gallager, R.: Data Networks, 2nd edn. Prentice-Hall Inc, Upper Saddle River, NJ (1992)

  4. 4.

    Blasco, P., Gunduz, D., Dohler, M.: A learning theoretic approach to energy harvesting communication system optimization. In: Proceedings of IEEE Globecom Workshops’12 (2012)

  5. 5.

    Buragohain, C., Agrawal, D., Suri, S.: Power aware routing for sensor databases. In: Proceedings of IEEE INFOCOM’05 (2005)

  6. 6.

    Chang, J.H., Tassiulas, L.: Maximum lifetime routing in wireless sensor networks. IEEE/ACM Trans. Netw. 12(4), 609–619 (2004)

  7. 7.

    Charny, A., Clark, D.D., Jain, R.: Congestion control with explicit rate indication. In: Proceedings of IEEE ICC’95 (1995)

  8. 8.

    Chen, S., Fang, Y., Xia, Y.: Lexicographic maxmin fairness for data collection in wireless sensor networks. IEEE Trans. Mob. Comput. 6(7), 762–776 (2007)

  9. 9.

    Chen, S., Sinha, P., Shroff, N., Joo, C.: Finite-horizon energy allocation and routing scheme in rechargeable sensor networks. In: Proceedings of IEEE INFOCOM’11 (2011)

  10. 10.

    Chen, S., Sinha, P., Shroff, N., Joo, C.: A simple asymptotically optimal energy allocation and routing scheme in rechargeable sensor networks. In: Proceedings of IEEE INFOCOM’12 (2012)

  11. 11.

    DeBruin, S., Campbell, B., Dutta, P.: Monjolo: an energy-harvesting energy meter architecture. In: ACM SenSys’13 (2013)

  12. 12.

    Gatzianas, M., Georgiadis, L., Tassiulas, L.: Control of wireless networks with rechargeable batteries. IEEE Trans. Wirel. Commun. 9(2), 581–593 (2010)

  13. 13.

    Gorlatova, M., Bernstein, A., Zussman, G.: Performance evaluation of resource allocation policies for energy harvesting devices. In: Proceedings of WiOpt’11 (2011)

  14. 14.

    Gorlatova, M., Kinget, P., Kymissis, I., Rubenstein, D., Wang, X., Zussman, G.: Challenge: ultra-low-power energy-harvesting active networked tags (EnHANTs). In: Proceedings of ACM MobiCom’09 (2009)

  15. 15.

    Gorlatova, M., Margolies, R., Sarik, J., Stanje, G., Zhu, J., Vigraham, B., Szczodrak, M., Carloni, L., Kinget, P., Kymissis, I., Zussman, G.: Energy harvesting active networked tags (EnHANTs): prototyping and experimentation. Technical Report 2012-07-27, Columbia University (2012)

  16. 16.

    Gorlatova, M., Wallwater, A., Zussman, G.: Networking low-power energy harvesting devices: measurements and algorithms. IEEE Trans. Mob. Comput. 12(9), 1853–1865 (2013)

  17. 17.

    Gurakan, B., Ozel, O., Yang, J., Ulukus, S.: Energy cooperation in energy harvesting two-way communications. In: Proceedings of IEEE ICC’13 (2013)

  18. 18.

    Huang, L., Neely, M.: Utility optimal scheduling in energy-harvesting networks. IEEE/ACM Trans. Netw. 21(4), 1117–1130 (2013)

  19. 19.

    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

  20. 20.

    Kleinberg, J.: Single-source unsplittable flow. In: Proceedings of IEEE FOCS’96 (1996)

  21. 21.

    Kleinberg, J., Rabani, Y., Tardos, E.: Fairness in routing and load balancing. In: Proceedings of IEEE FOCS’99 (1999)

  22. 22.

    Lenstra, J., Shmoys, D., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1–3), 259–271 (1990)

  23. 23.

    Lin, L., Shroff, N., Srikant, R.: Asymptotically optimal energy-aware routing for multihop wireless networks with renewable energy sources. IEEE/ACM Trans. Netw. 15(5), 1021–1034 (2007)

  24. 24.

    Liu, R.S., Fan, K.W., Zheng, Z., Sinha, P.: Perpetual and fair data collection for environmental energy harvesting sensor networks. IEEE/ACM Trans. Netw. 19(4), 947–960 (2011)

  25. 25.

    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)

  26. 26.

    Madan, R., Lall, S.: Distributed algorithms for maximum lifetime routing in wireless sensor networks. IEEE Trans. Wirel. Commun. 5(8), 2185–2193 (2006)

  27. 27.

    Mao, Z., Koksal, C., Shroff, N.: Near optimal power and rate control of multi-hop sensor networks with energy replenishment: Basic limitations with finite energy and data storage. IEEE Trans. Autom. Control 57(4), 815–829 (2012)

  28. 28.

    Megiddo, N.: Optimal flows in networks with multiple sources and sinks. Math. Program. 7(1), 97–107 (1974)

  29. 29.

    Ozel, O., Tutuncuoglu, K., Yang, J., Ulukus, S., Yener, A.: Transmission with energy harvesting nodes in fading wireless channels: optimal policies. IEEE J. Sel. Areas Commun. 29(8), 1732–1743 (2011)

  30. 30.

    Plotkin, S., Shmoys, D., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Math. Oper. Res. 20(2), 257–301 (1995)

  31. 31.

    Radunović, B., Boudec, J.Y.L.: A unified framework for max-min and min-max fairness with applications. IEEE/ACM Trans. Netw. 15(5), 1073–1083 (2007)

  32. 32.

    Sarkar, S., Khouzani, M., Kar, K.: Optimal routing and scheduling in multihop wireless renewable energy networks. IEEE Trans. Autom. Control 58(7), 1792–1798 (2013)

  33. 33.

    Sarkar, S., Tassiulas, L.: Fair allocation of discrete bandwidth layers in multicast networks. In: Proceedings of IEEE INFOCOM’00 (2000)

  34. 34.

    Srivastava, R., Koksal, C.: Basic performance limits and tradeoffs in energy-harvesting sensor nodes with finite data and energy storage. IEEE/ACM Trans. Netw. 21(4), 1049–1062 (2013)

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Acknowledgments

The authors are grateful to Mihalis Yannakakis and David Johnson for useful discussions.

Author information

Correspondence to Jelena Marašević.

Additional information

A partial and preliminary version of this paper appeared in Proc. ACM MobiHoc’14. This research was supported in part by NSF Grants CCF-1349602, CCF-1421161, CCF-09-64497, and CNS-10-54856 and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. [PIIF-GA-2013-629740].11.

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Marašević, J., Stein, C. & Zussman, G. Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis. Algorithmica 78, 521–557 (2017). https://doi.org/10.1007/s00453-016-0171-6

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Keywords

  • Energy harvesting
  • Energy adaptive networking
  • Network flows
  • Sensor networks
  • Routing
  • Fairness