# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018194

## Utility maximization for bandwidth allocation in peer-to-peer file-sharing networks

 School of Economics and Management, Yanshan University, Qinhuangdao 066004, China

* Corresponding author: Wei Sun

Received  October 2017 Revised  January 2018 Published  December 2018

Fund Project: The authors were supported in part by the National Natural Science Foundation of China (Nos. 71671159, 71301139 and 71671158), the Humanity and Social Science Foundation of Ministry of Education of China (No. 16YJC630106), the Natural Science Foundation of Hebei Province (Nos. G2018203302 and G2016203236), the project Funded by Hebei Education Department (Nos. BJ2017029 and BJ2016063) and Hebei Talents Program (No. A2017002108)

Peer-to-peer (P2P) networks have been commonly applied into many applications such as distributed storage, cloud computing and social networking. In P2P networks fairness fosters an incentive so as to encourage peers to offer resources (e.g, upload bandwidth) to the networks. In this paper, we consider fair bandwidth allocation of access links in P2P file-sharing networks and develop a coupled network-wide utility maximization model which aims at achieving several kinds of fairness among requesting peers. We provide a meaningful interpretation of the problem of maximizing social welfare and its sub-problems from an economic point of view. The coupled optimization problem is difficult to resolve in a distributed way because of its non-strict convexity and non-separation. We apply a modified successive approximation method to investigate the coupled problem and propose a distributed bandwidth allocation scheme to solve the approximation problems. Then, we investigate the convergence of the scheme by mathematical analysis and evaluate the performance through numerical examples, which validate that the scheme can achieve the global optimum within reasonable iterations.

Citation: Shiyong Li, Wei Sun, Quan-Lin Li. Utility maximization for bandwidth allocation in peer-to-peer file-sharing networks. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018194
##### References:
 [1] M. Analoui and M. H. Rezvani, Microeconomics-based resource allocation in overlay networks by using non-strategic behavior modeling, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 493-508. doi: 10.1016/j.cnsns.2010.04.005. [2] E. Antal and T. Vinkó, Modeling max-min fair bandwidth allocation in BitTorrent communities, Computational Optimization and Applications, 66 (2017), 1-18. doi: 10.1007/s10589-016-9866-5. [3] D. P. Bertsekas, Nonlinear Programming, Second edition. Athena Scientific Optimization and Computation Series. Athena Scientific, Belmont, MA, 1999. [4] M. Chen, M. Ponec, S. Sengupta, J. Li and P. A. Chou, Utility maximization in peer-to-peer systems with applications to video conferencing, IEEE/ACM Transactions on Networking, 20 (2012), 1681-1694. [5] M. Chiang, S. H. Low, A. R. Calderbank and J. C. Doyle, Layering as optimization decomposition: A mathematical theory of network architectures, Proceedings of the IEEE, 95 (2007), 255-312. [6] M. Chiang, C. W. Tan, D. P. Palomar, D. O'Neill and D. Julian, Power control by geometric programming, IEEE Transactions on Wireless Communications, 6 (2007), 2640-2650. [7] K. Eger and U. Killat, Fair resource allocation in peer-to-peer networks (extended version), Computer Communications, 30 (2007), 3046-3054. [8] A. S. Elmaghraby, A. Kumar, M. M. Kantardzic and M. G. Mostafa, A scalable pricing model for bandwidth allocation, Electronic Commerce Research, 5 (2005), 203-227. [9] S. Jin, Y. Zhao, W. Yue and L. Chen., Performance analysis of a P2P storage system with a lazy replica repair policy, Journal of Industrial and Management Optimization, 10 (2014), 151-166. doi: 10.3934/jimo.2014.10.151. [10] F. P. Kelly, A. Maulloo and D. Tan, Rate control in communication networks: Shadow prices, proportional fairness and stability, Journal of the Operational Research Society, 49 (1998), 237-252. [11] I. Koutsopoulos and G. Iosifidis, A framework for distributed bandwidth allocation in peer-to-peer networks, Performance Evaluation, 67 (2010), 285-298. [12] C. Kumar, K. Altinkemer and P. De, A mechanism for pricing and resource allocation in peer-to-peer networks, Electronic Commerce Research and Applications, 10 (2011), 26-37. [13] S. Li and W. Sun, A mechanism for resource pricing and fairness in peer-to-peer networks, Electronic Commerce Research, 16 (2016), 425-451. [14] S. Li, W. Sun and C. Hua, Fair resource allocation and stability for communication networks with multipath routing, International Journal of Systems Science, 45 (2014), 2342-2353. doi: 10.1080/00207721.2013.769073. [15] S. Li, W. Sun and N. Tian, Resource allocation for multi-class services in multipath networks, Performance Evaluation, 92 (2015), 1-23. [16] Z. Li and Q. Liao, Network pricing: Can both ISP and P2P benefit?, International Journal of Network Management, 24 (2014), 433-449. [17] F. Lin, X. Zhou, D. Huang, Y. Chen and J. Yuan, Hierarchical name system based on hybrid P2P for multimedia networks, Telecommunication Systems, 59 (2015), 393-400. [18] P. Lin, P.-C. Chung and Y. Fang, P2P-iSN: A peer-to-peer architecture for heterogeneous social networks, IEEE Network, 28 (2014), 56-64. [19] B. R. Marks and G. P. Wright, A general inner approximation algorithm for nonconvex mathematical programs, Operations Research, 26 (1978), 681-683. doi: 10.1287/opre.26.4.681. [20] M. J. Neely and L. Golubchik, Utility optimization for dynamic peer-to-peer networks with tit-for-tat constraints, IEEE INFOCOM, 2011, 1458-1466. [21] H. Nishida and T. Nguyen, A global contribution approach to maintain fairness in P2P networks, IEEE Transactions on Parallel and Distributed Systems, 21 (2010), 812-826. [22] B. Qureshi, G. Min and D. Kouvatsos, A distributed reputation and trust management scheme for mobile peer-to-peer networks, Computer Communications, 35 (2012), 608-618. [23] S. Rho, H. Chang, S. Kim and Y. S. Lee, An efficient peer-to-peer and distributed scheduling for cloud and grid computing, Peer-to-Peer Networking and Applications, 8 (2014), 863-871. [24] I. Rubin and R. Zhang, Max-min utility fair flow management for networks with route diversity, International Journal of Network Management, 20 (2010), 361-381. [25] A. Satsiou and L. Tassiulas, Reputation-based resource allocation in P2P systems of rational users, IEEE Transactions on Parallel and Distributed Systems, 21 (2010), 466-479. [26] S. Shakkottai and R. Srikant, Network optimization and control, Foundations and Trends in Networking, 2 (2007), 271-379. [27] F. Song, D. Huang, H. Zhou, H. Zhang and I. You, An optimization-based scheme for efficient virtual machine placement, International Journal of Parallel Programming, 42 (2014), 853-872. [28] F. Song, Y. Zhang, Z. An, H. Zhou and I. You, The correlation study for parameters in four tuples, International Journal of Ad Hoc and Ubiquitous Computing, 19 (2015), 38-49. [29] N. H. Tran and C. S. Hong, Joint rate and power control in wireless network: A novel successive approximations method, IEEE Communications Letters, 14 (2010), 872-874. [30] P. L. Vo, T. A. Le, S. Lee, C. S. Hong, B. Kim and H. Song, MReno: A practical multipath congestion control for communication networks, Computing, 96 (2014), 189-205. doi: 10.1007/s00607-013-0341-1. [31] P. L. Vo, T. A. Le, S. Lee, C. S. Hong, B. Kim and H. Song, Multi-path utility maximization and multi-path TCP design, Journal of Parallel and Distributed Computing, 74 (2014), 1848-1857. [32] K. Wang, H. Yin, W. Quan and G. Min, Enabling collaborative edge computing for software defined vehicular networks, IEEE Network, 32 (2018), 112-117. [33] H. Yan, D. Gao, W. Su, C. H. Foh, H. Zhang and A. V. Vasilakos, Caching strategy based on hierarchical cluster for named data networking, IEEE Access, 5 (2017), 8433-8443. [34] K. Zhang and N. Antonopoulos, A novel bartering exchange ring based incentive mechanism for peer-to-peer systems, Future Generation Computer Systems, 29 (2013), 361-369. [35] Y. Zheng, F. Lin, Y. Yang and T. Gan, Adaptive resource scheduling mechanism in P2P file sharing system, Peer-to-Peer Networking and Applications, 9 (2016), 1089-1100.

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##### References:
 [1] M. Analoui and M. H. Rezvani, Microeconomics-based resource allocation in overlay networks by using non-strategic behavior modeling, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 493-508. doi: 10.1016/j.cnsns.2010.04.005. [2] E. Antal and T. Vinkó, Modeling max-min fair bandwidth allocation in BitTorrent communities, Computational Optimization and Applications, 66 (2017), 1-18. doi: 10.1007/s10589-016-9866-5. [3] D. P. Bertsekas, Nonlinear Programming, Second edition. Athena Scientific Optimization and Computation Series. Athena Scientific, Belmont, MA, 1999. [4] M. Chen, M. Ponec, S. Sengupta, J. Li and P. A. Chou, Utility maximization in peer-to-peer systems with applications to video conferencing, IEEE/ACM Transactions on Networking, 20 (2012), 1681-1694. [5] M. Chiang, S. H. Low, A. R. Calderbank and J. C. Doyle, Layering as optimization decomposition: A mathematical theory of network architectures, Proceedings of the IEEE, 95 (2007), 255-312. [6] M. Chiang, C. W. Tan, D. P. Palomar, D. O'Neill and D. Julian, Power control by geometric programming, IEEE Transactions on Wireless Communications, 6 (2007), 2640-2650. [7] K. Eger and U. Killat, Fair resource allocation in peer-to-peer networks (extended version), Computer Communications, 30 (2007), 3046-3054. [8] A. S. Elmaghraby, A. Kumar, M. M. Kantardzic and M. G. Mostafa, A scalable pricing model for bandwidth allocation, Electronic Commerce Research, 5 (2005), 203-227. [9] S. Jin, Y. Zhao, W. Yue and L. Chen., Performance analysis of a P2P storage system with a lazy replica repair policy, Journal of Industrial and Management Optimization, 10 (2014), 151-166. doi: 10.3934/jimo.2014.10.151. [10] F. P. Kelly, A. Maulloo and D. Tan, Rate control in communication networks: Shadow prices, proportional fairness and stability, Journal of the Operational Research Society, 49 (1998), 237-252. [11] I. Koutsopoulos and G. Iosifidis, A framework for distributed bandwidth allocation in peer-to-peer networks, Performance Evaluation, 67 (2010), 285-298. [12] C. Kumar, K. Altinkemer and P. De, A mechanism for pricing and resource allocation in peer-to-peer networks, Electronic Commerce Research and Applications, 10 (2011), 26-37. [13] S. Li and W. Sun, A mechanism for resource pricing and fairness in peer-to-peer networks, Electronic Commerce Research, 16 (2016), 425-451. [14] S. Li, W. Sun and C. Hua, Fair resource allocation and stability for communication networks with multipath routing, International Journal of Systems Science, 45 (2014), 2342-2353. doi: 10.1080/00207721.2013.769073. [15] S. Li, W. Sun and N. Tian, Resource allocation for multi-class services in multipath networks, Performance Evaluation, 92 (2015), 1-23. [16] Z. Li and Q. Liao, Network pricing: Can both ISP and P2P benefit?, International Journal of Network Management, 24 (2014), 433-449. [17] F. Lin, X. Zhou, D. Huang, Y. Chen and J. Yuan, Hierarchical name system based on hybrid P2P for multimedia networks, Telecommunication Systems, 59 (2015), 393-400. [18] P. Lin, P.-C. Chung and Y. Fang, P2P-iSN: A peer-to-peer architecture for heterogeneous social networks, IEEE Network, 28 (2014), 56-64. [19] B. R. Marks and G. P. Wright, A general inner approximation algorithm for nonconvex mathematical programs, Operations Research, 26 (1978), 681-683. doi: 10.1287/opre.26.4.681. [20] M. J. Neely and L. Golubchik, Utility optimization for dynamic peer-to-peer networks with tit-for-tat constraints, IEEE INFOCOM, 2011, 1458-1466. [21] H. Nishida and T. Nguyen, A global contribution approach to maintain fairness in P2P networks, IEEE Transactions on Parallel and Distributed Systems, 21 (2010), 812-826. [22] B. Qureshi, G. Min and D. Kouvatsos, A distributed reputation and trust management scheme for mobile peer-to-peer networks, Computer Communications, 35 (2012), 608-618. [23] S. Rho, H. Chang, S. Kim and Y. S. Lee, An efficient peer-to-peer and distributed scheduling for cloud and grid computing, Peer-to-Peer Networking and Applications, 8 (2014), 863-871. [24] I. Rubin and R. Zhang, Max-min utility fair flow management for networks with route diversity, International Journal of Network Management, 20 (2010), 361-381. [25] A. Satsiou and L. Tassiulas, Reputation-based resource allocation in P2P systems of rational users, IEEE Transactions on Parallel and Distributed Systems, 21 (2010), 466-479. [26] S. Shakkottai and R. Srikant, Network optimization and control, Foundations and Trends in Networking, 2 (2007), 271-379. [27] F. Song, D. Huang, H. Zhou, H. Zhang and I. You, An optimization-based scheme for efficient virtual machine placement, International Journal of Parallel Programming, 42 (2014), 853-872. [28] F. Song, Y. Zhang, Z. An, H. Zhou and I. You, The correlation study for parameters in four tuples, International Journal of Ad Hoc and Ubiquitous Computing, 19 (2015), 38-49. [29] N. H. Tran and C. S. Hong, Joint rate and power control in wireless network: A novel successive approximations method, IEEE Communications Letters, 14 (2010), 872-874. [30] P. L. Vo, T. A. Le, S. Lee, C. S. Hong, B. Kim and H. Song, MReno: A practical multipath congestion control for communication networks, Computing, 96 (2014), 189-205. doi: 10.1007/s00607-013-0341-1. [31] P. L. Vo, T. A. Le, S. Lee, C. S. Hong, B. Kim and H. Song, Multi-path utility maximization and multi-path TCP design, Journal of Parallel and Distributed Computing, 74 (2014), 1848-1857. [32] K. Wang, H. Yin, W. Quan and G. Min, Enabling collaborative edge computing for software defined vehicular networks, IEEE Network, 32 (2018), 112-117. [33] H. Yan, D. Gao, W. Su, C. H. Foh, H. Zhang and A. V. Vasilakos, Caching strategy based on hierarchical cluster for named data networking, IEEE Access, 5 (2017), 8433-8443. [34] K. Zhang and N. Antonopoulos, A novel bartering exchange ring based incentive mechanism for peer-to-peer systems, Future Generation Computer Systems, 29 (2013), 361-369. [35] Y. Zheng, F. Lin, Y. Yang and T. Gan, Adaptive resource scheduling mechanism in P2P file sharing system, Peer-to-Peer Networking and Applications, 9 (2016), 1089-1100.
The resource allocation algorithm
Total number of iterations for the convergence of the proposed algorithm for coupled model
Performance of the resource allocation algorithm: fully coupled
Performance of the resource allocation algorithm: uncoupled
Performance of the resource allocation algorithm: half coupled
Optimal resource allocation obtained by the algorithm in three cases and LINGO
Optimal resource allocation for different fairness concepts
Aggregated utility of P2P networks with different number of peers
The optimum for the resource allocation model: fully coupled
 variable $x_{11}^*$ $x_{21}^*$ $x_{12}^*$ $x_{22}^*$ $x_{13}^*$ $x_{23}^*$ algorithm 5.4523 9.2144 3.8215 6.1785 2.7262 4.6072 LINGO 6.0114 8.6553 2.6028 7.3972 3.3859 3.9475
 variable $x_{11}^*$ $x_{21}^*$ $x_{12}^*$ $x_{22}^*$ $x_{13}^*$ $x_{23}^*$ algorithm 5.4523 9.2144 3.8215 6.1785 2.7262 4.6072 LINGO 6.0114 8.6553 2.6028 7.3972 3.3859 3.9475
The optimum for the resource allocation model: uncoupled
 variable $x_{11}^*$ $x_{21}^*$ $x_{12}^*$ $x_{22}^*$ $x_{13}^*$ $x_{23}^*$ algorithm 5.4534 9.2143 3.8233 6.1791 2.7272 4.6071 LINGO 5.4524 9.2143 3.8214 6.1786 2.7262 4.6071
 variable $x_{11}^*$ $x_{21}^*$ $x_{12}^*$ $x_{22}^*$ $x_{13}^*$ $x_{23}^*$ algorithm 5.4534 9.2143 3.8233 6.1791 2.7272 4.6071 LINGO 5.4524 9.2143 3.8214 6.1786 2.7262 4.6071
The optimum for the resource allocation model: half coupled
 variable $x_{11}^*$ $x_{21}^*$ $x_{12}^*$ $x_{22}^*$ $x_{13}^*$ $x_{23}^*$ algorithm 5.4516 9.2151 3.8227 6.1773 2.7248 4.6096 LINGO 5.4524 9.2143 3.8214 6.1786 2.7262 4.6071
 variable $x_{11}^*$ $x_{21}^*$ $x_{12}^*$ $x_{22}^*$ $x_{13}^*$ $x_{23}^*$ algorithm 5.4516 9.2151 3.8227 6.1773 2.7248 4.6096 LINGO 5.4524 9.2143 3.8214 6.1786 2.7262 4.6071

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