# Mean Field Games for Energy-Efficient Proactive Delay Tolerant Networks

**Proactive, Delay-Tolerant and Energy Efficient Networks**

Recent results have demonstrated that human usage of wireless networks were highly and accurately perdictable [1]. Exploiting predictions about the future link quality of a user has become a research topic of interest and several papers have proposed transmissions paradigms exploiting these network prediction capabilities. Exploiting such predictions especially makes sense in delay-tolerant transmission contexts, since it allows the network to adapt its transmission settings to the expected link qualities. For instance, in [2], the authors propose to exploit this future knowledge on the mobility of the user and its expected link quality, to buffer video streams and avoid outage, in anticipation to incoming tunnels or bad link quality zones. In my PhD, I proposed to exploit this future knowledge, into the well-known latency vs. energy efficiency trade-off [3], thus leading to so-called “Proactive delay-tolerant networks” [4].

**Mathematical Complexity of optimization and multi-user dynamic/stochastic games**

In such networks, the objective consists to define the transmission strategy of users, allowing to transmit a required data packet, before a given deadline, at a minimal cumulated power cost, assuming the system has prediction capabilities on the users mobilities and/or their expected future link qualities. The objective is then to adapt the transmssion powers, depending on the data remaining to transmit, the time remaining before the deadline, the future link qualities and the power strategies of other users which generate interference. Finding the optimal power strategy, i.e. the one that ensures the complete transmission of the required packet, on time, at a minimal power cost is hard. Mathematically speaking, the optimization problem can be modelled as a multi-user dynamic/stochastic game [5]. However, solving such a game can be difficult, especially when the number of users in the sytem grows large [6], which is the reason why most optimization problems have been left unsolved or solved in simple non-stochastic/constant scenarios only [7].

*Figure: The multi-user proactive delay-tolerant network under investigation.
Mobility of users and expected future link qualities are known.
*

*Figure: Cumulated power cost of the Mean Field strategy compared to two reference strategies. 50 Time slots, 100 users, sinusoidal channel evolution.*

**Mean Field Games, to the rescue !**

Recent works on Mean Field theory [8], allow to adress the mathematical complexity issue of multi-user games, in scenarios where the number of users *N* in the system is large enough to be considered infinite. Initially used in particle physics [9] and finance [10], the Mean Field theory allows to approximate a multi-user game into a mean field game. The new-built game appears to be 2-body game, instead of a *N*-body one, which drastically simplifies the analysis of the optimal configuration. In these papers [11][12][13], we exploit the mean field game theory and demonstrate the potential gains, in terms of energy efficiency, that this approach might offer, compared to reference transmission strategies, that are unable to exploit the latency and/or future knowledge. The gain is twofold: the first part is related to the offered latency, whereas the second one rellies on the capability of the system to exploit offered future predictions.

(2) M. Proebster, M. Kaschub, and S. Valentin, ‘Context-aware resource allocation to improve the quality of service of heterogeneous traffic,’ in Communications (ICC), 2011 IEEE International Conference on. IEEE, 2011, pp. 1–6.

(3) Y. Chen, S. Zhang, S. Xu, and G. Y. Li, ‘Fundamental trade-offs on green wireless networks,’ Communications Magazine, IEEE, vol. 49, no. 6, pp. 30–37, 2011.

(4) H. El Gamal, J. Tadrous, and A. Eryilmaz, ‘Proactive resource allocation: Turning predictable behavior into spectral gain,’ in Communication, Control, and Computing (Allerton), 2010 48th Annual Allerton Conference on. IEEE, 2010, pp. 427–434.

(5) T. Basar, G. J. Olsder, G. Clsder, T. Basar, T. Baser, and G. J. Olsder, ‘Dynamic noncooperative game theory.’ SIAM, 1995, vol. 200.

(6) O. Morgenstern and J. Von Neumann, ‘Theory of games and economic behavior,’ 1953.

(7) F. Mériaux, S. Lasaulce, and H. Tembine, ‘Stochastic differential games and energy-efficient power control,’ Dynamic Games and Applications, vol. 3, no. 1, pp. 3–23, 2013.

(8) J.-M. Lasry and P.-L. Lions, ‘Mean field games,’ Japanese Journal of Mathematics, vol. 2, no. 1, pp. 229–260, 2007.

(9) Rohringer, G., Toschi, A., Hafermann, H., Held, K., Anisimov, V. I., & Katanin, A. A. (2013). ‘One-particle irreducible functional approach: A route to diagrammatic extensions of the dynamical mean-field theory.’ Physical Review B, 88(11), 115112.

(10) O. Guéant, J.-M. Lasry, and P.-L. Lions, ‘Mean field games and applications,’ in Paris-Princeton Lectures on Mathematical Finance 2010. Springer, 2011, pp. 205–266.

(11) De Mari, M., Couillet, R., Calvanese Strinati, E., & Debbah, M. (2012, August), ‘Concurrent data transmissions in green wireless networks: When best send one's packets?.’ In Wireless Communication Systems (ISWCS), 2012 International Symposium on (pp. 596-600). IEEE.

(12) De Mari, M., Calvanese Strinati, E., & Debbah, M., ‘Energy-Efficiency in Proactive Delay-Tolerant Networks: A Mean Field Game Approach,’ to be submitted soon, 2016.

(13) De Mari, M., Calvanese Strinati, E., & Debbah, M., ‘Energy-Efficiency in Proactive Delay-Tolerant Networks with Mean Field Games,’ to be submitted to IEEE Trans. on Wireless Communications, 2016.