Abstract
A wireless sensor network (WSN) consists of a large number of unattended sensors with limited storage, battery power, computation, and communication capabilities, where battery power (or energy) is the most crucial resource for sensor nodes. The information sensed by sensors needs to be transmitted to sink quickly especially for the applications with delay restriction. However, it is difficult to achieve optimal energy efficiency and source-to-sink delay simultaneously. So it is very necessary to find a power control solution based tradeoff between energy and delay. In this paper, a cooperative differential game model is proposed, and a power solution is obtained which determines a fair distribution of the total cooperative cost among sources.
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Acknowledgements
This work is supported in part by the National Science Foundation of P.R. China under Grant no. 60773074 and by the National Science Foundation of P.R. China under Grant no. 60903004. Meanwhile, is supported by the Education Department of Henan province Science and Technology Research Projects no. 13A510030.
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Appendix: Proof of convexity of the cooperative game
Appendix: Proof of convexity of the cooperative game
To prove that the cooperative game Γ c (x,t) is convex, we need to show that
Let k=|K|, l=|L|, x=|K∩L|, m=|M|. Then, |K∪L|=k+l−x. Using (16), the left-hand side of the above inequality is given by
Straightforward calculations permit one to reduce the above expression to
The result follows from the fact that k≥1, l≥1 and x≤min(k,l).
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Miao, XN., Xu, G. Cooperative differential game model based on trade-off between energy and delay for wireless sensor networks. Ann Oper Res 206, 297–310 (2013). https://doi.org/10.1007/s10479-013-1354-z
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DOI: https://doi.org/10.1007/s10479-013-1354-z