Symbiot: Congestion-driven Multi-resource Fairness for Multi-User Sensor Networks

© 2015 IEEE.In this paper, we study the problem of multi-resource fairness in multi-user sensor networks with heterogeneous and time-varying resources. Particularly we focus on data gathering applications run on Wireless Sensor Networks (WSNs) or Internet of Things (IoT) in which users require to run a serious of sensing operations with various resource requirements. We consider both the resource demands of sensing tasks, and data forwarding tasks needed to establish multi-hop relay communications. By exploiting graph theory, queueing theory and the notion of dominant resource shares, we develop Symbiot, a light-weight, distributed algorithm that ensures multi-resource fairness between these users. With Symbiot, nodes can independently schedule its resources while maintaining network-level resource fairness through observing traffic congestion levels. Large-scale simulations based Contiki OS and Cooja network emulator show the effectiveness of Symbiot in adaptively utilizing available resources and reducing average completion times

Equations involving fractional Laplacian operator: Compactness and application

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_\alpha-2)$, $0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to $\varepsilon_n\in [0, 2^*_\alpha-2)$, satisfying $\|u_n\|_{H}\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided that $N>6\alpha$ and $\lambda>0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.Comment: 34 page