Orbits of linear operators and Banach space geometry

Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0. Then, the set of $x \in X$ such that $\|T^nx\| \geqslant a_n \|T^n\|$ for infinitely many $n$'s has a complement which is both $\sigma$-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents $q>0$, such that for every non nilpotent operator $T$, there exists $x \in X$ such that $(\|T^nx\|/\|T^n\|) \notin \ell^{q}(\mathbb{N})$, using techniques which involve the modulus of asymptotic uniform smoothness of $X$.Comment: 16 page

1-D Coordinate Based on Local Information for MAC and Routing Issues in WSNs

More and more critical Wireless Sensor Networks (WSNs) applications are emerging. Those applications need reliability and respect of time constraints. The underlying mechanisms such as MAC and routing must handle such requirements. Our approach to the time constraint problem is to bound the hop-count between a node and the sink and the time it takes to do a hop so the end-to-end delay can be bounded and the communications are thus real-time. For reliability purpose we propose to select forwarder nodes depending on how they are connected in the direction of the sink. In order to be able to do so we need a coordinate (or a metric) that gives information on hop-count, that allows to strongly differentiate nodes and gives information on the connectivity of each node keeping in mind the intrinsic constraints of WSWs such as energy consumption, autonomy, etc. Due to the efficiency and scalability of greedy routing in WSNs and the financial cost of GPS chips, Virtual Coordinate Systems (VCSs) for WSNs have been proposed. A category of VCSs is based on the hop-count from the sink, this scheme leads to many nodes having the same coordinate. The main advantage of this system is that the hops number of a packet from a source to the sink is known. Nevertheless, it does not allow to differentiate the nodes with the same hop-count. In this report we propose a novel hop-count-based VCS which aims at classifying the nodes having the same hop-count depending on their connectivity and at differentiating nodes in a 2-hop neighborhood. Those properties make the coordinates, which also can be viewed as a local identifier, a very powerful metric which can be used in WSNs mechanisms.Comment: (2011

RTXP : A Localized Real-Time Mac-Routing Protocol for Wireless Sensor Networks

Protocols developed during the last years for Wireless Sensor Networks (WSNs) are mainly focused on energy efficiency and autonomous mechanisms (e.g. self-organization, self-configuration, etc). Nevertheless, with new WSN applications, appear new QoS requirements such as time constraints. Real-time applications require the packets to be delivered before a known time bound which depends on the application requirements. We particularly focus on applications which consist in alarms sent to the sink node. We propose Real-Time X-layer Protocol (RTXP), a real-time communication protocol. To the best of our knowledge, RTXP is the first MAC and routing real-time communication protocol that is not centralized, but instead relies only on local information. The solution is cross-layer (X-layer) because it allows to control the delays due to MAC and Routing layers interactions. RTXP uses a suited hop-count-based Virtual Coordinate System which allows deterministic medium access and forwarder selection. In this paper we describe the protocol mechanisms. We give theoretical bound on the end-to-end delay and the capacity of the protocol. Intensive simulation results confirm the theoretical predictions and allow to compare with a real-time centralized solution. RTXP is also simulated under harsh radio channel, in this case the radio link introduces probabilistic behavior. Nevertheless, we show that RTXP it performs better than a non-deterministic solution. It thus advocates for the usefulness of designing real-time (deterministic) protocols even for highly unreliable networks such as WSNs

Publish/subscribe protocol in wireless sensor networks: improved reliability and timeliness

The rapidly-evolving demand of applications using wireless sensor networks in several areas such as building and industrial automation or smart cities, among other, makes it necessary to determine and provide QoS support mechanisms which can satisfy the requirements of applications. In this paper we propose a mechanism that establishes different QoS levels, based on Publish/Subscribe model for wireless networks to meet application requirements, to provide reliable delivery of packet and timeliness. The first level delivers packets in a best effort way. The second one intends to provide reliable packet delivery with a novel approach for Retransmission Timeout (RTO) calculation, which adjusts the RTO depending on the subscriber Packet Delivery Ratio (PDR). The third one provides the same reliable packet delivery as the second one, but in addition, it provides data aggregation trying to be efficient in terms of energy consumption and the use of network bandwidth. The last one provides timeliness in the packet delivery. We evaluate each QoS Level with several performance metrics such as PDR, Message Delivery Ratio, Duplicated and Retransmitted Packet Ratio and Packet Timeliness Ratio to demonstrate that our proposal provides significant improvements based on the increase of the PDR obtained.Peer ReviewedPostprint (author's final draft