We study a fixed-point formalization of the well-known analysis of Bianchi. We provide a significant simplification and generalization of the analysis. In this more general framework, the fixed-point solution and performance measures resulting from it are studied. Uniqueness of the fixed point is established. Simple and general throughput formulas are provided. It is shown that the throughput of any flow will be bounded by the one with the smallest transmission rate. The aggregate throughput is bounded by the reciprocal of the harmonic mean of the transmission rates. In an asymptotic regime with a large number of nodes, explicit formulas for the collision probability, the aggregate attempt rate, and the aggregate throughput are provided. The results from the analysis are compared with ns2 simulations and also with an exact Markov model of the backoff process. It is shown how the saturated network analysis can be used to obtain TCP transfer throughputs in some cases.

/pdf/new-insights-from-a-fixed-point-analysis-of-single-cell-ieee-4ykskikd6b.pdf

New insights from a fixed-point analysis of single cell IEEE 802.11 WLANs

The Projection Theorem. Theorems on Extension and Separation. Dual Spaces and Transposed Operators. The Banach Theorem and the Banach-Steinhaus Theorem. Construction of Hilbert Spaces. L ~ 2 Spaces and Convolution Operators. Sobolev Spaces of Functions of One Variable. Some Approximation Procedures in Spaces of Functions. Sobolev Spaces of Functions of Several Variables and the Fourier Transform. Introduction to Set-Valued Analysis and Convex Analysis. Elementary Spectral Theory. Hilbert-Schmidt Operators and Tensor Products. Boundary Value Problems. Differential-Operational Equations and Semigroups of Operators. Viability Kernels and Capture Basins. First-Order Partial Differential Equations. Selection of Results. Exercises. Bibliography. Index.

Applied functional analysis, by Jean-Pierre Aubin; translated by Carole Labrousse. Pp 423. £16·50. 1980. ISBN 0-471-02149-0 (Wiley)

Energy savings optimization becomes one of the major concerns in the wireless sensor network (WSN) routing protocol design, due to the fact that most sensor nodes are equipped with the limited nonrechargeable battery power. In this paper, we focus on minimizing energy consumption and maximizing network lifetime for data relay in one-dimensional (1-D) queue network. Following the principle of opportunistic routing theory, multihop relay decision to optimize the network energy efficiency is made based on the differences among sensor nodes, in terms of both their distance to sink and the residual energy of each other. Specifically, an Energy Saving via Opportunistic Routing (ENS_OR) algorithm is designed to ensure minimum power cost during data relay and protect the nodes with relatively low residual energy. Extensive simulations and real testbed results show that the proposed solution ENS_OR can significantly improve the network performance on energy saving and wireless connectivity in comparison with other existing WSN routing schemes.

Opportunistic Routing Algorithm for Relay Node Selection in Wireless Sensor Networks

We present a simple analytical model for IEEE 802.11 based wireless networks that effectively captures the important network performance characteristics such as throughput, delay and fairness. Several new insights relevant to network design are obtained by considering simple special cases of the model. Key features of our unified model include the ability to handle hidden/exposed terminals, directional antennas, multiple channels and arbitrary traffic matrices. The analytical framework is first applied to study the saturation throughput in a multi-hop setting. The associated flow starvation problem is investigated in detail and it is found that Exponential Backoff (EB) plays a much smaller role in flow starvation when compared to the minimum contention window (CWmin). We then illustrate applicability of our model to address key network design questions such as determining the feasibility of a particular traffic matrix and identifying bottlenecks. We also apply the framework to study the important problem of improving throughput and fairness in a multi-hop setting. It is observed from our results that a MAC protocol that adapts its minimum contention window CWmin is of greater utility than one that adapts to a value between fixed CWmin and CWmax (as the current EB in IEEE 802.11 does). Finally, guided by the model, we present a simple heuristic called Effective Contention Indicator (ECI) which identifies, to first-order, links experiencing severe contention.

Towards Performance Modeling of IEEE 802.11 Based Wireless Networks: A Unified Framework and Its Applications

Motivated partially by a control-theoretic viewpoint, we propose a game-theoretic model, called random access game, for contention control. We characterize Nash equilibria of random access games, study their dynamics, and propose distributed algorithms (strategy evolutions) to achieve Nash equilibria. This provides a general analytical framework that is capable of modeling a large class of system-wide quality-of-service (QoS) models via the specification of per-node utility functions, in which system-wide fairness or service differentiation can be achieved in a distributed manner as long as each node executes a contention resolution algorithm that is designed to achieve the Nash equilibrium. We thus propose a novel medium access method derived from carrier sense multiple access/collision avoidance (CSMA/CA) according to distributed strategy update mechanism achieving the Nash equilibrium of random access game. We present a concrete medium access method that adapts to a continuous contention measure called conditional collision probability, stabilizes the network into a steady state that achieves optimal throughput with targeted fairness (or service differentiation), and can decouple contention control from handling failed transmissions. In addition to guiding medium access control design, the random access game model also provides an analytical framework to understand equilibrium and dynamic properties of different medium access protocols.

/pdf/random-access-game-and-medium-access-control-design-8tt3nq1wgl.pdf

Random access game and medium access control design

We consider the vector fixed point equations arising out of the analysis of the saturation throughput of a single cell IEEE 802.11e (EDCA) wireless local area network with nodes that have different backoff parameters, including different arbitration interframe space (AIFS) values. We consider balanced and unbalanced solutions of the fixed point equations arising in homogeneous (i.e., one with the same backoff parameters) and nonhomogeneous networks. By a balanced fixed point, we mean one where all coordinates are equal. We are concerned, in particular, with: 1) whether the fixed point is balanced within a class, and 2) whether the fixed point is unique. Our simulations show that when multiple unbalanced fixed points exist in a homogeneous system then the time behavior of the system demonstrates severe short term unfairness (or multistability). We provide a condition for the fixed point solution to be balanced, and also a condition for uniqueness. We then extend our general fixed point analysis to capture AIFS based differentiation and the concept of virtual collision when there are multiple queues per station; again a condition for uniqueness is established. For the case of multiple queues per node, we find that a model with as many nodes as there are queues, with one queue per node, provides an excellent approximation. Implications for the use of the fixed point formulation for performance analysis are also discussed.

Fixed point analysis of single cell IEEE 802.11e WLANs: uniqueness and multistability

We consider the vector fixed point equations arising out of the analysis of the saturation throughput of a single cell IEEE 802.11e wireless local area network with nodes that have different back-off parameters, including different Arbitration InterFrame Space (AIFS) values. We consider balanced and unbalanced solutions of the fixed point equations arising in homogeneous and nonhomogeneous networks. We are concerned, in particular, with (i) whether the fixed point is balanced within a class, and (ii) whether the fixed point is unique. Our simulations show that when multiple unbalanced fixed points exist in a homogeneous system then the time behaviour of the system demonstrates severe short term unfairness (or multistability). Implications for the use of the fixed point formulation for performance analysis are also discussed. We provide a condition for the fixed point solution to be balanced within a class, and also a condition for uniqueness. We then provide an extension of our general fixed point analysis to capture AIFS based differentiation; again a condition for uniqueness is established. An asymptotic analysis of the fixed point is provided for the case in which packets are never abandoned, and the number of nodes goes to ∞. Finally the fixed point equations are used to obtain insights into the throughput differentiation provided by different initial back-offs, persistence factors, and AIFS, for finite number of nodes, and for differentiation parameter values similar to those in the standard.

Fixed point analysis of single cell IEEE 802.11e WLANs: uniqueness, multistability and throughput differentiation

We study the problem of scheduling in OFDMA-based relay networks with emphasis on IEEE 802.16j based WiMAX relay networks. In such networks, in addition to a base station, multiple relay stations are used for enhancing the throughput, and/or improving the range of the base station. We solve the problem of MAC scheduling in such networks so as to serve the mobiles in a fair manner while exploiting the multiuser diversity, as well as the frequency selectivity of the wireless channel. The scheduling resources consist of tiles in a two-dimensional scheduling frame with time slots along one axis, and frequency bands or sub-channels along the other axis. The resource allocation problem has to be solved once every scheduling frame which is about 5 - 10 ms long. While the original scheduling problem is computationally complex, we provide an easy-to-compute upper bound on the optimum. We also propose three fast heuristic algorithms that perform close to the optimum (within 99.5%), and outperform other algorithms such as OFDM2A proposed in the past. Through extensive simulation results, we demonstrate the benefits of relaying in throughput enhancement (an improvement in the median throughput of about 25%), and feasibility of range extension (for e.g., 7 relays can be used to extend the cell-radius by 60% but mean throughput reduces by 36%). Our algorithms are easy to implement, and have an average running time of less than 0.05 ms making them appropriate for WiMAX relay networks.

WiMAX relay networks: opportunistic scheduling to exploit multiuser diversity and frequency selectivity

We study a scheduling problem in a wireless network where vehicles are used as store-and-forward relays, a situation that might arise, for example, in practical rural communication networks. A fixed source node wants to transfer a file to a fixed destination node, located beyond its communication range. In the absence of any infrastructure connecting the two nodes, we consider the possibility of communication using vehicles passing by. Vehicles arrive at the source node at renewal instants and are known to travel towards the destination node with average speed v sampled from a given probability distribution. The source node communicates data packets (or fragments) of the file to the destination node using these vehicles as relays. We assume that the vehicles communicate with the source node and the destination node only, and hence, every packet communication involves two hops. In this setup, we study the source node's sequential decision problem of transferring packets of the file to vehicles as they pass by, with the objective of minimizing delay in the network. We study both the finite file size case and the infinite file size case. In the finite file size case, we aim to minimize the expected file transfer delay, i.e., expected value of the maximum of the packet sojourn times. In the infinite file size case, we study the average packet delay minimization problem as well as the optimal tradeoff achievable between the average queueing delay at the source node buffer and the average transit delay in the relay vehicle.

Delay Optimal Scheduling in a Two-Hop Vehicular Relay Network

We consider a dense, ad hoc wireless network, confined to a small region. The wireless network is operated as a single cell, i.e., only one successful transmission is supported at a time. Data packets are sent between source-destination pairs by multihop relaying. We assume that nodes self-organize into a multihop network such that all hops are of length d meters, where d is a design parameter. There is a contention-based multiaccess scheme, and it is assumed that every node always has data to send, either originated from it or a transit packet (saturation assumption). In this scenario, we seek to maximize a measure of the transport capacity of the network (measured in bit-meters per second) over power controls (in a fading environment) and over the hop distance d, subject to an average power constraint. We first motivate that for a dense collection of nodes confined to a small region, single cell operation is efficient for single user decoding transceivers. Then, operating the dense ad hoc wireless network (described above) as a single cell, we study the hop length and power control that maximizes the transport capacity for a given network power constraint. More specifically, for a fading channel and for a fixed transmission time strategy (akin to the IEEE 802.11 TXOP), we find that there exists an intrinsic aggregate bit rate (\Theta_{opt} bits per second, depending on the contention mechanism and the channel fading characteristics) carried by the network, when operating at the optimal hop length and power control. The optimal transport capacity is of the form d_{opt}(\bar{P_t}) \times \Theta_{opt} with d_{opt} scaling as \bar{P_t}^{{1\over \eta}}, where \bar{P_t} is the available time average transmit power and \eta is the path loss exponent. Under certain conditions on the fading distribution, we then provide a simple characterization of the optimal operating point. Simulation results are provided comparing the performance of the optimal strategy derived here with some simple strategies for operating the network.

Venkatesh Ramaiyan

Papers

Fixed point analysis of single cell IEEE 802.11e WLANs: uniqueness and multistability

Fixed point analysis of single cell IEEE 802.11e WLANs: uniqueness, multistability and throughput differentiation

WiMAX relay networks: opportunistic scheduling to exploit multiuser diversity and frequency selectivity

Delay Optimal Scheduling in a Two-Hop Vehicular Relay Network

Optimal Hop Distance and Power Control for a Single Cell, Dense, Ad Hoc Wireless Network