## Algorithms and Optimization for Wireless Networks

##### Abstract

Recently, many new types of wireless networks have emerged for both civil and military applications, such as wireless sensor networks, ad hoc networks, among others. To improve the performance of these wireless networks, many advanced communication techniques have been developed at the physical layer. For both theoretical and practical purposes, it is important for a network researcher to understand the performance limits of these new wireless networks. Such performance limits are important not only for theoretical understanding, but also in that they can be used as benchmarks for the design of distributed algorithms and protocols. However, due to some unique characteristics associated with these networks, existing analytical technologies may not be applied directly. As a result, new theoretical results, along with new mathematical techniques, need to be developed. In this dissertation, we focus on the design of new algorithms and optimization techniques to study theoretical performance limits associated with these new wireless networks.
In this dissertation, we mainly focus on sensor networks and ad hoc networks. Wireless sensor networks consist of battery-powered nodes that are endowed with a multitude of sensing modalities. A wireless sensor network can provide in-situ, unattended, high-precision, and real-time observation over a vast area. Wireless ad hoc networks are characterized by the absence of infrastructure support. Nodes in an ad hoc network are able to organize themselves into a multi-hop network. An ad hoc network can operate in a stand-alone fashion or could possibly be connected to a larger network such as the Internet (also known as mesh networks).
For these new wireless networks, a number of advanced physical layer techniques, e.g., ultra wideband (UWB), multiple-input and multiple-output (MIMO), and cognitive radio (CR), have been employed. These new physical layer technologies have the potential to improve network performance. However, they also introduce some unique design challenges. For example, CR is capable of reconfiguring RF (on the fly) and switching to newly-selected frequency bands. It is much more advanced than the current multi-channel multi-radio (MC-MR) technology. MC-MR remains hardware-based radio technology: each radio can only operate on a single channel at a time and the number of concurrent channels that can be used at a wireless node is limited by the number of radio interfaces. While a CR can use multiple bands at the same time. In addition, an MC-MR based wireless network typically assumes there is a set of "common channels" available for all nodes in the network. While for CR networks, each node may have a different set of frequency bands based on its particular location. These important differences between MC-MR and CR warrant that the algorithmic design for a CR network is substantially more complex than that under MC-MR.
Due to the unique characteristics of these new wireless networks, it is necessary to consider models and constraints at multiple layers (e.g., physical, link, and network) when we explore network performance limits. The formulations of these cross-layer problems are usually in very complex forms and are mathematically challenging. We aim to develop some novel algorithmic design and optimization techniques that provide optimal or near-optimal solutions.
The main contributions of this dissertation are summarized as follows.
1. Node lifetime and rate allocation
We study the sensor node lifetime problem by considering not only maximizing the time until the first node fails, but also maximizing the lifetimes for all the nodes in the network. For fairness, we maximize node lifetimes under the lexicographic max-min (LMM) criteria. Our contributions are two-fold. First, we develop a polynomial-time algorithm based on a parametric analysis (PA) technique, which has a much lower computational complexity than an existing state-of-the-art approach. We also present a polynomial-time algorithm to calculate the flow routing schedule such that the LMM-optimal node lifetime vector can be achieved. Second, we show that the same approach can be employed to address a different but related problem, called LMM rate allocation problem. More important, we discover an elegant duality relationship between the LMM node lifetime problem and the LMM rate allocation problem. We show that it is sufficient to solve only one of the two problems and that important insights can be obtained by inferring the duality results.
2. Base station placement
Base station location has a significant impact on sensor network lifetime. We aim to determine the best location for the base station so as to maximize the network lifetime. For a multi-hop sensor network, this problem is particularly challenging as data routing strategies also affect the network lifetime performance. We present an approximation algorithm that can guarantee $(1- \varepsilon)$-optimal network lifetime performance with any desired error bound $\varepsilon >0$. The key step is to divide the continuous search space into a finite number of
subareas and represent each subarea with a "fictitious cost point" (FCP). We prove that the largest network lifetime achieved by one of these FCPs is $(1- \varepsilon)$-optimal. This approximation algorithm offers a significant reduction in complexity when compared to a state-of-the-art algorithm, and represents the best known result to this problem.
3. Mobile base station
The benefits of using a mobile base station to prolong sensor network lifetime have been well recognized. However, due to the complexity of the problem (time-dependent network topology and traffic routing), theoretical performance limits and provably optimal algorithms remain difficult to develop. Our main result hinges upon a novel transformation of the joint base station movement and flow routing problem from the time domain to the space domain. Based on this transformation, we first show that if the base station is allowed to be present only on a set of pre-defined points, then we can find the optimal sojourn time for the base station on each of these points so that the overall network lifetime is maximized. Based on this finding, we show that when the location of the base station is un-constrained (i.e., can move to any point in the two-dimensional plane), we can develop an approximation algorithm for the joint mobile base station and flow routing problem such that the network lifetime is guaranteed to be at least $(1- \varepsilon)$ of the maximum network lifetime, where $\varepsilon$ can be made arbitrarily small. This is the first theoretical result with performance guarantee on this problem.
4. Spectrum sharing in CR networks
Cognitive radio is a revolution in radio technology that promises unprecedented flexibility in radio communications and is viewed as an enabling technology for dynamic spectrum access. We consider a cross-layer design of scheduling and routing with the objective of minimizing the required network-wide radio spectrum usage to support a set of user sessions. Here, scheduling considers how to use a pool of unequal size frequency bands for concurrent transmissions and routing considers how to transmit data for each user session. We develop a near-optimal algorithm based on a sequential fixing (SF) technique, where the determination of scheduling variables is performed iteratively through a sequence of linear programs (LPs). Upon completing the fixing of these scheduling variables, the value of the other variables in the optimization problem can be obtained by solving an LP.
5. Power control in CR networks
We further consider the case of variable transmission power in CR networks. Now, our objective is minimizing the total required bandwidth footprint product (BFP) to support a set of user sessions. As a basis, we first develop an interference model for scheduling when power control is performed at each node. This model extends existing so-called protocol models for wireless networks where transmission power is deterministic. As a result, this model can be used for a broad range of problems where power control is part of the optimization space. An efficient solution procedure based on the branch-and-bound framework and convex hull relaxations is proposed to provide $(1- \varepsilon)$-optimal solutions. This is the first theoretical result on this important problem.

##### Collections

- Doctoral Dissertations [12537]

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