Wireless ad hoc and sensor networking has been identified as a recent major success story in the field of communications. Today, it is emerging as a promising technology with an expansive range of applications. The full potential of these networks is yet to be realized as a result of design challenges in the related networking issues. This research seeks to develop a research area crossing frontiers in random graph theory, probabilistic methods, communications and networking. Focusing on the analysis and design of finite wireless networks (i.e., networks with small or moderate number of nodes), for which both existing asymptotic analysis based on infinite number of nodes or simulation based approaches are inadequate, the work has the potential to advance knowledge and understanding across several related fields.
Mathematical study of wireless networks serves as a theoretical foundation for this area. During the last ten years a variety of network properties such as network connectivity, coverage, reliability, delay, lifetime, sleep scheduling, throughput and capacity have been analytically studied to some extent. The majority of these analytical studies have concentrated on asymptotic scenarios, i.e., when the number of nodes tends to infinity. To study finite (small or moderate-size) networks researchers often resort to computer simulations and algorithmic approaches (such as linear programming). Thus, mathematical analysis of finite networks that could result in practically-useful formulas has been largely ignored. The goal of this research is to develop a general framework for design, study, and optimization of finite wireless ad hoc and sensor networks.
Wireless networking has been identified as a recent major success story in the field of communications. Today, it is emerging as a promising technology with an expansive range of applications. The full potential of these networks is yet to be realized as a result of design challenges in the related networking issues. Mathematical study of wireless networks serves as a theoretical foundation for this area. During the last decade, a variety of network properties such as network connectivity, coverage, reliability, delay, lifetime, sleep scheduling, throughput and capacity have been analytically studied to some extent. The majority of these analytical studies have concentrated on asymptotic scenarios, i.e., when the number of nodes in the network tends to infinity. However, in real world we have to face small or moderate-size networks which consist of a limited number of nodes. To study finite (small or moderate-size) networks researchers often resort to computer simulations and algorithmic approaches (such as linear programming). Thus, mathematical analysis of finite networks that could result in practically-useful formulas has been largely ignored. Using this grant, the PIs showed the need for a general methodology to study finite networks. They showed that the existing asymptotic or simulation based approaches are inadequate. Thus, it is very crucial from the practical point of view to analyze finite networks. These analytic results will essentially help us to understand, design, and analyze practical wireless networks, and also to design more suitable communication protocols. The PIs studied several new approaches to attack this important problem. The results are very promising and suggest that this key open problem is apparently solvable. The PIs considered different models of wireless networks as well as different properties of these networks such as connectivity, coverage, and capacity. They showed several scenarios in which they have been able to obtain simple and closed-form formulas that accurately predict the connectivity and coverage of finite networks. These results are helpful when we need to design a wireless sensor network consisting of an arbitrary deployment of a hundred sensor nodes. Some fundamental questions that have been answered are as follows. What is the transport capacity of the network? What are the connectivity and coverage probabilities of such networks? How do network parameters such as the communication radius of nodes, number of nodes, and so on, affect these properties? The PIs also studied the capacity of random wireless networks. The capacity is defined as the maximum possible number of concurrent transmissions that can occur at the network without interfering each other. An algorithm which finds the exact value for the capacity was developed. This algorithm runs in a very reasonable time. The PIs also derived closed form formulas to bound the exact value of the capacity given the number of nodes participating in the network and the communication radii of these nodes. These formulas could be used to design real-life networks with optimum transmission power that support maximum number of clients.
