Random walks and green
- 1. Random Walks and Green’s Function on Digraphs: A Framework for Estimating Wireless Transmission Costs Abstract: Various applications in wireless networks, such as routing and query processing, can be formulated as random walks on graphs. Many results have been obtained for such applications by utilizing the theory of random walks (or spectral graph theory), which is mostly developed for undirected graphs. However, this formalism neglects the fact that the underlying (wireless) networks in practice contain asymmetric links, which are best characterized by directed graphs (digraphs). Therefore, random walk on digraphs is a more appropriate model to consider for such networks. In this paper, by generalizing the random walk theory (or spectral graph theory) that has been primarily developed for undirected graphs to digraphs, we show how various transmission costs in wireless networks can be formulated in terms of hitting times and cover times of random walks on digraphs. Using these results, we develop a unified theoretical framework for estimating various transmission costs in wireless networks. Our framework can be applied to random walk query processing strategy and the three routing paradigms best path routing, opportunistic routing, and stateless routing to which nearly all existing routing Protocol ls belong. Extensive simulations demonstrate that the proposed Digraph-based analytical model can achieve more accurate transmission cost estimation over existing methods. Reasons of Proposal: Due to the unique characteristics of wireless technologies and the dynamics in the environments (e.g., mobility and interference) they operate in, wireless channels are known to be time-varying, unreliable, and asymmetric. Furthermore, wireless networks are often designed to support certain applications or missions, and deployed in specific environments. For these reasons, a plethora of wireless mechanism especially, routing algorithms and protocols have been proposed and developed to achieve a range of different objectives such as throughput, latency, energy consumption, network lifetime, and so forth. Evaluating the efficacy of wireless protocols in terms of various transmission cost metrics, and deciding on which one to employ in a specific environment so as to attain certain performance
- 2. objective, can be a challenging task in practice. The ability to analyze, estimate, and quantify various transmission costs is therefore imperative in the design of wireless networks. Existing System: While experimentation and testing in realistic wireless environments are indispensable and provide the most definite and authoritative means to evaluate the efficacy of wireless routing protocols, they are in general very expensive and are typically utilized in the later stage of the network design and evaluation process. Simulation-based evaluation is also important and necessary. However, conducting realistic simulations is hard, and simulation results often hinge on the settings and parameters used. We believe that analytical models and theories also play a critical role in the design of wireless networks, complementing the roles played by real- world experimentation and simulations. By generating performance bounds and theoretical limits, they provide important insights on what is achievable and under what conditions and produce useful metrics for understanding the key design tradeoffs. Such insights and understanding are particularly important in the early stage of wireless network design. Proposed System: Guided by this belief, in this paper we develop a unified theoretical framework to quantify and estimate various transmission costs of wireless routing protocols. To account for the stochastic and asymmetric natures of wireless channels, we model a wireless network as a directed graph (in short, digraph), where each directed edge (link) is associated with a packet delivery probability. We consider three wireless routing paradigms, the (traditional) best path routing (e.g., AODV, DSR, and several energy-aware routing protocols), opportunistic routing (e.g., ExOR), and stateless (stochastic) routing nearly all existing routing protocols fall under one of these paradigms, or use a combination thereof. Under the (simplifying) assumption that packet delivery probabilities are independent, we demonstrate how packet forwarding under each paradigm can be modeled as a Markov chain on a digraph with an appropriately defined transition probability matrix capturing the specifics of the routing algorithm under consideration. In other words, the traversal of a packet being forwarded in a wireless network can be viewed as a random walk on a digraph. Consequently, various transmission costs of end-to-end packet delivery (e.g., the expected number of transmissions, end-to-
- 3. end packet delivery ratio, throughput, latency, energy consumptions) can therefore be formulated using well-known notions such as hitting times, sojourn times associated with random walks. The main contributions of this paper are summarized as follows. By building on top of our previous work in , we utilize the random walk (Markov chain) model to formulate the end-to-end transmission costs for various types of wireless routing strategies. • The theory of random walk (and the closely related spectral graph theory) has been developed primarily for undirected graphs. We successfully extend the theory of random walks on undirected graphs to directed graphs (digraphs), with a more general definition of normalized (graph) Laplacian matrix. We also show how the hitting times, commute times, sojourn times (or hitting costs), and (partial) cover time can be computed using the Moore–Penrose pseudo-inverse of . • Using three representative routing protocols (one from each routing paradigm) and a query processing protocol as examples, we systematically illustrate how our proposed theoretical framework based on random walks on digraphs can be used to estimate various transmission costs. Our analysis subsumes earlier results obtained using more ad hoc methods. We also perform extensive simulations to show the relative errors in estimation when asymmetric links are artificially symmetrized and undirected graphs are used.