First-passage-time problems in time-aware networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 183-194).First passage time or the first time that a stochastic process crosses a boundary is a random variable whose probability distribution is sought in engineering, statistics, finance, and other disciplines. The probability distribution of the first passage time has practical utility but is difficult to obtain because the values of the stochastic process at different times often constitute dependent random variables. As a result, most first-passage-time problems are still open and few of them are explicitly solved. In this thesis, we solve a large class of first-passage-time problems and demonstrate the applications of our solutions to networks that need to maintain common-time references. Motivated by rich applications of first passage time, we solve first-passage-time problems, which are divided into four categories according to the form of stochastic processes and the type of the boundaries. The four categories cover Brownian motion with quadratic drift and the boundary that consists of two constants; Brownian motion with polynomial drift of an arbitrary degree and the boundary that consists of two constants; multi-dimensional Brownian motion with polynomial drift and a class of boundaries that are characterized by open sets in the Euclidean space; and a discrete-time process with a class of correlations and the boundary that consists of one constant. These first-passage-time problems are challenging yet important for practical utility. The solutions to these first-passage-time problems range from an explicit expression to a bound of the first-passage-time distribution, reflecting the inherent difficulty in these first-passage-time problems. For Brownian motion with quadratic drift, the solution is explicit, consisting of elementary functions and functions that are characterized by Laplace transforms. For Brownian motion with polynomial drift of an arbitrary order, the solution involves analytical and numerical methods. For multi-dimensional Brownian motion, the solution is explicit for a certain shape of the boundary and is given by an upper bound and a lower bound for the other shapes. For the discrete-time process, the solution is explicit. The strength of our solutions is that they cover a large class of first-passage-time problems and are easy to use. The primary approach that allows us to solve these first-passage-time problems is transformation methodology. We apply various types of transformations, including transformation of probability measure, transformation of time, and integral transformation. Although these transformations are known, the combination of them in an appropriate order enables the solutions to previously-unsolved first-passage-time problems. We also discuss other problems that can be solved as consequences of the transformation methodology, including first-passage-time problems that involve a one-sided constant boundary, a moving boundary, and drifts such as logarithmic, exponential, sinusoidal, and square-root functions. A large class of first-passage-time problems confirms the utility of the transformation methodology. We demonstrate an application of the first-passage-time problems in the context of network synchronization. In the first setting that we consider, the first passage time is the first time that a network loses synchronization with a reference clock. At the first passage time, clocks in the network need to be calibrated. In the second setting, the first passage time represents the first time that a node achieves a correct synchronization of frames or packets. At the first passage time, a node in the network is able to process the packets that are transmitted as parts of the calibration. In both settings, we consider two performance metrics-the average and the outage-which succinctly summarize the first passage time. These metrics give insight, for example, into the amount of time for networks to lose synchronization as a function of key parameters such as noise in the clocks and the number of nodes in the network. Given the large class of first-passage-time problems being solved, we expect the thesis results to be useful in many disciplines where first-passage-time problems appear.by Watcharapan Suwansantisuk.Ph.D