Absorbing Markov Chain

The paper develops DILOC, a \emph{distributive}, \emph{iterative} algorithm that locates M sensors in $\mathbb{R}^m, m\geq 1$, with respect to a minimal number of m+1 anchors with known locations. The sensors exchange data with their neighbors only; no centralized data processing or communication occurs, nor is there centralized knowledge about the sensors' locations. DILOC uses the barycentric coordinates of a sensor with respect to its neighbors that are computed using the Cayley-Menger determinants. These are the determinants of matrices of inter-sensor distances. We show convergence of DILOC by associating with it an absorbing Markov chain whose absorbing states are the anchors. We introduce a stochastic approximation version extending DILOC to random environments when the knowledge about the intercommunications among sensors and the inter-sensor distances are noisy, and the communication links among neighbors fail at random times. We show a.s. convergence of the modified DILOC and characterize the error between the final estimates and the true values of the sensors' locations. Numerical studies illustrate DILOC under a variety of deterministic and random operating conditions.

American universities use a procedure based on a rolling six-year graduation rate to calculate statistics regarding their students' final educational outcomes (graduating or not graduating). As an alternative to the six-year graduation rate method, many studies have applied absorbing Markov chains for estimating graduation rates. In both cases, a frequentist approach is used. For the standard six-year graduation rate method, the frequentist approach corresponds to counting the number of students who finished their program within six years and dividing by the number of students who entered that year. In the case of absorbing Markov chains, the frequentist approach is used to compute the underlying transition matrix, which is then used to estimate the graduation rate. In this paper, we apply a sensitivity analysis to compare the performance of the standard six-year graduation rate method with that of absorbing Markov chains. Through the analysis, we highlight significant limitations with regards to the estimation accuracy of both approaches when applied to small sample sizes or cohorts at a university. Additionally, we note that the Absorbing Markov chain method introduces a significant bias, which leads to an underestimation of the true graduation rate. To overcome both these challenges, we propose and evaluate the use of a regularly updating multi-level absorbing Markov chain (RUML-AMC) in which the transition matrix is updated year to year. We empirically demonstrate that the proposed RUML-AMC approach nearly eliminates estimation bias while reducing the estimation variation by more than 40%, especially for populations with small sample sizes.

In spite of many applications of evolutionary algorithms in optimisation, theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few. It is still unclear when an evolutionary algorithm is expected to solve an optimisation problem efficiently or otherwise. This paper gives a general analytic framework for analysing first hitting times of evolutionary algorithms. The framework is built on the absorbing Markov chain model of evolutionary algorithms. The first step towards a systematic comparative study among different EAs and their first hitting times has been made in the paper.

The paper introduces DILOC, a distributed, iterative algorithm to locate M sensors (with unknown locations) in Rm, m ges 1, with respect to a minimal number of m + 1 anchors with known locations. The sensors and anchors, nodes in the network, exchange data with their neighbors only; no centralized data processing or communication occurs, nor is there a centralized fusion center to compute the sensors' locations. DILOC uses the barycentric coordinates of a node with respect to its neighbors; these coordinates are computed using the Cayley-Menger determinants, i.e., the determinants of matrices of internode distances. We show convergence of DILOC by associating with it an absorbing Markov chain whose absorbing states are the states of the anchors. We introduce a stochastic approximation version extending DILOC to random environments, i.e., when the communications among nodes is noisy, the communication links among neighbors may fail at random times, and the internodes distances are subject to errors. We show a.s. convergence of the modified DILOC and characterize the error between the true values of the sensors' locations and their final estimates given by DILOC. Numerical studies illustrate DILOC under a variety of deterministic and random operating conditions.

To cope with the increasing demand of wireless communication services multi-carrier systems are being used. Radio resources are very limited and efficient usages of these resources are inevitable to get optimum performance of the system. Paging channel is a low-bandwidth channel and one of the most important channels on which system performance depends significantly. Therefore it is vulnerable to even moderate overloads. In this paper, an efficient paging algorithm, Concurrent Search, is proposed for efficient use of paging channel in Multi- carrier CDMA system instead of existing sequential searching algorithm. It is shown by the simulation that the paging performance in proposed algorithm is far better than the existing system.