The morphing of fluid queues into Markov-modulated Brownian motion

Ramaswami showed recently that standard Brownian motion arises as the limit of a family of Markov-modulated linear fluid processes. We pursue this analysis with a fluid approximation for Markov-modulated Brownian motion. Furthermore, we prove that the stationary distribution of a Markov-modulated Brownian motion reflected at zero is the limit from the well-analyzed stationary distribution of approximating linear fluid processes. Key matrices in the limiting stationary distribution are shown to be solutions of a new quadratic equation, and we describe how this equation can be efficiently solved. Our results open the way to the analysis of more complex Markov-modulated processes.Comment: 20 page; the material on p7 (version 1) has been removed, and pp.8-9 replaced by Theorem 2.7 and its short proo

Two-dimensional fluid queues with temporary assistance

We consider a two-dimensional stochastic fluid model with $N$ ON-OFF inputs and temporary assistance, which is an extension of the same model with $N = 1$ in Mahabhashyam et al. (2008). The rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, and the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates modifications in the original rules of output-capacity sharing from Mahabhashyam et al. (2008) and considerably complicates both the theoretical analysis and the numerical computation of various performance measures

Bringing Students Back to Mathematics: Classroom Knowledge and Motivation

This paper reports part of a larger research study that investigated how teachers motivate students to learn mathematics at the college level. Findings from the study indicated that teachers have the power to influence and reinvigorate students who had given up learning mathematics. In the framework of Self-Determination Theory (SDT), the researchers analyzed five students’ motivational levels based on intrinsic and extrinsic motivation to see how each student was motivated by their teacher. Findings from the study could provide some directions for future research on students’ motivation to learn mathematics