The Research On The Random Walk In The Quarter-plane With Flexible Boundaries

In this thesis,we consider the random walk in the quarter-plane.First,we use the theoretical result from the random walk with narrow boundary in the quarter-plane to analyze the problem in real life.Then we extend this theoretical result to the random walk with one extra layer flexible boundary.Our thesis is organized as follows:In Chapter 1,firstly,we provide the research background and the state-of-the-art literature review of random walks in the quarter-plane and queueing theory.Secondly,we introduce the main results of the thesis.Finally,the preliminary knowledge of our research is demonstrated.In Chapter 2,we consider the random walk with narrow boundary in the quarter-plane.We introduce an algorithm to check whether the invariant measure of any given random walk can be expressed as a sum of geometric terms.Based on the results,we have obtained the performance measure for a two-stage production system.Moreover,the performance measure can be used to optimize the production system.In Chapter 3,we consider the random walk with a one-layer flexible boundary in the quarter-plane.Firstly,we describe the model of random walk with a one-layer flexible boundary in the quarter-plane,and also provide the equilibrium distribution equation of this random walk.Then,we obtain the necessary condition for the invariant measure of such random walks to be expressed as a sum of three geometric terms.In Chapter 4,we have extended our results from Chapter 3.We describe the random walk with two-layer flexible boundaries and the equilibrium equations of this random walk.Then,we provide the necessary condition of the random walks with two-layer boundaries of which the invariant measure can be expressed as a sum of three geometric terms.