Random walk on a quadrant: mapping to a one-dimensional level-dependent Quasi-Birth-and-Death process

We consider a neighborhood random walk on a quadrant {(X1(t),X2(t),φ(t)):t≥0} with environment phase variable φ(t) modeled by a continuous-time Markov chain with φ(t)∈Snm when X1(t) = n, X2(t) = m. We describe this random walk using a two-dimensional level-dependent Quasi-Birth-and-Death process (2D-LD-QBD) with phase variable φ(t) and level variables X1(t),X2(t)∈{0,1,2,…} which change in a skip-free manner at the moments of jump in the process. We transform this random walk into a one-dimensional LD-QBD {(Z(t),χ(t)):t≥0} with level variable Z(t)∈{0,1,2,…} recording the maximum of the two level variables and phase variable χ(t)=(χ1(t),χ2(t),φ(t)) recording the remaining information about the random walk. Using this transformation, we perform transient and stationary analysis of the random walk, including first hitting times for various sample paths, using matrix-analytic methods. We also construct a sequence of neighborhood random walks, represented as two-dimensional QBDs ({(X1(k)(t),X2(k)(t),φ(t)):t≥0})k=1,2,…, converging in distribution to a two-dimensional stochastic fluid model (SFM) {(Y1(t),Y2(t),φ(t)):t≥0}, which describes a movement on a quadrant in which the position changes in a continuous manner according to rates dY1(t)/dt=c1,φ(t) and dY2(t)/dt=c2,φ(t) modulated by the underlying phase process {φ(t):t≥0}. Numerical examples are provided to illustrate the application of the methodology.

我们研究定义于象限上的邻域随机游走{(X₁(t),X₂(t),φ(t)): t≥0}，其环境相位变量φ(t)由连续时间马尔可夫链（Continuous-time Markov Chain, CTMC）建模，且当X₁(t)=n、X₂(t)=m时，φ(t)∈Sₙₘ。我们借助带相位变量的二维相依水平拟生灭过程（2D-LD-QBD）描述该随机游走，其中水平变量X₁(t)、X₂(t)∈{0,1,2,…}，且过程在跳跃时刻以无跳越方式变化。我们将该随机游走转换为一维相依水平拟生灭过程{(Z(t),χ(t)): t≥0}，其中水平变量Z(t)∈{0,1,2,…}用于记录两个水平变量的最大值，相位变量χ(t)=(χ₁(t),χ₂(t),φ(t))用于记录该随机游走的其余信息。基于该转换，我们采用矩阵解析方法（Matrix-analytic Methods, MAM）对该随机游走开展暂态与稳态分析，包括各类样本轨道的首达时计算。我们还构造了一列邻域随机游走，其表示为二维拟生灭过程序列{(X₁⁽ᵏ⁾(t),X₂⁽ᵏ⁾(t),φ(t)): t≥0}（k=1,2,…），该序列依分布收敛于二维随机流体模型（SFM）{(Y₁(t),Y₂(t),φ(t)): t≥0}，后者描述了象限上的运动：位置变化遵循由底层相位过程{φ(t): t≥0}调制的速率dY₁(t)/dt = c_{1,φ(t)}与dY₂(t)/dt = c_{2,φ(t)}，且变化过程连续。本文给出数值示例以阐明该方法的应用。