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.