We consider a neighbourhood random walk on a quadrant with environment phase-variable modelled by a continuous-time Markov chain. We describe this random walk using a two-dimensional level-dependent Quasi-Birth-and-Death process (2D-LD-QBD) with level-variables 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 with level variable recording the maximum of the two level-variables and phase-variable 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.

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