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This work deals with the stationary analysis of two-dimensional partiallyhomogeneous nearest-neighbour random walks. Such type of random walks arecharacterized by the fact that the one-step transition probabilities arefunctions of the state-space. We show that its stationary behaviour isinvestigated by solving a finite system of linear equations, two matrixfunctional equations, and a functional equation with the aid of the theory ofRiemann (-Hilbert) boundary value problems. This work is strongly motivated byemerging applications in flow level performance of wireless networks that giverise in queueing models with scalable service capacity, as well as inqueue-based random access protocols, where the network's parameters arefunctions of the queue lengths. A simple numerical illustration, along withsome details on the numerical implementation are also presented.

On partially homogeneous nearest-neighbour random walks in the quarter  plane and their application in the analysis of two-dimensional queues with  limited state-dependency