This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such random walks are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that stationary behaviour is investigated by solving a finite system of linear equations, two matrix functional equations, and a functional equation with the aid of the theory of Riemann (–Hilbert) boundary value problems. This work is strongly motivated by emerging applications in flow level performance of wireless networks that give rise in queueing models with scalable service capacity, as well as in queue-based random access protocols, where the network’s parameters are functions of the queue lengths. A simple numerical illustration, along with some details on the numerical implementation are also presented.

这项工作涉及二维部分同质最近邻随机游动的平稳分析。这种随机游走的特征在于，单步转移概率是状态空间的函数。我们表明，借助黎曼（–Hilbert）边值问题理论，通过求解线性方程组，两个矩阵泛函方程和泛函方程的有限系统，研究了稳态行为。这项工作受到无线网络流量级别性能中新兴应用程序的强烈推动，这些应用程序产生了具有可扩展服务容量的排队模型以及基于队列的随机访问协议，其中网络的参数是队列长度的函数。一个简单的数字插图