论文标题
在四分之一平面中对某些马尔可夫调制的反射随机步行的固定分析
Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane
论文作者
论文摘要
在这项工作中，我们专注于对特定类别的连续时间的固定分析，马尔可夫修饰的反射随机步行在四分之一平面中，并在两节点马尔可夫修饰的排队网络进行建模中应用，并用耦合队列进行了建模。二维过程的过渡速率取决于有限状态马尔可夫背景过程的状态。这种调制在二维晶格的内部状态集中是均匀的，但在边界处的一组状态可能有所不同。为了获得固定分布，我们应用了功率系列近似方法以及Riemann边界价值问题的理论。我们还根据某些对称性假设获得了固定分布的第一瞬间的明确表达式。还提出了在用耦合轨道队列的优先重试系统建模中的应用。使用排队网络示例，我们通过数值验证了理论发现。
In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. An application in the modelling of a priority retrial system with coupled orbit queues is also presented. Using a queueing network example, we numerically validated the theoretical findings.
