论文标题
在二维离散时间QBD过程中，固定分布的任何方向的任何方向尾巴渐近学
Tail Asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process
论文作者
论文摘要
我们考虑一个离散的时间二维准外事和死亡过程（简短的2D-qbd过程）$ \ {（\ boldsymbol {\ boldsymbol {x} _n，j_n）\} $ $ \ boldsymbol {x} _n =（x_ {1，n}，x_ {2，n}）$是级别状态，$ j_n $相位状态（背景状态）和$ s_0 $ a a s_0 $ a a有限套件，以及固定尾部分布的研究无关。 2D-QBD过程是通常的一维QBD过程的扩展。通过使用排队理论的矩阵分析方法和复杂的分析方法，我们在任何方向上获得了固定尾巴分布的渐近衰变速率。该结果是通过使用大偏差技术获得的一定二维反映随机工作的相应结果的扩展。我们还提出了一个条件，以确保固定概率的序列几何地衰减无功率，从而渐近。在文献中已经研究了在2D-QBD过程中坐标方向上固定尾部分布的渐近特性。本文的结果也是这些结果的重要补充。
We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) $\{(\boldsymbol{X}_n,J_n)\}$ on $\mathbb{Z}_+^2\times S_0$, where $\boldsymbol{X}_n=(X_{1,n},X_{2,n})$ is the level state, $J_n$ the phase state (background state) and $S_0$ a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in any direction. This result is an extension of the corresponding result for a certain two-dimensional reflecting random work without background processes, obtained by using the large deviation techniques. We also present a condition ensuring the sequence of the stationary probabilities geometrically decays without power terms, asymptotically. Asymptotic properties of the stationary tail distribution in the coordinate directions in a 2d-QBD process have already been studied in the literature. The results of this paper are also important complements to those results.
