This paper considers the reliability approximation of the queueing system PH/PH/1 with server breakdown and repair, where the life time of the server has the exponential distribution, and its repair time has the phase type distribution. First, we obtain the steady availability and steady failure frequency of the server. Then, we get the reliability function of the server, its probability density function and its Laplace-Stiltjes transform, which also gives an effective approximation of the reliability function of the server, where the error of the approximation can be easily bounded and is uniform in t. Finally, we discuss a special case and we can show that the reliability approximation has a wider application to various Markovian queueing systems with a repairable server.

this paper, we first introduce a symmetric phase type (PH) distribution, and then propose a symmetric exponential-polynomial type (EP) distribution based on the symmetric PH distribution. The class of symmetric EP distributions is illustrated to be so large that an arbitrary symmetric probability density function in L2 (-∞, +∞), which is the space of square integrable functions on the real line, can be approximated in (-∞, +∞) by a sequence of symmetric EP probability density functions. We prove this result by means of a constructive approach based on two orthogonal decompositions: Laguerre spectrum decomposition and Hermite spectrum decomposition. We also provide a moment-based approach for simply determining a symmetric EP probability density function to approximatively express a symmetric random variable under which some moments of the symmetric random variable are given. We further propose multidimensional symmetric PH and EP distributions and provide their useful structures and properties. Finally, we apply the symmetric PH and EP distributions to study the probability Hough transform.

系统地研究了两个不同并行服务台的可修排队系统MAP/PH（M/PH）/2，其中两个不同的服务台拥有一个修理工，若其中一台处于修理状态，则另一台失效后就处于待修状态，利用拟生灭过程理论，我们首先讨论了两个服务台的广义服务时间的相依性，然后给出了系统的稳态可用度和稳态故障度，最后得到了系统首次失效前的时间分布及其均值。

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.

该文研究了基于有限个数据点的随机Hough变换的概率模型。在这个模型中，主要讨论了在随机Hough变换的基本算法中起相当关键作用的两个量：累加器数组的控制阈值以及从图像中提取全部基元所需随机抽样的总次数这两个量对随机Hough变换的算法设计及其终止规则是相当有用的。该文的主要结果包括两部分：其一是对累加器数组引入了多项分布，系统地研究了累加器数组的概率结构及其相互关系。同时也计算了提取全部基元所需的随机抽样总次数的分布、均值和方差。另者是基于不断的随机抽样而使得累加器数组的随机变化作者引入了多维纯生过程，证明了提取全部基元所需随机抽样的总次数服从离散的PH分布，从而它的各阶矩都可用简洁的矩阵形式统一地表出。针对于图像的固有因素，作者也讨论了基元的平稳提取概率。该文的结果为随机Hough变换的进一步研究和应用提供了较为严格的理论依据。