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On Markovian traffic with applications to TES processes

机译：关于马尔可夫交通及其在TES流程中的应用

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Markov processes are an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose inter-arrival times (separating the time points of discrete arrivals) form a real-valued Markov chain. As such this paper aims to extend the classical results of renewal traffic, where interarrival times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The paper derives integral equations for these functions in the transform domain. These are then specialized to a subclass,TES+, of a versatile class of random sequences, calledTES(Transform-Expand-Sample), consisting of marginally uniform autoregressive schemes with modulo-1 reduction, followed by various transformations.TESmodels are designed to simultaneously capture both first-order and second-order statistics of empirical records, and consequently can produce high-fidelity models. Two theoretical solutions forTES+traffic functions are derived: an operator-based solution and a matric solution, both in the transform domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to obtain instructive closed-form representations for two measures of traffic burstiness: peakedness and index of dispersion, elucidating the relationship between them.
机译：马尔可夫过程是多种随机应用中的重要组成部分。值得注意的实例包括提供给他们的排队系统和交通流程。本文关注的是马尔可夫交通，即其到达间隔时间（分隔离散到达时间点）形成实值马尔可夫链的交通过程。因此，本文旨在扩展更新流量的经典结果，在该结果中，到达时间被认为是独立的，分布均匀的。遵循传统的更新理论，解决了三个函数：给定时间间隔内到达次数的概率，相应的均值和未来到达次数的概率。本文在变换域中导出了这些函数的积分方程。然后将它们专门化为通用序列的子类TES +的子类TES（Transform-Expand-Sample），该子序列由具有模1减少的边际均匀自回归方案组成，然后进行各种变换.TES模型旨在同时捕获经验记录的一阶和二阶统计量，因此可以生成高保真模型。推导了TES +交通功能的两个理论解：基于算子的解和矩阵解，都在变换域中。说明并解决了一种特殊情况，允许将积分方程转换为微分方程。最后，将结果应用于获得针对交通突发性的两种度量的指导性封闭形式表示形式：峰度和分散指数，阐明了它们之间的关系。

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