One-sided Markov additive processes with lattice and non-lattice increments

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications Pub Date : 2025-09-11 DOI:10.1016/j.spa.2025.104771

Jevgenijs Ivanovs , Guy Latouche , Peter Taylor

{"title":"One-sided Markov additive processes with lattice and non-lattice increments","authors":"Jevgenijs Ivanovs , Guy Latouche , Peter Taylor","doi":"10.1016/j.spa.2025.104771","DOIUrl":null,"url":null,"abstract":"<div><div>Dating from the work of Neuts in the 1980s, the field of matrix-analytic methods has been developed to analyse discrete or continuous-time Markov chains with a two-dimensional state space in which the increment of a <em>level</em> variable is governed by an auxiliary <em>phase</em> variable. More recently, matrix-analytic techniques have been applied to general Markov additive models with a finite phase space. The basic assumption underlying these developments is that the process is skip-free (in the case of QBDs or fluid queues) or that it is <em>one-sided</em>, that is it is jump-free in one direction.</div><div>From the Markov additive perspective, traditional matrix-analytic models can be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments in the level are constrained to be <em>lattice</em> random variables and for fluid queues, they have to be piecewise linear.</div><div>In this paper we discuss one-sided lattice and non-lattice Markov additive processes in parallel. Results that are standard in one tradition are interpreted in the other, and new perspectives emerge. In particular, using three fundamental matrices, we address hitting, two-sided exit, and creeping probabilities.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104771"},"PeriodicalIF":1.2000,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414925002157","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}

引用次数: 0

Abstract

Dating from the work of Neuts in the 1980s, the field of matrix-analytic methods has been developed to analyse discrete or continuous-time Markov chains with a two-dimensional state space in which the increment of a level variable is governed by an auxiliary phase variable. More recently, matrix-analytic techniques have been applied to general Markov additive models with a finite phase space. The basic assumption underlying these developments is that the process is skip-free (in the case of QBDs or fluid queues) or that it is one-sided, that is it is jump-free in one direction.

From the Markov additive perspective, traditional matrix-analytic models can be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments in the level are constrained to be lattice random variables and for fluid queues, they have to be piecewise linear.

In this paper we discuss one-sided lattice and non-lattice Markov additive processes in parallel. Results that are standard in one tradition are interpreted in the other, and new perspectives emerge. In particular, using three fundamental matrices, we address hitting, two-sided exit, and creeping probabilities.

查看原文本刊更多论文

具有格和非格增量的单侧马尔可夫加性过程

从Neuts在20世纪80年代的工作开始，矩阵解析方法领域已经发展到分析具有二维状态空间的离散或连续时间马尔可夫链，其中水平变量的增量由辅助相位变量控制。最近，矩阵解析技术已应用于有限相空间的一般马尔可夫加性模型。这些发展背后的基本假设是，该过程是无跳变的（在qbd或流体队列的情况下），或者它是片面的，即它在一个方向上是无跳变的。从马尔可夫可加性的角度来看，传统的矩阵解析模型可以看作是特例：对于M/G/1和GI/M/1型马尔可夫链，水平上的增量被约束为点阵随机变量，对于流体队列，它们必须是分段线性的。本文讨论了单侧格和非格马尔可夫加性过程的并行问题。在一种传统中是标准的结果在另一种传统中得到解释，新的观点出现了。特别是，使用三个基本矩阵，我们处理命中，双边退出和爬行概率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Stochastic Processes and their Applications 数学-统计学与概率论

CiteScore

2.90

自引率

7.10%

发文量

180

审稿时长

23.6 weeks

期刊介绍： Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests. Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.