A Review of Regenerative Processes

Sigman, Karl; Wolff, Ronald W.

doi:10.1137/1035046

“…This assumption has been relaxed significantly and the notion of regenerative processes that we understand nowadays is wider; we will sketch these ideas in the following sections. This presentation is influenced by many books, which can be found in the references, and draws heavily from Sigman and Wolff [21].…”

Section: Classical Regenerative Processesmentioning

confidence: 99%

“…Because of property (i), R 1 is a stopping time for {X(t), t 0}. For a rigorous definition of property (i) see also Sigman and Wolff [21]. A standard assumption that is often made is that the paths of X(t) are rightcontinuous with left-hand limits almost surely.…”

Section: Classical Regenerative Processesmentioning

confidence: 99%

“…There is a version of the central limit theorem for regenerative processes that can be used to construct confidence intervals when simulating regenerative processes; see Sigman and Wolff [21] and Wolff [26] for details.…”

Section: Remarkmentioning

confidence: 99%

“…Many-server queues formed the main motivation to extend the classical notion of regeneration to this definition [21]. Previously, for the GI/GI/1 queue we had taken the times when the system empties as the regeneration epochs.…”

Section: One-dependent Regenerative Processesmentioning

confidence: 99%

“…This assumption has been relaxed significantly and the notion of regenerative processes that we understand nowadays is wider; we will sketch these ideas in the following sections. This presentation is influenced by many books, which can be found in the references, and draws heavily from Sigman and Wolff [21].The power of the concept of regenerative processes lies in the existence of a limiting distribution under conditions that are mild and usually easy to verify. For example, in continuous time it is only required that the mean cycle length of the embedded renewal process is finite, that the cycle length distribution is nonlattice and that the sample paths of the regenerative process satisfy some conditions (e.g.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Regenerative Processes

Vlasiou

¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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We review the theory of regenerative processes, which are processes that can be intuitively seen as comprising of i.i.d. cycles. Although we focus on the classical definition, we present a more general definition that allows for some form of dependence between two adjacent cycles, and mention two further extensions of the second definition. We mention the connection of regenerative processes to the single‐server queue, to multiserver queues, and more generally to Harris ergodic Markov chains and processes. In the main theorem, we pay some attention to the conditions under which a limiting distribution exists and provide references that should serve as a starting point for the interested reader.

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Regenerative Processes

Wolff

¹

2004

Encyclopedia of Actuarial Science

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A regenerative process is a stochastic process with the property that after some (usually) random time, it starts over in the sense that, from the random time on, the process is stochastically equivalent to what it was at the beginning. In many applied fields such as telecommunications, finance, and actuarial science, important probability models turn out to be regenerative. Properties of regenerative processes and the mathematical methods used in their analysis are fundamental in the analysis and understanding of these models. For classical regenerative processes, cycles and cycle lengths are i.i.d. (independent and identically distributed). This assumption has been weakened over time. Now, processes satisfying the first paragraph above are regarded as regenerative, without any independent assumptions. This article introduces the classical case in the section ‘Classical Regenerative Processes’, presents their time‐average properties in the section ‘Time‐average Properties, Classical Case’, and briefly treats more advanced topics in the section ‘Existence of Limits; Combining Processes’.

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Regenerative Processes

Wolff¹

2014

Wiley StatsRef: Statistics Reference Online

0

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A regenerative process is a stochastic process with the property that after some (usually) random time, it starts over in the sense that, from the random time on, the process is stochastically equivalent to what it was at the beginning. In many applied fields such as telecommunications, finance, and actuarial science, important probability models turn out to be regenerative. Properties of regenerative processes and the mathematical methods used in their analysis are fundamental in the analysis and understanding of these models. For classical regenerative processes, cycles and cycle lengths are i.i.d. (independent and identically distributed). This assumption has been weakened over time. Now, processes satisfying the first paragraph above are regarded as regenerative, without any independent assumptions. This article introduces the classical case in the section ‘Classical Regenerative Processes’, presents their time‐average properties in the section ‘Time‐average Properties, Classical Case’, and briefly treats more advanced topics in the section ‘Existence of Limits; Combining Processes’.

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A Review of Regenerative Processes

Cited by 76 publications

References 27 publications

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