Light tail asymptotics in multidimensional reflecting processes for queueing networks

Miyazawa, Masakiyo

doi:10.1007/s11750-011-0179-7

Cited by 72 publications

(102 citation statements)

References 86 publications

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“…We illustrate a typical example of reflected MAP below, which is a generalization of the 0-partially homogeneous chain studied in [5] (see also [22]). Then,Ŝ F ≡Ŝ 0 X ×Ŝ F Y stands for the boundary face for F = D. We refer to it as the F -face.…”

Section: Reflected Markov Additive Processmentioning

confidence: 99%

“…This form looks to be simpler than the corresponding formula (4.13), but they must be identical. When there is no background state, (4.17) is nothing but the stationary equation, which is stated in [22].…”

Section: Reflected Markov Additive Processmentioning

confidence: 99%

“…However, there are few results for computing the decay rates from primitive modeling parameters. Moreover, most of them are limited to the two dimensional case without background state although exact asymptotics are also obtained (e.g., see [5,21,22]). We are interested to find the rough asymptotics for the multidimensional reflected MAP which may have background states, where reflection at the boundary may be arbitrarily given.…”

Section: Multidimensional Formulationmentioning

confidence: 99%

“…The upper bound (5.15) may not be easy to use since it involves the unknown factor µ 0 . There have been some studies to find bounds for µ 0 * (θ) for the two dimensional reflecting process without background state (e.g., see [7,21,22,23]). 3.…”

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confidence: 99%

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Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Miyazawa

Zwart

2012

Stochastic Systems

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We extend the framework of Neuts' matrix analytic approach to a reflected process generated by a discrete time multidimensional Markov additive process. This Markov additive process has a general background state space and a real vector valued additive component, and generates a multidimensional reflected process. Our major interest is to derive a closed form formula for the stationary distribution of this reflected process. To this end, we introduce a real valued level, and derive new versions of the Wiener-Hopf factorization for the Markov additive process with the multidimensional additive component. In particular, it is represented by moment generating functions, and we consider the domain for it to be valid.Our framework is general enough to include multi-server queues and/or queueing networks as well as non-linear time series which are currently popular in financial and actuarial mathematics. Our results yield structural results for such models. As an illustration, we apply our results to extend existing results on the tail behavior of reflected processes.A major theme of this work is to connect recent work on matrix analytic methods to classical probabilistic studies on Markov additive processes. Indeed, using purely probabilistic methods such as censoring, duality, level crossing and time-reversal (which are known in the matrix analytic methods community but date back to Arjas & Speed [2] and Pitman [29]), we extend and unify existing results in both areas.

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Section: Reflected Markov Additive Processmentioning

confidence: 99%

Section: Reflected Markov Additive Processmentioning

confidence: 99%

Section: Multidimensional Formulationmentioning

confidence: 99%

mentioning

confidence: 99%

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Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Miyazawa

Zwart

2012

Stochastic Systems

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“…First, we provide necessary and sufficient conditions on the arrival rates for which there exists a rate allocation scheme that results in a stable network. These results are based on existing stability results for random walks in the quarter-plane [FIM99,Miy11]. We present results that provide significantly more insight for the special case of a coupled queue with forwarding.…”

Section: Introductionmentioning

confidence: 87%

Modeling the impact of interference on wireless ad hoc network performance

Coenen¹

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Reversibility in Queueing Models

Miyazawa

2011

Wiley Encyclopedia of Operations Research and Management Science

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In stochastic models for queues and their networks, random events evolve in time. A process for their backward evolution is referred to as a time‐reversed process . If some property is unchanged under time reversal, we may better understand the model. A concept of reversibility is invented for this invariance. Local balance for a stationary Markov chain has been used, but it is still too strong for queueing applications. We are concerned with a continuous‐time Markov chain, but do not assume it has a stationary distribution. We define reversibility in structure as an invariant property of a family of the set of models under a certain operation. Any member of this set is a pair of transition rate functions and its supporting measure, and each set represents dynamics of queueing systems such as arrivals and departures. We use a permutation Γ of the family members to describe the change of the dynamics under time reversal. This reversibility is called Γ ‐reversibility in structure . To apply these definitions, we introduce new classes of models, called reacting systems and self‐reacting systems . Using them, we give a unified view for queues and their networks. They include symmetric service, batch movements, and state‐dependent routing.

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Light tail asymptotics in multidimensional reflecting processes for queueing networks

Cited by 72 publications

References 86 publications

Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Modeling the impact of interference on wireless ad hoc network performance

Reversibility in Queueing Models

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