Equilibrium distribution of block-structured Markov chains with repeating rows

This paper considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For the case (i), we present a weaker sufficient condition for the subexponential asymptotics than those given in the literature. As for the case (ii), the subexponential asymptotics has not been studied, as far as we know. We show that the subexponential asymptotics in the case (ii) is different from that in the case (i). We also study the locally subexponential asymptotics of the stationary distributions in both cases (i) and (ii).

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“…Remark 2.1.1 The proof of Theorem 1 in [10] is based on induction and probabilistic interpretation, which are valid without the recurrence of T .…”

Section: R-and G-matricesmentioning

confidence: 99%

Subexponential Asymptotics of the Stationary Distributions of GI/G/1-Type Markov Chains

Kimura

Masuyama

2013

Stochastic Models

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“…See Asmussen [3] for more about this type of WienerHopf factorization. The formulas replacingG + * △ in (3.27) by R + * △ (s, θ) is also known as R-G decomposition when Y n is integer valued and S X is a finite or countable set (e.g., see [13,33]). …”

Section: 4mentioning

confidence: 99%

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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“…The solution time-complexity of system (2) by conventional methods is of the order O((N F ) 3 ). It is however known that for block tridiagonal matrices with repetitive structure, such a complexity can be reduced to order O(NF 3 ) (Le Boudec 1989;Grassman and Heyman 1990;Ye and Li 1994;de Nitto Personè and Grassi 1996). In the next section, an appropriate arrangement of matrix S is given in order to obtain the repetitive block tridiagonality, thus reducing the order O((N F ) 3 ) problem to an order of O(NF 3 ).…”

Section: The Markov Process Definitionmentioning

confidence: 99%

“…Being of the type defined in Le Boudec (1989), Grassman and Heyman (1990), Ye andLi (1994), de Nitto Personè andGrassi (1996), the Fig. 2 repetitive block tridiagonal structure reduces the order O ((N F ) 3 ) problem to an order O(NF 3 ) one.…”

Section: Theoremmentioning

confidence: 99%

Analysis of cyclic queueing networks with parallelism and vacation

Personè

2008

Ann Oper Res

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The aim of this paper is to improve the machine interference model with vacation to deal with more recent problems of the communication area. To this scope the model is extended to include parallelism in the vacation station. The underlying Markov process is analyzed and a state arrangement is found that yields an efficient matrix-analytic technique that substantially lowers down the time-and space-complexity of standard methods. A numerical example of the method effectiveness is presented, and an example of resource allocation is introduced that finds applications in the QoS management of wireless networks.

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Equilibrium distribution of block-structured Markov chains with repeating rows

Cited by 77 publications

References 11 publications

Subexponential Asymptotics of the Stationary Distributions of GI/G/1-Type Markov Chains

Subexponential Asymptotics of the Stationary Distributions of GI/G/1-Type Markov Chains

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

Analysis of cyclic queueing networks with parallelism and vacation

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