Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process

Ozawa, Toshihisa

doi:10.1007/s11134-012-9323-9

Cited by 31 publications

(62 citation statements)

References 22 publications

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“…This N 1 is well-defied since G 1 (1) is the G-matrix of the transition probability matrix A (1) * (see Proposition 3.5 of Ozawa [6]). Furthermore, N 1 satisfies G 1 (1) = N 1 A * ,−1 and R (1) * = A * ,1 N 1 .…”

Section: Proof Of Lemma 25 Of Ref [7]mentioning

confidence: 95%

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Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Ozawa

Kobayashi

2018

Queueing Syst

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A discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process), denoted by {Y n } = {(X 1,n , X 2,n , J n )}, is a two-dimensional skip-free random walk {(X 1,n , X 2,n )} on Z 2 + with a supplemental process {J n } on a finite set S 0 . The supplemental process {J n } is called a phase process. The 2d-QBD process {Y n } is a Markov chain in which the transition probabilities of the two-dimensional process {(X 1,n , X 2,n )} vary according to the state of the phase process {J n }. This modulation is assumed to be space homogeneous except for the boundaries of Z 2 + . Under certain conditions, the directional exact asymptotic formulae of the stationary distribution of the 2d-QBD process have been obtained in Ref. [7]. In this paper, we give an example of 2d-QBD process and proofs of some lemmas and propositions appeared in Ref. [7].

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Section: Proof Of Lemma 25 Of Ref [7]mentioning

confidence: 95%

“…We consider the same queueing model as that used in Ozawa [6]. It is a single-server two-queue model in which the server visits the queues alternatively, serves one queue (denoted by Q 1 ) according to a 1-limited service and the other queue (denoted by Q 2 ) according to an exhaustive-type Klimited service (see Fig.…”

Section: An Examplementioning

confidence: 99%

Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Ozawa

Kobayashi

2018

Queueing Syst

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“…If a (+) 2 < 0, then L (1) has a unique stationary distribution π (1) * = (π (1) * ,l , l ∈ Z + ) given as 8) and the mean transition rate vector with respect to L (1) , a (1) = (a…”

Section: Induced Ctmcs and Mean Transition Rate Vectorsmentioning

confidence: 99%

Stability condition of a two-dimensional QBD process and its application to estimation of efficiency for two-queue models

Ozawa

2019

Performance Evaluation

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In order to analyze stability of a two-queue model, we consider a two-dimensional quasibirth-and-death process (2d-QBD process), denoted by {Y (t)} = {((L 1 (t), L 2 (t)), J(t))}. The two-dimensional process {(L 1 (t), L 2 (t))} on Z 2 + is called a level process, where the individual processes {L 1 (t)} and {L 2 (t)} are assumed to be skip free. The supplemental process {J(t)} is called a phase process and it takes values in a finite set. The 2d-QBD process is a CTMC, in which the transition rates of the level process vary according to the state of the phase process like an ordinary (one-dimensional) QBD process. In this paper, we first state the conditions ensuring a 2d-QBD process is positive recurrent or transient and then demonstrate that the efficiency of a two-queue model can be estimated by using the conditions we obtained.

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“…According to Ozawa [36], we assume the following transition structure. The state space of the front process is composed of the inside of the quarter plane and three boundary faces, the origin and the two half coordinate axes.…”

Section: Introductionmentioning

confidence: 99%

“…See Figure 1 in Section 3.1 for their details. This Markov modulated two dimensional random walk is called a discrete-time 2d-QBD process, 2d-QBD process for short, in [36]. We adopt the same terminology.…”

Section: Introductionmentioning

confidence: 99%

A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov-modulated two-dimensional reflecting process

Miyazawa

2015

Queueing Syst

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Markov modulation is versatile in generalization for making a simple stochastic model which is often analytically tractable to be more flexible in application. In this spirit, we modulate a two dimensional reflecting skip-free random walk in such a way that its state transitions in the boundary faces and interior of a nonnegative integer quadrant are controlled by Markov chains. This Markov modulated model is referred to as a 2d-QBD process according to Ozawa [36]. We are interested in the tail asymptotics of its stationary distribution, which has been well studied when there is no Markov modulation.Ozawa studied this tail asymptotics problem, but his answer is not analytically tractable. We think this is because Markov modulation is so free to change a model even if the state space for Markov modulation is finite. Thus, some structure, say, extra conditions, would be needed to make the Markov modulation analytically tractable while minimizing its limitation in application.The aim of this paper is to investigate such structure for the tail asymptotic problem. For this, we study the existence of a right subinvariant positive vector, called a superharmonic vector, of a nonnegative matrix with QBD block structure, where each block matrix is finite dimensional. We characterize this existence under a certain extra assumption. We apply this characterization to the 2d-QBD process, and derive the tail decay rates of its marginal stationary distribution in an arbitrary direction. This solves the tail decay rate problem for a two node generalized Jackson network, which has been open for many years.

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Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process

Cited by 31 publications

References 22 publications

Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Stability condition of a two-dimensional QBD process and its application to estimation of efficiency for two-queue models

A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov-modulated two-dimensional reflecting process

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