arXiv subjects

Toshihisa Ozawa

Publications and source records attributed to Toshihisa Ozawa.

11 recordsLinked to original sources

arXiv2025

Asymptotics of the occupation measure defined on a nonnegative matrix of two-dimensional quasi-birth-and-death type

We consider a nonnegative matrix having the same block structure as that of the transition probability matrix of a two-dimensional quasi-birth-and-death process (2d-QBD process for short) and define two kinds of measure for the nonnegative matrix. One corresponds to the mean number of visits to each state before the 2d-QBD process starting from the level zero returns to the level zero for the first time. The other corresponds to the probabilities that the 2d-QBD process starting from each state visits the level zero. We call the former the occupation measure and the latter the hitting measure. We obtain asymptotic properties of the occupation measure such as the asymptotic decay rate in an arbitrary direction. Those of the hitting measure can be obtained from the results for the occupation measure by using a kind of duality between the two measures.

math.PR↗

arXiv2023

Exact asymptotics of the stationary tail probabilities in an arbitrary direction in a two-dimensional discrete-time QBD process

We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) on $\mathbb{Z}_+^2\times S_0$, where $S_0$ is a finite set, and give a complete expression for the asymptotic decay function of the stationary tail probabilities in an arbitrary direction. The 2d-QBD process is a kind of random walk in the quarter plane with a background process. In our previous paper (Queueing Systems, vol. 102, pp. 227-267, 2022), we have obtained the asymptotic decay rate of the stationary tail probabilities in an arbitrary direction and clarified that if the asymptotic decay rate $ξ_{\boldsymbol{c}}$, where $\boldsymbol{c}$ is a direction vector in $\mathbb{N}^2$, is less than a certain value $θ_{\boldsymbol{c}}^{max}$, the sequence of the stationary tail probabilities in the direction $\boldsymbol{c}$ geometrically decays without power terms, asymptotically. In this paper, we give the function according to which the sequence asymptotically decays, including the case where $ξ_{\boldsymbol{c}}=θ_{\boldsymbol{c}}^{max}$. When $ξ_{\boldsymbol{c}}=θ_{\boldsymbol{c}}^{max}$, the function is given by an exponential function with power term $k^{-\frac{1}{2}}$ except for two boundary cases, where it is given by just an exponential function without power terms. This result coincides with the existing result for a random walk in the quarter plane without background processes, obtained by Malyshev (Siberian Math. J., vol. 12, ,pp. 109-118, 1973).

math.PR↗

arXiv2022

Tail Asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process

We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) $\{(\boldsymbol{X}_n,J_n)\}$ on $\mathbb{Z}_+^2\times S_0$, where $\boldsymbol{X}_n=(X_{1,n},X_{2,n})$ is the level state, $J_n$ the phase state (background state) and $S_0$ a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in any direction. This result is an extension of the corresponding result for a certain two-dimensional reflecting random work without background processes, obtained by using the large deviation techniques. We also present a condition ensuring the sequence of the stationary probabilities geometrically decays without power terms, asymptotically. Asymptotic properties of the stationary tail distribution in the coordinate directions in a 2d-QBD process have already been studied in the literature. The results of this paper are also important complements to those results.

math.PR↗

arXiv2020

Asymptotic property of the occupation measures in a multi-dimensional skip-free Markov modulated random walk

We consider a discrete-time $d$-dimensional process $\{\boldsymbol{X}_n\}=\{(X_{1,n},X_{2,n},...,X_{d,n})\}$ on $\mathbb{Z}^d$ with a background process $\{J_n\}$ on a countable set $S_0$, where individual processes $\{X_{i,n}\},i\in\{1,2,...,d\},$ are skip free. We assume that the joint process $\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ is Markovian and that the transition probabilities of the $d$-dimensional process $\{\boldsymbol{X}_n\}$ vary according to the state of the background process $\{J_n\}$. This modulation is assumed to be space homogeneous. We refer to this process as a $d$-dimensional skip-free Markov modulate random walk. For $\boldsymbol{y}, \boldsymbol{y}'\in \mathbb{Z}_+^d\times S_0$, consider the process $\{\boldsymbol{Y}_n\}_{n\ge 0}$ starting from the state $\boldsymbol{y}$ and let $\tilde{q}_{\boldsymbol{y},\boldsymbol{y}'}$ be the expected number of visits to the state $\boldsymbol{y}'$ before the process leaves the nonnegative area $\mathbb{Z}_+^d\times S_0$ for the first time. For $\boldsymbol{y}=(\boldsymbol{x},j)\in \mathbb{Z}_+^d\times S_0$, the measure $(\tilde{q}_{\boldsymbol{y},\boldsymbol{y}'}; \boldsymbol{y}'=(\boldsymbol{x}',j')\in \mathbb{Z}_+^d\times S_0)$ is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as $\boldsymbol{x}'$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.

math.PR↗

arXiv2020

Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk

We consider a discrete-time two-dimensional process $\{(X_{1,n},X_{2,n})\}$ on $\mathbb{Z}^2$ with a background process $\{J_n\}$ on a finite set $S_0$, where individual processes $\{X_{1,n}\}$ and $\{X_{2,n}\}$ are both skip free. We assume that the joint process $\{\boldsymbol{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$ vary according to the state of the background process $\{J_n\}$. This modulation is assumed to be space homogeneous. We refer to this process as a two-dimensional skip-free Markov modulate random walk. For $\boldsymbol{Y}, \boldsymbol{Y}'\in \mathbb{Z}_+^2\times S_0$, consider the process $\{\boldsymbol{Y}_n\}_{n\ge 0}$ starting from the state $\boldsymbol{Y}$ and let $\tilde{q}_{\boldsymbol{Y},\boldsymbol{Y}'}$ be the expected number of visits to the state $\boldsymbol{Y}'$ before the process leaves the nonnegative area $\mathbb{Z}_+^2\times S_0$ for the first time. For $\boldsymbol{Y}=(x_1,x_2,j)\in \mathbb{Z}_+^2\times S_0$, the measure $(\tilde{q}_{\boldsymbol{Y},\boldsymbol{Y}'}; \boldsymbol{Y}'=(x_1',x_2',j')\in \mathbb{Z}_+^2\times S_0)$ is called an occupation measure. Our main aim is to obtain asymptotic decay rate of the occupation measure as the values of $x_1'$ and $x_2'$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.

math.PR↗

arXiv2019

Convergence parameters of nonnegative block tri-diagonal matrices and their application to multi-dimensional QBD processes

First, we consider a nonnegative homogeneous block tri-diagonal matrix and obtain its convergence parameter, where some results in the field of matrix analytic method are extended to the case where block matrices have countably infinite dimension. Second, we apply our results to a multi-dimensional QBD process and obtain lower bounds for the directional asymptotic decay rates of the stationary distribution.

math.PR↗

arXiv2018

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

In order to analyze stability of a two-queue model, we consider a two-dimensional quasi-birth-and-death process (2d-QBD process), denoted by $\{\boldsymbol{Y}(t)\}=\{((L_1(t),L_2(t)),J(t))\}$. The two-dimensional process $\{(L_1(t),L_2(t))\}$ on $\mathbb{Z}_+^2$ 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 obtain.

math.PR↗

arXiv2018

Exact asymptotic formulae of the stationary distribution of a discrete-time 2d-QBD process: an example and additional proofs

A discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process), $\{\boldsymbol{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$, is a two-dimensional skip-free random walk $\{(X_{1,n},X_{2,n})\}$ on $\mathbb{Z}_+^2$ 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 $\{\boldsymbol{Y}_n\}$ 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 $\mathbb{Z}_+^2$. Under certain conditions, the directional exact asymptotic formulae of the stationary distribution of the 2d-QBD process have been obtained in "T. Ozawa and M. Kobayashi, Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process, Queueing Systems (2018) DOI:10.1007/s11134-018-9586-x." In this paper, we give an example of 2d-QBD process and proofs of some lemmas and propositions appeared in that paper.

math.PR↗

arXiv2017

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

We consider a discrete-time two-dimensional process $\{(L_{1,n},L_{2,n})\}$ on $\mathbb{Z}_+^2$ with a supplemental process $\{J_n\}$ on a finite set, where individual processes $\{L_{1,n}\}$ and $\{L_{2,n}\}$ are both skip free. We assume that the joint process $\{Y_n\}=\{(L_{1,n},L_{2,n},J_n)\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{1,n},L_{2,n})\}$ are modulated depending on the state of the background process $\{J_n\}$. This modulation is space homogeneous except for the boundaries of $\mathbb{Z}_+^2$. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process and, under several conditions, obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.

math.PR↗

arXiv2015

Stability of multidimensional skip-free Markov modulated reflecting random walks: Revisit to Malyshev and Menshikov's results and application to queueing networks

Let $\{\boldsymbol{X}_n\}$ be a discrete-time $d$-dimensional process on $\mathbb{Z}_+^d$ with a supplemental (background) process $\{J_n\}$ on a finite set and assume the joint process $\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ to be Markovian. Then, the process $\{\boldsymbol{X}_n\}$ can be regarded as a kind of reflecting random walk (RRW for short) in which the transition probabilities of the RRW are modulated according to the state of the background process $\{J_n\}$; we assume this modulation is space-homogeneous inside $\mathbb{Z}_+^d$ and on each boundary face of $\mathbb{Z}_+^d$. Further we assume the process $\{\boldsymbol{X}_n\}$ is skip free in all coordinates and call the joint process $\{\boldsymbol{Y}_n\}$ a $d$-dimensional skip-free Markov modulated reflecting random walk (MMRRW for short). The MMRRW is an extension of an ordinary RRW and stability of ordinary RRWs have been studied by Malyshev and Menshikov. Following their results, we obtain stability and instability conditions for MMRRWs and apply our results to stability analysis of a two-station network.

math.PR↗

arXiv2013

Positive recurrence and transience of a two-station network with server states

We study positive recurrence and transience of a two-station network in which the behavior of the server in each station is governed by a Markov chain with a finite number of server states; this service process can represent various service disciplines such as a non-preemptive priority service and K-limited service. Assuming that exogenous customers arrive according to independent Markovian arrival processes (MAPs), we represent the behavior of the whole network as a continuous-time Markov chain and, by the uniformization technique, obtain the corresponding discrete-time Markov chain, which is positive recurrent (transient) if and only if the original continuous-time Markov chain is positive recurrent (resp. transient). This discrete-time Markov chain is a four-dimensional skip-free Markov modulated reflecting random walk (MMRRW) and, applying several existing results of MMRRWs to the Markov chain, we obtain conditions on which the Markov chain is positive recurrent and on which it is transient. The conditions are represented in terms of the difference of the input rate and output rate of each queue in each induced Markov chain. In order to demonstrate how our results work in two-station networks, we give several examples.

math.PR↗