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We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) { (X n , J n) } on Z + 2 × S 0 , where 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 an arbitrary 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 and the singularity analysis methods. 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. [ABSTRACT FROM AUTHOR]

We consider a discrete-time two-dimensional process {(X1,n,X2,n)} on Z+2 with a supplemental process {Jn} on a finite set, where the individual processes {X1,n} and {X2,n} are both skip-free. We assume that the joint process {Yn}={(X1,n,X2,n,Jn)} is Markovian and that the transition probabilities of the two-dimensional process {(X1,n,X2,n)} are modulated depending on the state of the supplemental process {Jn}. This modulation is space homogeneous except for the boundaries of Z+2. We call this process a discrete-time two-dimensional quasi-birth-and-death process. Under several conditions, we obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions. [ABSTRACT FROM AUTHOR]

We consider a discrete-time two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ on $\mathbb{Z}_{+}^{2}$ with a background process { J} on a finite set, where individual processes $\{L_{n}^{(1)}\}$ and $\{L_{n}^{(2)}\}$ are both skip free. We assume that the joint process $\{Y_{n}\}=\{(L_{n}^{(1)},L_{n}^{(2)},J_{n})\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ are modulated depending on the state of the background process { J}. This modulation is space homogeneous, but the transition probabilities in the inside of $\mathbb{Z}_{+}^{2}$ and those around the boundary faces may be different. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process, and obtain the decay rates of the stationary distribution in the coordinate directions. We also distinguish the case where the stationary distribution asymptotically decays in the exact geometric form, in the coordinate directions. [ABSTRACT FROM AUTHOR]