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

We deal with a two-dimensional discrete-time quasi-birth-and-death process (2d-QBD process for short), which is an extension of ordinary one dimensional QBD process (see, for example, Latouche and Ramaswami [10]), and study asymptotic properties of the stationary tail distribution in any direction. The 2d-QBD process is also a two-dimensional skip-free Markov modulated reflecting random walk (2d-MMRRW for short), and the 2d-MMRRW is a two-dimensional skip-free reflecting random walk (2d-RRW for short) having a background process. Asymptotics of the stationary distributions in various 2d-RRWs without background processes have been investigated in the literature for several decades, especially, by Masakiyo Miyazawa and his colleagues (see a survey paper [13] of Miyazawa and references therein). Some of their results have been extended to the 2d-QBD process in Ozawa [20], Miyazawa [14] and Ozawa and Kobayashi [21], where the asymptotic decay rates and exact asymptotic formulae of the stationary distribution in the coordinate directions were obtained (cf. results in Miyazawa [12] and Kobayashi and Miyazawa [9]). In this paper, we further extend it to an arbitrary direction. In Miyazawa [14], the tail decay rates of the marginal stationary distribution in an arbitrary direction have also been obtained.

Let a Markov chain $\{\boldsymbol{Y}_{n}\}=\{(\boldsymbol{X}_{n},J_{n})\}$ be a 2d-QBD process on the state space $\mathbb{Z}_{+}^{2}\times S_{0}$, where $\boldsymbol{X}_{n}=(X_{1,n},X_{2,n})$, $S_{0}$ is a finite set with cardinality $s_{0}$, i.e., $S_{0}=\{1,2,...,s_{0}\}$, and $\mathbb{Z}_{+}$ is the set of all nonnegative integers. The process $\{\boldsymbol{X}_{n}\}$ is called the level process, $\{J_{n}\}$ the phase process (background process), and the transition probabilities of the level process vary according to the state of the phase process. This modulation is space homogeneous except for the boundaries of $\mathbb{Z}_{+}^{2}$. The level process is assumed to be skip free, i.e., for any $n\geq 0$, $\boldsymbol{X}_{n+1}-\boldsymbol{X}_{n}\in\{-1,0,1\}^{2}$. Stochastic models arising from various Markovian two-queue models and two-node queueing networks such as two-queue polling models and generalized two-node Jackson networks with Markovian arrival processes and phase-type service processes can be represented as two-dimensional continuous-time QBD processes, and their stationary distributions can be analyzed through the corresponding two-dimensional discrete-time QBD processes obtained by the uniformization technique; See, for example, Refs. [14, 20, 23]. In that sense, 2d-QBD processes are more versatile than 2d-RRWs, which have no background processes. This is a reason why we are interested in stochastic models with a background process. Here we emphasize that the assumption of skip-free is not so restricted since any 2d-MMRRW with bounded jumps can be represented as a 2d-MMRRW with skip-free jumps (i.e., 2d-QBD process); See Introduction of Ozawa [24].

Denote by $\mathscr{I}_{2}$ the set of all the subsets of $\{1,2\}$, i.e., $\mathscr{I}_{2}=\{\emptyset,\{1\},\{2\},\{1,2\}\}$, and we use it as an index set. Divide $\mathbb{Z}_{+}^{2}$ into $2^{2}=4$ exclusive subsets defined as

The class $\{\mathbb{B}^{\alpha};\alpha\in\mathscr{I}_{2}\}$ is a partition of $\mathbb{Z}_{+}^{2}$. $\mathbb{B}^{\emptyset}$ is the set containing only the origin, and $\mathbb{B}^{\{1,2\}}$ is the set of all positive points in $\mathbb{Z}_{+}^{2}$. Let $P$ be the transition probability matrix of the 2d-QBD process $\{\boldsymbol{Y}_{n}\}$ and represent it in block form as $P=\left(P_{\boldsymbol{x},\boldsymbol{x}^{\prime}};\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{Z}_{+}^{2}\right)$, where $P_{\boldsymbol{x},\boldsymbol{x}^{\prime}}=(p_{(\boldsymbol{x},j),(\boldsymbol{x}^{\prime},j^{\prime})};j,j^{\prime}\in S_{0})$ and $p_{(\boldsymbol{x},j),(\boldsymbol{x}^{\prime},j^{\prime})}=\mathbb{P}(\boldsymbol{Y}_{1}=(\boldsymbol{x}^{\prime},j^{\prime})\,|\,\boldsymbol{Y}_{0}=(\boldsymbol{x},j))$. For $\alpha\in\mathscr{I}_{2}$ and $i_{1},i_{2}\in\{-1,0,1\}$, let $A^{\alpha}_{i_{1},i_{2}}$ be a one-step transition probability block from a state in $\mathbb{B}^{\alpha}$, which is defined as

where we assume the blocks corresponding to impossible transitions are zero (see Fig. 1).

For example, if $\alpha=\{1\}$, we have $A^{\alpha}_{i,-1}=O$ for $i\in\{-1,0,1\}$. Since the level process is skip free, for every $\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{Z}_{+}^{2}$, $P_{\boldsymbol{x},\boldsymbol{x}^{\prime}}$ is given by

We assume the following condition throughout the paper.

Next, we define several Markov chains derived from the 2d-QBD process. For a nonempty set $\alpha\in\mathscr{I}_{2}$, let $\{\boldsymbol{Y}^{\alpha}_{n}\}=\{(\boldsymbol{X}^{\alpha}_{n},J^{\alpha}_{n})\}$ be a process derived from the 2d-QBD process $\{\boldsymbol{Y}_{n}\}$ by removing the boundaries that are orthogonal to the $x_{i}$-axis for each $i\in\alpha$. To be precise, the process $\{\boldsymbol{Y}^{\{1\}}_{n}\}$ is a Markov chain on $\mathbb{Z}\times\mathbb{Z}_{+}\times S_{0}$ whose transition probability matrix $P^{\{1\}}=(P^{\{1\}}_{\boldsymbol{x},\boldsymbol{x}^{\prime}};\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{Z}\times\mathbb{Z}_{+})$ is given as

where $\mathbb{N}$ is the set of all positive integers. The process $\{\boldsymbol{Y}^{\{2\}}_{n}\}$ on $\mathbb{Z}_{+}\times\mathbb{Z}\times S_{0}$ and its transition probability matrix $P^{\{2\}}=(P^{\{2\}}_{\boldsymbol{x},\boldsymbol{x}^{\prime}};\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{Z}_{+}\times\mathbb{Z})$ are analogously defined. The process $\{\boldsymbol{Y}^{\{1,2\}}_{n}\}$ is a Markov chain on $\mathbb{Z}^{2}\times S_{0}$, whose transition probability matrix $P^{\{1,2\}}=(P^{\{1,2\}}_{\boldsymbol{x},\boldsymbol{x}^{\prime}};\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{Z}^{2})$ is given as

Regarding $X^{\{1\}}_{1,n}$ as the additive part, we see that the process $\{\boldsymbol{Y}^{\{1\}}_{n}\}=\{(X^{\{1\}}_{1,n},(X^{\{1\}}_{2,n},J^{\{1\}}_{n}))\}$ is a Markov additive process (MA-process for short) with the background state $(X^{\{1\}}_{2,n},J^{\{1\}}_{n})$ (see, for example, Ney and Nummelin [18]). The process $\{\boldsymbol{Y}^{\{2\}}_{n}\}=\{(X^{\{2\}}_{2,n},(X^{\{2\}}_{1,n},J^{\{2\}}_{n}))\}$ is also an MA-process, where $X^{\{2\}}_{2,n}$ is the additive part and $(X^{\{2\}}_{1,n},J^{\{2\}}_{n})$ the background state, and $\{\boldsymbol{Y}^{\{1,2\}}_{n}\}=\{(X^{\{1,2\}}_{1,n},X^{\{1,2\}}_{2,n}),J^{\{1,2\}}_{n})\}$ an MA-process, where $(X^{\{1,2\}}_{1,n},X^{\{1,2\}}_{2,n})$ the additive part and $J^{\{1,2\}}_{n}$ the background state. We call them the induced MA-processes derived from the original 2d-QBD process. Their background processes are called induced Markov chains in Fayolle et. al. [4]. Let $\{\bar{A}^{\{1\}}_{i};i\in\{-1,0,1\}\}$ be the Markov additive kernel (MA-kernel for short) of the induced MA-process $\{\boldsymbol{Y}^{\{1\}}_{n}\}$, which is the set of transition probability blocks and defined as, for $i\in\{-1,0,1\}$,

Let $\{\bar{A}^{\{2\}}_{i};i\in\{-1,0,1\}\}$ be the MA-kernel of $\{\boldsymbol{Y}^{\{2\}}_{n}\}$, defined in the same manner. With respect to $\{\boldsymbol{Y}^{\{1,2\}}_{n}\}$, the MA-kernel is given by $\{A^{\{1,2\}}_{i_{1},i_{2}};i_{1},i_{2}\in\{-1,0,1\}\}$. We assume the following condition throughout the paper.

Let $\bar{A}^{\{1\}}_{*}(z)$ and $\bar{A}^{\{2\}}_{*}(z)$ be the matrix generating functions of the kernels of $\{\boldsymbol{Y}^{\{1\}}_{n}\}$ and $\{\boldsymbol{Y}^{\{2\}}_{n}\}$, respectively, defined as

The matrix generating function of the kernel of $\{\boldsymbol{Y}^{\{1,2\}}_{n}\}$ is given by $A^{\{1,2\}}_{*,*}(z_{1},z_{2})$, defined as

Note that we use generating functions instead of moment generating functions in the paper because the generating functions are more suitable for complex analysis. Let $\Gamma^{\{1\}}$, $\Gamma^{\{2\}}$ and $\Gamma^{\{1,2\}}$ be regions in which the convergence parameters of $\bar{A}^{\{1\}}_{*}(e^{\theta_{1}})$, $\bar{A}^{\{2\}}_{*}(e^{\theta_{2}})$ and $A^{\{1,2\}}_{*,*}(e^{\theta_{1}},e^{\theta_{2}})$ are greater than $1$, respectively, i.e.,

By Lemma A.1 of Ozawa [24], $\mbox{\rm cp}(\bar{A}^{\{1\}}_{*}(e^{\theta}))^{-1}$ and $\mbox{\rm cp}(\bar{A}^{\{2\}}_{*}(e^{\theta}))^{-1}$ are log-convex in $\theta$, and the closures of $\Gamma^{\{1\}}$ and $\Gamma^{\{2\}}$ are convex sets; $\mbox{\rm cp}(\bar{A}^{\{1,2\}}_{*}(e^{\theta_{1}},e^{\theta_{2}}))^{-1}$ is also log-convex in $(\theta_{1},\theta_{2})$, and the closure of $\Gamma^{\{1,2\}}$ is a convex set. Furthermore, by Proposition B.1 of Ozawa [24], $\Gamma^{\{1,2\}}$ is bounded under Assumption 1.2.

Assuming $\{\boldsymbol{Y}_{n}\}$ is positive recurrent (a condition for this will be given in the next section), we denote by $\boldsymbol{\nu}$ the stationary distribution of $\{\boldsymbol{Y}_{n}\}$, where $\boldsymbol{\nu}=(\boldsymbol{\nu}_{\boldsymbol{x}},\boldsymbol{x}\in\mathbb{Z}_{+}^{2})$, $\boldsymbol{\nu}_{\boldsymbol{x}}=(\nu_{(\boldsymbol{x},j)},j\in S_{0})$ and $\nu_{(\boldsymbol{x},j)}$ is the stationary probability that the 2d-QBD process is in the state $(\boldsymbol{x},j)$ in steady state. Let $\boldsymbol{c}=(c_{1},c_{2})\in\mathbb{N}^{2}$ be an arbitrary discrete direction vector and, for $i\in\{1,2\}$, define a real value $\theta_{\boldsymbol{c},i}^{\dagger}$ as

where $\langle\boldsymbol{a},\boldsymbol{b}\rangle$ is the inner product of vectors $\boldsymbol{a}$ ad $\boldsymbol{b}$. Our main aim is to demonstrate under certain conditions that, for any $j\in S_{0}$,

i.e., the asymptotic decay rate of the stationary distribution in direction $\boldsymbol{c}$ is given by the smaller of $\theta_{\boldsymbol{c},1}^{\dagger}$ and $\theta_{\boldsymbol{c},2}^{\dagger}$. We also present a condition ensuring the sequence $\{\nu_{(k\boldsymbol{c},j)}\}_{k\geq 0}$ geometrically decays without power terms. We prove them by using the matrix analytic method of the queueing theory as well as the complex analytic method; the former has been introduced by Marcel Neuts and developed in the literature; See, for example, Refs. [1, 10, 16, 17]. Our model is a kind of multidimensional reflecting process, and asymptotics in various multidimensional reflecting processes have been investigated in the literature for several decades; See Miyazawa [13] and references therein. $0$-partially homogeneous ergodic Markov chains discussed in Borovkov and Mogul’skiĭ [2] are Markov chains on the positive quadrant including 2d-RRWs as a special case. For those Markov chains, a formula corresponding to (1.5) have been obtained by using the large deviations techniques; See Theorem 3.1 of Borovkov and Mogul’skiĭ [2] and also see Proposition 5.1 of Miyazawa [12] for the case of 2d-RRW. They have considered only models without background processes. In Dai and Miyazawa [3], results parallel to ours have been obtained for a two-dimensional continuous-state Markov process, named semimartingale-reflecting Brownian motion (SRBM for short). The 2d-SRBM is also a model without background processes. With respect to models with a background process, asymptotics of the stationary distribution in a Markov modulated fluid network with a finite number of stations have recently been studied in Miyazawa [15], where upper and lower bounds for the stationary tail decay rate in various directions were obtained by using so-called Dynkin’s formula.

The rest of the paper is organized as follows. In Section 2, we give a stability condition for the 2d-QBD process and define the asymptotic decay rates of the stationary distribution. In the same section, we introduce a key formula representing the stationary distribution in terms of the fundamental (potential) matrix of the induced MA-process $\{\boldsymbol{Y}_{n}^{\{1,2\}}\}$. We call it a compensation equation. Furthermore, we define block state processes derived from the original 2d-QBD process, which will be used for proving propositions in the following sections. A summary of their properties is given in Appendix A. In Section 3, we obtain the asymptotic decay rate of the stationary distribution in any direction. First, we obtain it in the case where the direction vector is given by $\boldsymbol{c}=(1,1)$. The asymptotic decay rate for a general direction vector is obtained from the results in the case of $\boldsymbol{c}=(1,1)$, by using the block state process. In Section 4, we explain a geometric property of the asymptotic decay rates and give an example of two-queue model. In the two-queue model, the asymptotic decay rate corresponds to the decreasing rate of the joint queue length probability in steady state when the queue lengths of both the queues simultaneously enlarge. The paper concludes with a remark about the relation between our analysis and the large deviation techniques in Section 5.

Notation for vectors and matrices. For a matrix $A$, we denote by $[A]_{i,j}$ the $(i,j)$-entry of $A$ and by $A^{\top}$ the transpose of $A$. If $A=(a_{i,j})$, $|A|=(|a_{i,j}|)$. Similar notations are also used for vectors. The convergence parameter of a nonnegative square matrix $A$ with a finite or countable dimension is denoted by $\mbox{\rm cp}(A)$, i.e., $\mbox{\rm cp}(A)=\sup\{r\in\mathbb{R}_{+};\sum_{n=0}^{\infty}r^{n}A^{n}<\infty,\ \mbox{entry-wise}\}$. For a finite square matrix $A$, we denote by $\mbox{\rm spr}(A)$ the spectral radius of $A$, which is the maximum modulus of eigenvalue of $A$. If $A$ is nonnegative, $\mbox{\rm spr}(A)$ corresponds to the Perron-Frobenius eigenvalue of $A$ and we have $\mbox{\rm spr}(A)=\mbox{\rm cp}(A)^{-1}$. $O$ is a matrix of $0$’s, $\mathbf{1}$ is a column vector of $1$’s and $\mathbf{0}$ is a column vector of $0$’s; their dimensions, which are finite or countably infinite, are determined in context. $I$ is the identity matrix. For an $n_{1}\times n_{2}$ matrix $A=(a_{i,j})$, $\mbox{\rm vec}(A)$ is the vector of stacked columns of $A$, i.e., $\mbox{\rm vec}(A)=(a_{1,1},\cdots,a_{n_{1},1},a_{1,2},\cdots,a_{n_{1},2},\cdots,a_{1,n_{2}},\cdots,a_{n_{1},n_{2}})^{\top}$.