Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

Abstract

In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. An application in the modelling of a priority retrial system with coupled orbit queues is also presented. Using a queueing network example, we numerically validated the theoretical findings.

Appendices

Appendix A

Theorem 5

For every $|x|=1$, $x\ne 1$, $G(x,y)=0$ has a unique zero, say Y(x), in the disc $|y|<1$.

Proof

In order to enhance the readability, we consider the case where $\theta _{i,j}=0$, $i,j=1,\ldots ,N$, so that T(x, y) is given in (16) (Sect. 3.2). Analogous arguments can be applied even in the general case, where $\theta _{i,j}>0$, $i,j=1,\ldots ,N$.

The proof is based on the application of Rouché’s theorem (Titchmarsh 1939; Adan et al. 2006). Note that $G(x,y)=0$ is rewritten as $G_{0}(x,y)=h(x,y):=y\sum _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}A_{0,j}(x,y)A_{j,0}(x,y)}{D_{j}(x,y)}$, where

$$\begin{aligned} G_{0}(x,y)=&yD_{0}(x,y)+\sum \nolimits _{i=0}^{\infty }q^{(2)}_{i,-1}(y-x^{i}). \end{aligned}$$

It is easy to see that $G_{0}(x,y)=f(x,y)-g(x,y)$, with

$$\begin{aligned} f(x,y)=&y\left[ \theta _{0,.}+\sum \nolimits _{i=1}^{\infty }q_{i,0}(0)(1-x^{i})+\sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0) \right. \\&\left. +\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)\right] ,\\ g(x,y)=&\sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)x^{i}y^{j}+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)y^{j}\\&+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)x^{i}. \end{aligned}$$

We first show that $G_{0}(x,y)=0$ has a unique root in $|y|<1$, for $|x|=1$, $x\ne 1$. Then, for $|x|=1$, $x\ne 1$,

$$\begin{aligned} |f(x,y)|&=|y||\theta _{0,.}+\sum \nolimits _{i=1}^{\infty }q_{i,0}(0)(1-x^{i})+\sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)\\&+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)|\\&\ge |y|\left[ \theta _{0,.}+\sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)\right] \\&>|y|\left[ \sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)\right] , \\ |g(x,y)|&\le \sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)|x|^{i}|y|^{j}+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)|y|^{j}+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)|x|^{i}\\&=\sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)|y|^{j}+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)|y|^{j}+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0) \end{aligned}$$

Then, for all y such that $|y|=1$ have that

$$\begin{aligned} |g(x,y)|\le & {} \sum \nolimits _{i=1}^{\infty }\sum \nolimits _{j=1}^{\infty }q_{i,j}(0)+\sum \nolimits _{j=1}^{\infty }q_{0,j}(0)\\&+\sum \nolimits _{i=0}^{\infty }q_{i,-1}(0)<|f(x,y)|,\,|y|=1,|x|=1,x\ne 1, \end{aligned}$$

which implies by Rouché’s theorem, see, e.g. Titchmarsh (1939), that $G_{0}(x,y)$ has as many zeros, counted according to their multiplicity, inside $|y| = 1$ as f(x, y). Since f(x, y) has only one zero of multiplicity 1 at $y = 0$, yields that for every x with $|x| = 1$, $x\ne 1$, $G_{0}(x,y) = 0$ has one root inside $|y| = 1$.

Now, note that

$$\begin{aligned} |G_{0}(x,y)|=|f(x,y)-g(x,y)|\ge ||f(x,y)|-|g(x,y)||>\theta _{0,.}. \end{aligned}$$

Moreover,

$$\begin{aligned} |-h(x,y)|=&|-y\sum \nolimits _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}A_{0,j}(x,y)A_{j,0}(x,y)}{D_{j}(x,y)} |\\ \le&|y|\sum \nolimits _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}|A_{0,j}(x,y)||A_{j,0}(x,y)|}{|D_{j}(x,y)|}\\ <&\sum \nolimits _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}}{\theta _{j,0}}=\theta _{0,.}, \end{aligned}$$

since $|A_{0,j}(x,y)|=1$, $|A_{j,0}(x,y)|=1$, $|D_{j}(x,y)|\ge \theta _{j,0}$, for $|x|=1$, $|y|=1$, $x\ne 1$. Thus,

$$\begin{aligned} |-h(x,y)|<\theta _{0,.}<|G_{0}(x,y)|,\,|x|=1, |y|=1, x\ne 1. \end{aligned}$$

Therefore, Rouché’s theorem implies that $G_{0}(x,y)=y\sum _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}}{D_{j}(x,y)}$, i.e., $G(x,y)=0$ has the same number of zeros for every x with $|x| = 1$, $x\ne 1$, inside $|y|=1$, as $G_{0}(x,y)$, which we shown that has exactly one. Thus, denote this zero as $y=Y(x)$, $|x| = 1$, $x\ne 1$, with $|Y(x)|<1$. $\square$

Appendix B

Proof of Theorem 2:

To enhance the readability, we consider the case where $\theta _{i,j}=0$, $i,j=1,\ldots ,N$, and considered the QBD version of $\{Z(t);t\ge 0\}$, i.e, ${\mathbb {H}}=\{-1,0,1\}$, ${\mathbb {H}}^{+}=\{0,1\}$. Analogous arguments can be applied even in the general case. The proof is based on the application of Rouché’s theorem (Titchmarsh 1939; Adan et al. 2006).

Note that $U(gs,gs^{-1})=0$ is written as $R(gs,gs^{-1})=g^{2}\sum _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}}{D_{j}(gs,gs^{-1})}$. We first show that $R(gs,gs^{-1})=0$ has a single root in $|g|\le 1$, $|s|=1$. Indeed, $R(gs,gs^{-1}=0)$ is rewritten as

$$\begin{aligned}&g^{2}=L(gs,gs^{-1})\\&\quad :=\frac{q_{-1,0}(0)gs^{-1}+q_{0,-1}(0)gs}{D_{0}(gs,gs^{-1})+q_{-1,0}(0)+q_{0,-1}(0)+q_{-1,1}(0)(1-s^{-2})+q_{1,-1}(0)(1-s^{2})}. \end{aligned}$$

Note that the denominator of $L(gs,gs^{-1})$ never vanishes for $|g|\le 1$, $|s|=1$ and in particular,

$$\begin{aligned}&|D_{0}(gs,gs^{-1})+q_{-1,0}(0)+q_{0,-1}(0)+q_{-1,1}(0)(1-s^{-2})+q_{1,-1}(0)(1-s^{2})|\\&\quad >\theta _{0,.}+q_{-1,0}(0)+q_{0,-1}(0). \end{aligned}$$

Therefore,

$$\begin{aligned}&|L(gs,gs^{-1})|\\ \le&\frac{q_{-1,0}(0)|g|+q_{0,-1}(0)|g|}{|D_{0}(gs,gs^{-1})+q_{-1,0}(0)+q_{0,-1}(0)+q_{-1,1}(0)(1-s^{-2})+q_{1,-1}(0)(1-s^{2})|}\\<&\frac{q_{-1,0}(0)+q_{0,-1}(0)}{\theta _{0,.}+q_{-1,0}(0)+q_{0,-1}(0)}<1=|g|^{2}. \end{aligned}$$

Thus, by applying Rouché’s theorem $R(gs,gs^{-1})=0$ has a single zero in $|g|<1$, $|s|=1$. Note also that

$$\begin{aligned} |R(gs,gs^{-1})|\ge |g|^{2}(\theta _{0.}+q_{1,0}(1-gs)+q_{0,1}(1-gs^{-1}+q_{1,1}(0))(1-|g|^{2}))>\theta _{0,.}. \end{aligned}$$

On the other hand,

$$\begin{aligned}&|g^{2}\sum \nolimits _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}A_{0,j}(gs,gs^{-1})A_{j,0}(gs,gs^{-1})}{D_{j}(gs,gs^{-1})} |\\&\quad \le \sum \nolimits _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}}{|D_{j}(gs,gs^{-1})|}<\theta _{0,.}<|R(gs,gs^{-1})|. \end{aligned}$$

Therefore, since $R(gs,gs^{-1})$ has a single zero in $|g|<1$, $|s=1|$, Rouché’s theorem states that $R(gs,gs^{-1})=g^{2}\sum _{j=1}^{N}\frac{\theta _{j,0}\theta _{0,j}A_{0,j}(gs,gs^{-1})A_{j,0}(gs,gs^{-1})}{D_{j}(gs,gs^{-1})}$ has a single root in $|g|<1$, $|s=1|$. Equivalently, $U(gs,gs^{-1})=0$ has a a single root in $|g|<1$, $|s=1|$. $\square$