Revisiting the Tail Asymptotics of the Double QBD Process: Refinement and Complete Solutions for the Coordinate and Diagonal Directions

Abstract

We consider a two-dimensional skip-free reflecting random walk on a nonnegative integer quadrant. We are interested in the tail asymptotics of its stationary distribution, provided its existence is assumed. We derive exact tail asymptotics for the stationary probabilities on the coordinate axis. This refines the asymptotic results in the literature and completely solves the tail asymptotic problem on the stationary marginal distributions in the coordinate and diagonal directions. For this, we use the so-called analytic function method in such a way that either generating functions or moment-generating functions are suitably chosen. The results are exemplified by a two-node network with simultaneous arrivals.

Appendix

Proof of Lemma 8.6

Note that u ² D ₂(u) is a polynomial of order 2 at least and order 4 at most. For k = 1, 3, let c _k be the coefficients of u ^k in the polynomial u ² D ₂(u). Then,

$$\begin{array}{rcl} & & {c}_{1} = -2(1 - {p}_{00}){p}_{(-1)0} - 4({p}_{(-1)(-1)}{p}_{01} + {p}_{(-1)1}{p}_{0(-1)}) \leq 0, \\ & & {c}_{3} = -2(1 - {p}_{00}){p}_{10} - 4({p}_{1(-1)}{p}_{01} + {p}_{11}{p}_{0(-1)}) \leq 0. \end{array}$$

Hence, if both u ₁ ^(1, max) and − u ₁ ^(1, max) are the solutions of u ² D ₂(u) = 0, then

$$\begin{array}{rcl} 2({c}_{1}{u}_{1}^{(1,\max )} + {c}_{ 3}{({u}_{1}^{(1,\max )})}^{3}) = {({u}_{ 1}^{(1,\max )})}^{2}({D}_{ 2}({u}_{1}^{(1,\max )}) - {D}_{ 2}(-{u}_{1}^{(1,\max )})) = 0.& & \\ \end{array}$$

Since u ₁ ^(1, max) > 0, this holds true if and only if ${c}_{1} = {c}_{3} = 0$, which is equivalent to ${p}_{01} = {p}_{0(-1)} = {p}_{(-1)0} = {p}_{10} = 0$ because p ₀₀ = 1 is impossible. Hence, u ² D ₂(u) = 0 has the two solutions u ₁ ^(1, max) and − u ₁ ^(1, max) if and only if (v-a) does not hold. In this case, we have ${c}_{1} = {c}_{3} = 0$, which implies that u ² D ₂(u) is an even function. Since u ² D ₂(u) = 0 has only real solutions including u ₁ ^(1, max) by Lemmas 8.4 and 8.5, we complete the proof. □ 

Proof of Lemma 8.7

By (8.17), we have

$$\begin{array}{rcl} {\sum \limits_{i\in \{-1,0,1\}}}{ \sum \limits_{j\in \{-1,0,1\}}}{p}_{ij}{x}^{i}{y}^{j} = 1,\qquad {\sum \limits_{i\in \{-1,0,1\}}}{ \sum \limits_{j\in \{-1,0,1\}}}{(-1)}^{i+j}{p}_{ ij}{x}^{i}{y}^{j} = 1.& & \\ \end{array}$$

Subtracting both sides of these equations we have

$$\begin{array}{rcl}{ p}_{10}x + {p}_{01}y + {p}_{0(-1)}{y}^{-1} + {p}_{ (-1)0}{x}^{-1} = 0.& & \\ \end{array}$$

Since x, y are positive, this equation holds true if and only if

$$\begin{array}{rcl}{ p}_{10} = {p}_{01} = {p}_{0(-1)} = {p}_{(-1)0} = 0.& & \\ \end{array}$$

This is the condition that (v-a) does not hold. □