Departure processes and busy periods of a tandem network

Abstract

Consider a single-server tandem queueing network with a Markovian arrival process (MAP) to the first station and exponential service times. Using the dimension-reduction and expanding blocks (DREB) scheme, we formulate the joint queue length process into a single-dimensional level-dependent quasi-birth-death (LDQBD) process with expanding blocks. This allows us to show that the departure process from each station is a MAP with infinite phases or an IMAP, which implies that the departure process of any IMAP/M/1 is an IMAP. It also allows us to study the inter-departure time and the busy period in each station easily.

Appendix: proof of theorem 2

As we can see from (5) that $\tilde{{\bf X}}$ is the very special M/G/1 type of Markov chain. Apply RG-factorization method (see Li and Zhao 2002; Li et al. 2005) to $\tilde{{\bf X}},$ we have

$$ \tilde{{\bf X}} - I = (I-{\bf R}_U){\bf U}_D(I-{\bf G}_L), $$

(29)

where

$$ I-{\bf R}_U=\left( \begin{array}{ccccc} I & -{\bf R}_1^0 & -{\bf R}_2^0 &-{\bf R}_3^0 & \cdots\cr &I &-{\bf R}_1 & -{\bf R}_2 & \cdots\cr && I & -{\bf R}_1 & \cdots\cr && & I & \cdots\cr &&&& \ddots \end{array} \right), $$

(30)

$$ {\bf U}_D = diag({\bf U}_0, {\bf U}, {\bf U}, \ldots), $$

(31)

$$ I-{\bf G}_L = \left( \begin{array}{ccccc} I &&&&\cr -{\bf G} & I&&&\cr &-{\bf G} & I&&\cr &&-{\bf G} & I&\cr && & \ddots & \ddots \end{array} \right). $$

(32)

Similar to Lemma 2 of Li, Lian and Liu [Li et al. 2005], by comparing (29) with (5), we have

$$ {\bf U}_0+{\bf R}_1^0{\bf UG}={\bf X}_0{\bf X}-I, $$

(33)

$$ -{\bf R}_k^0{\bf U}+{\bf R}_{k+1}^0{\bf UG} ={\bf X}_0{\bf X}^{k+1},\quad k\geq 1, $$

(34)

$$- {\bf UG} ={\bf X}_1 , $$

(35)

$$ {\bf U}+{\bf R}_1{\bf UG}={\bf X}_1{\bf X}-I, $$

(36)

$$ -{\bf R}_k{\bf U}+{\bf R}_{k+1}{\bf G}={\bf X}_1{\bf X}^{k+1},\quad k\geq 1. $$

(37)

Notice that $(I-{\bf X}{\bf G})^{-1}=\sum\limits_{n=0}^{\infty} {\bf X}^n{\bf G}^n,$ it is easy to obtain that

$$ {\bf U}_0={\bf X}_0{\bf X}(I-{\bf X}{\bf G})^{-1} -I , $$

(38)

for all k ≥ 0, 

$$ -{\bf R}_k^0{\bf U}={\bf X}_0{\bf X}^{k+1}(I-{\bf X}{\bf G})^{-1},\quad k\geq 0, $$

(39)

so

$$ {\bf R}_k^0 = -{\bf X}_0{\bf X}^{k+1}(I-{\bf X}{\bf G})^{-1}{\bf U}^{-1}. $$

(40)

Similarly, we have

$$ {\bf U}={\bf X}_1{\bf X}(I-{\bf X}{\bf G})^{-1} -I , $$

(41)

$$ {\bf R}_k = -{\bf X}_1{\bf X}^{k+1}(I-{\bf X}{\bf G})^{-1}{\bf U}^{-1}, k\geq 0. $$

(42)

Compare (35) and (41), we obtain

$$ {\bf G}={\bf X}_1+{\bf X}_1{\bf X}(I-{\bf X}{\bf G})^{-1}{\bf G}. $$

(43)

A discussion similar to the proof of theorem 2.2.2 in Neuts [Neuts 1989] leads to the conclusion that G is the minimal nonnegative solution to the system of matrix equations in (43).

Similar to Theorem 3.2 of (Li et al. 2005), we can easily obtain (6) and (7). $\square$