A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $GI ^{[X]}/C$-$MSP /1/\infty$

Abstract

We consider a batch arrival infinite-buffer single-server queue with generally distributed inter-batch arrival times with arrivals occurring in batches of random sizes. The service process is correlated and its structure is governed by a Markovian service process in continuous time. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at a pre-arrival epoch. We also obtain the steady-state probability distribution at an arbitrary epoch using the classical argument based on Markov renewal theory. Some important performance measures such as the average number of customers in the system and the mean sojourn time have also been obtained. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

Appendices

Appendix 1

Theorem 7.1

Every function $z^{\widehat{r}}-(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})S_{i,i}(z)$, $1\le i\le m$ has exactly $\widehat{r}$ zeroes inside the unit circle.

Proof

Consider absolute values of $f(z)=z^{\widehat{r}}$ and $\bar{F}(z)=-(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})S_{i,i}(z)$ on the circle $|z|=1-\delta$, where $\delta$ is positive and sufficiently small. First, note that $\mathbf{S}\equiv \mathbf{S}(1)$ is the imbedded transition probability matrix of J(t) in a random amount of time with the distribution of an inter-batch arrival time. Since the state space of J(t) is finite, $\mathbf{S}$ is an irreducible and aperiodic discrete-time Markov chain, which is necessarily ergodic. This implies, in particular, that $S_{i,i}(1)\le 1,$ for $1\le i\le m.$ As Eq. (7) yields $\mathbf{S}^{\prime }(1)=\sum _{n=1}^{\infty }n\mathbf{S}_n$ which represents the mean number of customers served and the phase changes of the underlying Markov chain, we have $S^{\prime }_{i,i}(1)\ge 0$. Now let us consider the following inequality for $|\bar{F}(z)|$, with $|z|=1-\delta$,

$$\begin{aligned} |\bar{F}(z)|\le & {} \Big (g_1(1-\delta )^{\widehat{r}-1}+g_2(1-\delta )^{\widehat{r}-2}+\cdots +g_{\widehat{r}}\Big )\{S_{i,i}(1)-\delta S^{\prime }_{i,i}(1)+o(\delta )\}\nonumber \\= & {} \Big ((g_1-g_1\delta ({\widehat{r}-1}) +o(\delta ))+(g_2-g_2\delta ({\widehat{r}-2})+o(\delta ))+\cdots +(g_{\widehat{r}-1}-\delta g_{\widehat{r}-1})\nonumber \\&+\,g_{\widehat{r}}\Big )\Big \{S_{i,i}(1)-\delta S^{\prime }_{i,i}(1)+o(\delta )\Big \}\nonumber \\= & {} \Big ( 1 - \delta (\widehat{r}-\bar{g})+o(\delta )\Big )\Big \{S_{i,i}(1)-\delta S^{\prime }_{i,i}(1)+o(\delta )\Big \} \nonumber \\= & {} S_{i,i}(1) -\delta \Big ((\widehat{r}-\bar{g})S_{i,i}(1) + S^{\prime }_{i,i}(1)\Big ) +o(\delta ). \end{aligned}$$

(87)

$$\begin{aligned}&\text{ Also }\quad ~ |f(z)|=|z|^{\widehat{r}}=1-\delta \widehat{r} +o(\delta ). \end{aligned}$$

(88)

We note that $\mathbf{S} = \mathbf{S}(1)$ is a stochastic matrix which represents the probabilities of phase changes of the underlying Markov chain during a busy period of an inter-batch arrival time period. Since $\mathbf{S}(1-\delta )\mathbf{e}\le \mathbf{S}(1)\mathbf{e}=\mathbf{e}$, we have $S_{ii}(1-\delta )\le 1-\sum \limits _{j\ne i}S_{i,j}(1-\delta )$. Thus, (87) yields

$$\begin{aligned} |\bar{F}(z)|\le & {} S_{i,i}(1) -\delta \Big ((\widehat{r}-\bar{g})S_{i,i}(1) + S^{\prime }_{i,i}(1)\Big ) +o(\delta ) \nonumber \\= & {} S_{i,i}(1)-\delta S^{\prime }_{i,i}(1)+o(\delta ) -\delta (\widehat{r}-\bar{g})S_{i,i}(1), \nonumber \\= & {} S_{i,i}(1-\delta )-\delta (\widehat{r}-\bar{g})S_{i,i}(1)+ o(\delta ), \quad \hbox {using Taylor's series expansion}\nonumber \\\le & {} 1-\sum \limits _{j=1,~j\ne i}^mS_{i,j}(1-\delta )-\delta (\widehat{r}-\bar{g})S_{i,i}(1)+o(\delta ) \end{aligned}$$

(89)

$$\begin{aligned}< & {} 1-\delta \widehat{r}=|f(z)|, \quad \text{ since }~\sum \limits _{ j\ne i}S_{i,j}(1-\delta )> 0,\quad \widehat{r}\ge \bar{g}\quad \text{ and }\quad 0\le S_{i,i}(1)\le 1.\nonumber \\ \end{aligned}$$

(90)

Hence, using the well-known Rouché’s theorem, f(z) and $f(z)+\bar{F}(z)$ have the same number of zeros inside the unit circle. It is obvious that $f(z)=z^{\widehat{r}}$ has exactly $\widehat{r}$ zeroes inside the unit circle. Thus, $f(z)+\bar{F}(z)= z^{\widehat{r}}-(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2} +g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})S_{i,i}(z)$ has exactly $\widehat{r}$ zeroes inside the unit circle.

It may be noted that this theorem is similar to Lemma 2.2 of Adan et al. (2006).

Theorem 7.2

The following inequalities hold on the circle $|z|=1-\delta$:

$$\begin{aligned} |z^{\widehat{r}}-F(z)S_{i,i}(z)|>\sum \limits _{ j=1,~j\ne i}^m|F(z)S_{i,j}(z)|,~1\le i\le m, \end{aligned}$$

(91)

where $F(z)=(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}}).$

Proof

On the circle $|z|=1-\delta$, using the Taylor series expansion, we have

$$\begin{aligned} |z^{\widehat{r}}-F(z)S_{i,i}(z)|\ge & {} |z^{\widehat{r}}|-|(g_1z^{\widehat{r}-1} +g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})S_{i,i}(z)|\\\ge & {} 1-\delta \widehat{r}-\Big (S_{i,i}(1) -\delta ((\widehat{r}-\bar{g})S_{i,i}(1) + S^{\prime }_{i,i}(1)\Big )+o(\delta )\\= & {} \sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)-\delta \Big (\widehat{r}(1-S_{i,i}(1))+\bar{g}S_{i,i}(1)-S^{\prime }_{i,i}(1)\Big )+o(\delta ), \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{ j=1,~j\ne i}^m|F(z)S_{i,j}(z)|\le & {} \sum \limits _{ j=1,~j\ne i}^m\left( g_1|z|^{\widehat{r}-1}+g_2|z|^{\widehat{r}-2}+g_3|z|^{\widehat{r}-3}+\cdots +g_{\widehat{r}}\right) S_{i,j}(|z|)\\\le & {} \sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)-\delta \left( (\widehat{r}-\bar{g})\sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)\right. \\&+\left. \sum \limits _{ j=1,~j\ne i}^mS^{\prime }_{i,j}(1)\right) +o(\delta ). \end{aligned}$$

Now, assume that the following inequality holds:

$$\begin{aligned} |z^{\widehat{r}}-F(z)S_{i,i}(z)| \le \sum \limits _{ j=1,~j\ne i}^m|F(z)S_{i,j}(z)|,\quad 1\le i\le m. \end{aligned}$$

This implies that

$$\begin{aligned}&\sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)-\delta \left( \widehat{r}(1-S_{i,i}(1))+\bar{g}S_{i,i}(1)-S^{\prime }_{i,i}(1)\right) +o(\delta )\\&\quad \le \sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)-\delta \left( (\widehat{r}-\bar{g})\sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)\right. \nonumber \\&\qquad +\left. \sum \limits _{ j=1,~j\ne i}^mS^{\prime }_{i,j}(1)\right) +o(\delta )\\&\quad \Rightarrow \bar{g}S_{i,i}(1)+\widehat{r}-\widehat{r}S_{i,i}(1)-S^{\prime }_{i,i}(1)\ge (\widehat{r}-\bar{g})\sum \limits _{ j=1,~j\ne i}^mS_{i,j}(1)\nonumber \\&\qquad +\sum \limits _{ j=1,~j\ne i}^mS^{\prime }_{i,j}(1)\\&\quad \Rightarrow \bar{g}+\widehat{r} - \widehat{r}\sum \limits _{ j=1}^mS_{i,j}(1) \ge \sum \limits _{ j=1}^mS^{\prime }_{i,j}(1) \\&\quad \Rightarrow \bar{g} \ge \sum \limits _{j=1}^mS^{\prime }_{i,j}(1)\\&\quad \Rightarrow 1 \ge \frac{1}{\bar{g}}\cdot \sum \limits _{ j=1}^mS^{\prime }_{i,j}(1) \\&\quad \Rightarrow \overline{\pi }_i \ge \frac{1}{\bar{g}}\cdot \overline{\pi }_i\sum \limits _{ j=1}^mS^{\prime }_{i,j}(1) \\&\quad \Rightarrow \sum \limits _{ i=1}^m \overline{\pi }_i\ge \frac{1}{\bar{g}}\cdot \sum \limits _{ i=1}^m \overline{\pi }_i\sum \limits _{j=1}^mS^{\prime }_{i,j}(1)= \frac{1}{\bar{g}}\cdot \overline{\varvec{\pi }}\mathbf{S}^{\prime }(1)\mathbf{e}\\&\quad \Rightarrow 1\ge \frac{1}{\rho }\Rightarrow \rho \ge 1. \end{aligned}$$

This contradicts the system stability condition $\rho <1$, and hence Eq. (91) is satisfied.

Theorem 7.3

The determinant $det[z^{\widehat{r}}\mathbf{I}_m-(g_1z^{\widehat{r}-1} +g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})\mathbf{S}(z)]$ has exactly $m\widehat{r}$ zeros inside the unit circle.

Proof

Mathematical induction is used to prove this theorem. Let us denote

$$\begin{aligned} \mathbf{D}_n(z)=det\left[ z^{\widehat{r}}\mathbf{I}_n-(g_1z^{\widehat{r}-1} +g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})\mathbf{S}_n(z)\right] , \quad 1\le n\le m, \nonumber \\ \end{aligned}$$

(92)

where $\mathbf{S}_n(z)$ is the principal minor of order n of the matrix $\mathbf{S}(z)$ starting from the element $S_{1,1}(z)$.

First, we show that the statement is true for $n=1$.

For $n=1$, Eq. (92) becomes $\mathbf{D}_1(z)=z^{\widehat{r}}-(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})S_{1,1}(z)$. It has been proved at several places that $z^{\widehat{r}}=(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2}+g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})S(z)$ has exactly $\widehat{r}$ roots inside the unit circle as $\rho <1$, see, e.g., Chaudhry and Templeton (1983).

Next for $n=2$, Eq. (92) becomes

$$\begin{aligned} \mathbf{D}_2(z)= & {} \left| \begin{array}{cc} z^{\widehat{r}}-F(z)S_{1,1}(z) &{} -F(z)S_{1,2}(z) \\ -F(z)S_{2,1}(z) &{} z^{\widehat{r}}-F(z)S_{2,2}(z) \end{array} \right| \nonumber \\= & {} -F(z)S_{2,1}(z)C_{2,1}(z)+\left[ z^{\widehat{r}}-F(z)S_{2,2}(z)\right] \mathbf{D}_1(z), \end{aligned}$$

(93)

where $F(z)=(g_1z^{\widehat{r}-1}+g_2z^{\widehat{r}-2} +g_3z^{\widehat{r}-3}+\cdots +g_{\widehat{r}})$ and $C_{2,1}(z)=-F(z)S_{1,2}(z)$ is the cofactor of $-F(z)S_{2,1}(z)$.

Again, we can write (93) as

$$\begin{aligned} \left| \frac{\mathbf{D}_2(z)-\left[ z^{\widehat{r}}-F(z)S_{2,2}(z)\right] \mathbf{D}_1(z)}{\left[ z^{\widehat{r}}-F(z)S_{2,2}(z)\right] \mathbf{D}_1(z)}\right|= & {} \left| \frac{-F(z)S_{2,1}(z)C_{2,1}(z)}{\left[ z^{\widehat{r}}-F(z)S_{2,2}(z)\right] \mathbf{D}_1(z)}\right| \nonumber \\= & {} \frac{\left| F(z)S_{2,1}(z)\right| \left| y_{2,1}(z)\right| }{\left| z^{\widehat{r}}-F(z)S_{2,2}(z)\right| }<1, \end{aligned}$$

(94)

where

$$\begin{aligned} |y_{2,1}(z)|=\frac{|C_{2,1}(z)|}{|\mathbf{D}_1(z)|}=\frac{|F(z)S_{1,2}(z)|}{|z^{\widehat{r}}-F(z)S_{1,1}(z)|}<1 \end{aligned}$$

and

$$\begin{aligned} \frac{\left| F(z)S_{2,1}(z)\right| }{\left| z^{\widehat{r}} -F(z)S_{2,2}(z)\right| }<1 \end{aligned}$$

by Theorem 7.2 for $i=1$ and $i=2$, respectively.

Hence, by Rouché’s theorem $\mathbf{D}_2(z)$ has $2\widehat{r}$ roots inside the unit circle, $|z|=1$, since $(z^{\widehat{r}}-F(z)S_{2,2}(z))\mathbf{D}_1(z)$ has $2\widehat{r}$ roots.

Finally, we assume that the statement is true for $n=m-1.$ It must then be shown that the statement holds for $n=m$.

The determinant $\mathbf{D}_n(z)$ for $n=m$ is given by

$$\begin{aligned}&\mathbf{D}_m(z) \nonumber \\&\quad =\left| \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} z^{\widehat{r}}-F(z)S_{1,1}(z) &{} -F(z)S_{1,2}(z) &{}\cdots &{} -F(z)S_{1,m-1}(z) &{}-F(z)S_{1,m}(z) \\ -F(z)S_{2,1}(z) &{} z^{\widehat{r}}-F(z)S_{2,2}(z) &{}\cdots &{} -F(z)S_{2,m-1}(z)&{}-F(z)S_{2,m}(z) \\ \vdots &{}\vdots &{}\vdots &{} \vdots &{}\vdots \\ -F(z)S_{m-1,1}(z) &{} -F(z)S_{m-1,2}(z) &{}\cdots &{} z^{\widehat{r}}-F(z)S_{m-1,m-1}(z) &{}-F(z)S_{m-1,m}(z)\\ -F(z)S_{m,1}(z) &{} -F(z)S_{m,2}(z) &{}\cdots &{}-F(z)S_{m,m-1}(z) &{} z^{\widehat{r}}-F(z)S_{m,m}(z) \end{array} \right| .\nonumber \\ \end{aligned}$$

(95)

Now, we can rewrite (95) in the following way

$$\begin{aligned} \mathbf{D}_m(z)=-\sum _{j=1}^{m-1}F(z)S_{m,j}(z)C_{m,j}(z)+\left[ z^{\widehat{r}}-F(z)S_{m,m}(z)\right] \mathbf{D}_{m-1}(z), \end{aligned}$$

(96)

where $C_{m,j}(z)$ is the cofactor of $-F(z)S_{m,j}(z)$.

Again, we can write (96) as

$$\begin{aligned} \left| \frac{\mathbf{D}_m(z)-\left[ z^{\widehat{r}}-F(z)S_{m,m}(z)\right] \mathbf{D}_{m-1}(z)}{\left[ z^{\widehat{r}}-F(z)S_{m,m}(z)\right] \mathbf{D}_{m-1}(z)}\right|= & {} \left| \frac{-\sum _{j=1}^{m-1}F(z)S_{m,j}(z)C_{m,j}(z)}{\left[ z^{\widehat{r}}-F(z)S_{m,m}(z)\right] \mathbf{D}_{m-1}(z)}\right| \nonumber \\\le & {} \frac{\sum _{j=1}^{m-1}\left| F(z)S_{m,j}(z)\right| \left| y_{m,j}(z)\right| }{\left| z^{\widehat{r}}-F(z)S_{m,m}(z)\right| }, \end{aligned}$$

(97)

where $|y_{m,j}(z)|=\frac{|C_{m,j}(z)|}{|\mathbf{D}_{m-1}(z)|}$ is the unique solution (by Cramer’s rule, provided $\mathbf{D}_{m-1}(z)\ne 0$) of the system of equations

$$\begin{aligned}&\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} z^{\widehat{r}}-F(z)S_{1,1}(z) &{} -F(z)S_{1,2}(z) &{}\cdots &{} -F(z)S_{1,m-1}(z) \\ -F(z)S_{2,1}(z) &{} z^{\widehat{r}}-F(z)S_{2,2}(z) &{}\cdots &{} -F(z)S_{2,m-1}(z)\\ \vdots &{}\vdots &{}\vdots &{} \vdots \\ -F(z)S_{m-1,1}(z) &{} -F(z)S_{m-1,2}(z) &{}\cdots &{} z^{\widehat{r}}-F(z)S_{m-1,m-1}(z) \end{array} \right) \cdot \left( \begin{array}{c} y_{m,1}(z) \\ y_{m,2}(z)\\ \vdots \\ y_{m,m-1}(z) \end{array} \right) \nonumber \\&\quad =\left( \begin{array}{c} -F(z)S_{1,m}(z) \\ -F(z)S_{2,m}(z)\\ \vdots \\ -F(z)S_{m-1,m}(z) \end{array} \right) . \end{aligned}$$

(98)

The k-th equation of (98) is given by

$$\begin{aligned}&\left[ z^{\widehat{r}}-F(z)S_{k,k}(z)\right] y_{m,k}(z)-\sum \limits _{j=1,j\ne k}^{m-1}F(z)S_{k,j}(z)y_{m,j}(z)\nonumber \\&\quad =-F(z)S_{k,m}(z),\quad 1\le k\le m-1. \end{aligned}$$

(99)

Now, the entries of the matrix $\mathbf{D}_n(z)~(1\le n\le m)$ satisfy the condition that the modulus of each entry of the matrix is less than or equal to one and the modulus of the diagonal element is greater than the sum of the moduli of all other elements in that row on the circle $|z|=1-\delta$. It implies that the matrix $\mathbf{D}_{m-1}(z)$ is nonsingular and the system (98) has a unique solution $y_{m,j}(z)$ with $|y_{m,j}(z)|< 1,~1\le j\le m-1$ on the circle $|z|=1-\delta$. Let us assume the contrary that

$$\begin{aligned} \text{ Max }_j|y_{m,j}(z)|=|y_{m,k}(z)|\ge 1. \end{aligned}$$

(100)

Because of our assumption $\left| \frac{y_{m,j}(z)}{y_{m,k}(z)}\right| \le 1$ and $\left| \frac{1}{y_{m,k}(z)}\right| \le 1$, we can rewrite (99) in the form

$$\begin{aligned} |z^{\widehat{r}}-F(z)S_{k,k}(z)|\le & {} \sum \limits _{j=1,j\ne k}^{m-1}\left| F(z)S_{k,j}(z)\right| \left| \frac{y_{m,j}(z)}{y_{m,k}(z)}\right| +\left| F(z)S_{k,m}(z)\right| \left| \frac{1}{y_{m,k}(z)}\right| \\\le & {} \sum \limits _{j=1,j\ne k}^m\left| F(z)S_{k,j}(z)\right| . \end{aligned}$$

This contradicts Theorem 7.2. Thus we have $|y_{m,j}(z)|<1$.

Hence, using Theorem 7.2 and $|y_{m,j}(z)|<1$, the right-hand side expression of Eq. (97) is smaller than one, and therefore $|\bar{G}(z)|<|f(z)|$, where $f(z)=[z^{\widehat{r}}-F(z)S_{m,m}(z)]\mathbf{D}_{m-1}(z)$ and $\bar{G}(z)=\mathbf{D}_m(z)-[z^{\widehat{r}}-F(z)S_{m,m}(z)]\mathbf{D}_{m-1}(z)$. By Rouché’s theorem f(z) and $f(z)+\bar{G}(z)$ have the same number of zeros inside the unit circle, $|z|=1$. Since by our assumption, $\mathbf{D}_{m-1}(z)$ has $(m-1)\widehat{r}$ zeros, and $[z^{\widehat{r}}-F(z)S_{m,m}(z)]$ has $\widehat{r}$ zeros by Theorem 7.1, f(z) has $m\widehat{r}$ zeros inside the unit circle. This implies that $f(z)+\bar{G}(z)=\mathbf{D}_m(z)$ has exactly $m\widehat{r}$ zeros inside the unit circle, $|z|=1$, if we let $\delta \rightarrow 0.$

Appendix 2

For the sake of completeness, we provide procedure for obtaining $\mathbf{S}_n$.

As presented by Lucantoni (1991), applying the uniformization argument, P(n,t) is of the form:

$$\begin{aligned} \mathbf{P}(n,t)= & {} \sum _{l=n}^{\infty }e^{-s t}\frac{(s t)^{l}}{l!}\mathbf{U}^{(l)}_n \quad n\ge 0, \end{aligned}$$

(101)

where $s=max_i\{-[L_0]_{ii}\}~(1\le i\le m)$ and $\mathbf{U}^{(l)}_n$ is given by

$$\begin{aligned} \mathbf{U}^{(0)}_0= & {} \mathbf{I}_m,\quad \mathbf{U}^{(0)}_n=\mathbf{0},~ \quad n\ge 1, \\ ~\mathbf{U}^{(l+1)}_0= & {} \mathbf{U}^{(l)}_0(\mathbf{I}_m+s^{-1}\mathbf{L}_0),\quad l\ge 0,\\ \mathbf{U}^{(l+1)}_n= & {} \mathbf{U}^{(l)}_n(\mathbf{I}_m+s^{-1}\mathbf{L}_0)+s^{-1}\mathbf{U}^{(l)}_{n-1}\mathbf{L}_{1}, \quad n\ge 1, l\ge 0. \end{aligned}$$

By substituting the values of $\mathbf{P}(n,t)$ in (101), we obtain

$$\begin{aligned} \mathbf{S}_n=\sum _{l=0}^{\infty }v_l \mathbf{U}^{(l)}_n, \quad n\ge 0, \end{aligned}$$

(102)

where $v_l=\int _0^{\infty }e^{-s t}\frac{(s t)^{l}}{l!}dA(t).$

However, when the inter-batch arrival time distributions are of phase type ($PH$-distribution), these matrices can be evaluated using a procedure given by Neuts (1981). The following theorem gives a procedure for the computation of the matrices $\mathbf{S}_n.$

Theorem 7.4

If A(t) follows a $PH$-distribution with irreducible representation $(\varvec{\alpha },\mathbf{T})$, where $\varvec{\alpha }$ and $\mathbf{T}$ are of dimension $\nu$, then the matrices $\mathbf{S}_n$ are given by

$$\begin{aligned} \mathbf{S}_n=\mathbf{U}_n\left( \mathbf{I}_m\otimes \mathbf{T}^0\right) ,\quad n\ge 0, \end{aligned}$$

where

$$\begin{aligned} \mathbf{U}_0= & {} -(\mathbf{I}_m\otimes \varvec{\alpha })\left[ \mathbf{L}_0\otimes \mathbf{I}_{\nu }+\mathbf{I}_m\otimes \mathbf{T}\right] ^{-1},\\ \mathbf{U}_n= & {} -\mathbf{U}_{n-1}(\mathbf{L}_1\otimes \mathbf{I}_{\nu })\left[ \mathbf{L}_0\otimes \mathbf{I}_{\nu }+\mathbf{I}_m\otimes \mathbf{T}\right] ^{-1},\quad n\ge 1, \end{aligned}$$

with $\mathbf{T}^0=-\mathbf{Te}_{\nu }$ and the symbol $\otimes$ denotes the Kronecker product of two matrices.

Proof

The proof is given by Neuts (1981) for $PH$-type service.

Theorem 7.5

If inter-batch arrival time distribution is of phase type having the parameters $(\alpha ,\mathbf{T})$, then the matrix-generating function $\mathbf{S}(z)$ of the $\mathbf{S}_n$’s is given by

$$\begin{aligned} \mathbf{S}(z)= & {} \sum \limits _{n=0}^{\infty }\mathbf{S}_nz^n=\int _0^{\infty }e^{\mathbf{L}(z)t}\otimes a(t)dt\end{aligned}$$

(103)

$$\begin{aligned}= & {} (\mathbf{I}_m\otimes \alpha )(\mathbf{L}(z)\oplus \mathbf{T})^{-1}(\mathbf{I}_m\otimes \mathbf{T}\mathbf{e}_{\nu }), \end{aligned}$$

(104)

where $\mathbf{L}(z)\oplus \mathbf{T}=(\mathbf{L}(z)\otimes \mathbf{I}_{\nu })+(\mathbf{I}_m\otimes \mathbf{T})$.

Proof

A similar proof was presented by Chaudhry et al. (2013) for C-MAP arrival and service time distribution with rational Laplace–Stieltjes transform and is applicable in our case with some modification. We present it as follows. As inter-batch arrival time distribution is of phase type with representation $(\varvec{\alpha },\mathbf{T})$, we can write its density function as $a(t)=\varvec{\alpha }e^{\mathbf{T}t}\mathbf{T}^0$ and $\mathbf{T}^0=-\mathbf{Te}_\nu$. Now write (7) as

$$\begin{aligned} \mathbf{S}(z)=\int _0^{\infty }e^{\mathbf{L}(z)t}\otimes a(t)dt, \end{aligned}$$

(105)

where $\otimes$ is a Kronecker product of two square matrices of order m and 1, respectively. We write (105) in a form such that each square matrix is the product of three different matrices (not necessarily square matrices) as

$$\begin{aligned} \mathbf{S}(z)=\int _0^{\infty }\left( \mathbf{I}_me^{\mathbf{L}(z)t}\mathbf{I}_m\right) \otimes \left( \varvec{\alpha }e^{\mathbf{T}t}\mathbf{T}^0\right) dt. \end{aligned}$$

Using the well known rule of Kronecker product $(\mathbf{E}_1\mathbf{E}_2\mathbf{E}_3)\otimes (\mathbf{F}_1\mathbf{F}_2\mathbf{F}_3)=(\mathbf{E}_1\otimes \mathbf{F}_1)(\mathbf{E}_2\otimes \mathbf{F}_2)(\mathbf{E}_3\otimes \mathbf{F}_3)$, Now write

$$\begin{aligned} \mathbf{S}(z)= & {} \int _0^{\infty }\left( \mathbf{I}_m\otimes \varvec{\alpha }\right) \left( e^{\mathbf{L}(z)t}\otimes e^{\mathbf{T}t}\right) \left( \mathbf{I}_m\otimes \mathbf{T}^0\right) dt\\= & {} \left( \mathbf{I}_m\otimes \varvec{\alpha }\right) \left( \int _0^{\infty }\left( e^{\mathbf{L}(z)t}\otimes e^{\mathbf{T}t}\right) dt\right) \left( \mathbf{I}_m\otimes \mathbf{T}^0\right) . \end{aligned}$$

We use another important rule of Kronecker product, i.e., $e^{\mathbf{E}_1}\otimes e^{\mathbf{F}_1}=e^{\mathbf{E}_1\oplus \mathbf{F}_1}$, where $\oplus$ is the Kronecker sum defined as $\mathbf{E}_1\oplus \mathbf{F}_1=\mathbf{E}_1\otimes \mathbf{I}_\nu +\mathbf{I}_m\otimes \mathbf{F}_1$. Here, it is assumed that $\mathbf{E}_1$ and $\mathbf{F}_1$ are square matrices of order m and $\nu$, respectively. We let $\mathbf{E}=(\mathbf{L}(z)\oplus \mathbf{T})$ and assume that $\mathbf{E}^{-1}$ exists and, in fact, this is the case here. From the fact that $\int _0^{\infty }e^{\mathbf{E}t}dt=-\mathbf{E}^{-1}$, we finally get

$$\begin{aligned} \mathbf{S}(z)=(\mathbf{I}_m\otimes \varvec{\alpha })\left( -(\mathbf{L}(z)\oplus \mathbf{T})^{-1}\right) (\mathbf{I}_m\otimes \mathbf{T}^0), \end{aligned}$$

where $\mathbf{L}(z)\oplus \mathbf{T}=(\mathbf{L}(z)\otimes \mathbf{I}_{\nu })+(\mathbf{I}_m\otimes \mathbf{T})$.

Using $\mathbf{T}^0=-\mathbf{Te}_\nu$, we can finally write

$$\begin{aligned} \mathbf{S}(z)=(\mathbf{I}_m\otimes \varvec{\alpha })(\mathbf{L}(z)\oplus \mathbf{T})^{-1}(\mathbf{I}_m\otimes \mathbf{Te}_\nu ). \end{aligned}$$

Appendix 3

The roots used in Tables 1 and 2 are given as follows: $\gamma _1=-0.407817,~\gamma _2= -0.465012,~\gamma _3= 0.514488,~\gamma _4= -0.578216,~\gamma _5= 0.635947,~\gamma _{6}= 0.955556,~\gamma _7= -0.056530+ 0.474062\mathrm {i},~\gamma _8=-0.056530- 0.474062\mathrm {i},\quad \gamma _9= -0.063981 - 0.522104\mathrm {i},~\gamma _{10}= -0.063981 + 0.522104\mathrm {i},~\gamma _{11}= -0.166860 + 0.730216\mathrm {i},~\gamma _{12}= -0.166860 - 0.730216\mathrm {i}.$ The corresponding $k_{ij}~(1\le i\le 12,~1\le j\le 3)$ values are as follows: $k_{1,1} = -0.089208,~k_{2,1} = -0.064613,~k_{3,1} = -0.080747,~k_{4,1} = -0.000008,~k_{5,1} = 0.245860,~k_{6,1} = 0.000620,~k_{7,1} = -0.141483-0.002068\mathrm {i},~k_{8,1} =-0.141483 +0.002068\mathrm {i},~k_{9,1} = 0.135531+0.155512\mathrm {i},~k_{10,1} =0.135531- 0.155512\mathrm {i},~k_{11,1} = -0.000000-0.000004\mathrm {i},~k_{12,1} =-0.000000+ 0.000004\mathrm {i},~ k_{1,2} = 0.000082,~k_{2,2} = 0.000650,~k_{3,2} = -0.000083,~k_{4,2} = -0.000683,~k_{5,2} = -0.003945,~k_{6,2} = 0.004963,~k_{7,2} = 0.000033+ 0.000128\mathrm {i},~k_{8,2} = 0.000033-0.000128\mathrm {i},~k_{9,2} = -0.000570-0.000724\mathrm {i},~k_{10,2} =-0.000570 +0.000724\mathrm {i},~k_{11,2} = 0.000045-0.000727\mathrm {i},~k_{12,2} = 0.000045+0.000727\mathrm {i},~ k_{1,3} =-0.176925,~k_{2,3} =0.008388,~k_{3,3} = 0.131913,~ k_{4,3} =0.000001,~k_{5,3} =0.040980,~k_{6,3} = 0.000139,~k_{7,3} = -0.026025-0.203224\mathrm {i},~k_{8,3} = -0.026025+0.203224\mathrm {i},~k_{9,3} = 0.023776-0.028263\mathrm {i},~k_{10,3} = 0.023776+0.028263\mathrm {i},~k_{11,3} = 0.000000-0.000000\mathrm {i},~k_{12,3} =0.000000+0.000000\mathrm {i}.$

The roots used in Tables 3 and 4 are as given below:

$\gamma _1=0.077940,~\gamma _2=0.155208,~\gamma _3=0.960551,~\gamma _4=-0.014089-0.013831\mathrm {i} ,~\gamma _5=-0.014089+0.013831\mathrm {i},~\gamma _{6}=0.019264-0.042765\mathrm {i},~\gamma _7=0.019264 +0.042765\mathrm {i},~\gamma _8=0.036447-0.054298\mathrm {i},\quad \gamma _9=0.036447+0.054298\mathrm {i}.$

The roots used in Tables 5 and 6 are given as follows:

$\gamma _1=-0.004418,~\gamma _2=-0.013638,~\gamma _3=0.139687,~\gamma _4=0.219339 ,~\gamma _5=0.873012,~\gamma _{6}=-0.002207-0.013973\mathrm {i},~\gamma _7=-0.002207+0.013973\mathrm {i}, ~\gamma _8=0.001703-0.017126\mathrm {i},~\gamma _9=0.001703+0.017126\mathrm {i}.$

The roots used in Tables 7 and 8 are given by:

$\gamma _1=-0.425046,~\gamma _2=-0.437159,~\gamma _3=0.539541,~\gamma _4= 0.543699,~\gamma _5=-.666644,~\gamma _{6}=0.718361,~\gamma _7=-0.041788+0.445149\mathrm {i},~\gamma _8=-0.041788-0.445149\mathrm {i},\quad \gamma _9= -0.040755+0.457173\mathrm {i},~\gamma _{10}= -0.040755-0.457173\mathrm {i},~\gamma _{11}=-0.018507 +0.664232\mathrm {i},~\gamma _{12}=-0.018507 -0.664232\mathrm {i}.$

The roots used in Tables 9 1nd 10 are given by: $\gamma _1=0.180975,~\gamma _2=0.944434,~\gamma _3=0.278271+0.075281\mathrm {i},~\gamma _4=0.278271-0.075281\mathrm {i}.$