Appendix Sub-matrices of the infinitesimal generator matrix \(\bar {\Theta }\)
$$\begin{array}{@{}rcl@{}} &&\quad \begin{array}{cccccc}(((0))) & (((1))) & (((2))) & \cdots& (((S-1))) & (((S))) \end{array} \\ \bar{\Theta} = \begin{array}{l} (((0))) \\ (((1))) \\ (((2))) \\ {\vdots} \\ (((S-1))) \\ (((S))) \end{array}&& \left( \begin{array}{cccccc} {\Theta}_{0,0} & {\Theta}_{0,1} & {\Theta}_{0,2} & {\cdots} & {\Theta}_{0,S-1} & {\Theta}_{0,S}\\ {\Theta}_{1,0} & {\Theta}_{1,1} & {\Theta}_{1,2} & {\cdots} & {\Theta}_{1,S-1} & {\Theta}_{1,S}\\ {\Theta}_{2,0} &{\Theta}_{2,1} & {\Theta}_{2,2} & {\cdots} & {\Theta}_{2,S-1} & {\Theta}_{2,S}\\ {\vdots} & {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} \\ {\Theta}_{S-1,0} & {\Theta}_{S-1,1} & {\Theta}_{S-1,2} & {\cdots} & {\Theta}_{S-1,S-1} & {\Theta}_{S-1,S}\\ {\Theta}_{S,0} & {\Theta}_{S,1} & {\Theta}_{S,2} & {\cdots} & {\Theta}_{S,S-1} & {\Theta}_{S,S} \end{array}\right) \end{array} $$
Due to the assumptions of the model, we note that
$${\Theta}_{i_{1},j_{1}} = \textbf{0} , \quad \text{for} \quad j_{1} \neq i_{1}, i_{1} - 1, i_{1} + Q.$$
First consider the case \({\Theta }_{i_{1},i_{1}+Q}\) . This will happen only when the inventory level is replenished. We consider the inventory level is zero, that is Θ0,Q
. Then
$$\begin{array}{@{}rcl@{}} [{\Theta}_{0,Q}]_{{i_{2}}{j_{2}}} & = & \left\{ \begin{array}{lll} \bar{C_{00}^{(0)}}, & j_{2} = i_{2}, & i_{2} = 0,\\ \bar{{C_{01}^{(0)}}}, & j_{2} = 1, & i_{2} = 0,\\ \bar{C_{22}^{(0)}}, & j_{2} = i_{2}, & i_{2} = 2,\\ \textbf{0}, & \text{otherwise}. \end{array}\right. \end{array} $$
We denote Θ0,Q
as C
0 and its sub matrices are
$$\begin{array}{@{}rcl@{}} \left[\bar{C}_{00}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{E}_{0}, & j_{3} = 0, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \qquad [\bar{E}_{0}]_{i_{4},i_{4}} = \beta I_{(M+1)} \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{C}_{01}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{E}_{0}, & j_{3} = i_{3}, & i_{3} = 1,2,\ldots, N, \\ \textbf{0}, & \text{otherwise}. \end{array}\right.\\ \left[\bar{C}_{22}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{E}_{0}, & j_{3} = i_{3}, & i_{3} \in {V_{1}^{N}} ,\\ \textbf{0}, & \text{otherwise}. \end{array}\right.\\ \end{array} $$
When the inventory level lies between one to s . In this case, only the inventory level changes from i
1 to i
1 + Q . Hence \({\Theta }_{i_{1},i_{1}+Q} = \beta I_{(3N+2)(M+1)}\) . We denote \({\Theta }_{i_{1},i_{1}+Q}\) as \(\bar {C}\) .
Next, we discuss the case \({\Theta }_{i_{1},i_{1}-1}, i_{1} = 1, 2, \ldots , S\) . It will happen only when the essential service completion of a high/low priority customer. If the inventory level is one, we get
$$\begin{array}{@{}rcl@{}} [{\Theta}_{i_{1},i_{1}-1}]_{{i_{2}}{j_{2}}} & = & \left\{ \begin{array}{lll} \bar{A_{10}^{(0)}}, & j_{2} = 0, & i_{2} = 1,\\ \bar{A_{12}^{(0)}}, & j_{2} = 2, & i_{2} = 1,\\ \bar{A_{30}^{(0)}}, & j_{2} =0, & i_{2} = 3,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \end{array} $$
We denote Θ1,0 as A
0 and its sub matrices are
$$\begin{array}{@{}rcl@{}} \left[\bar{A}_{10}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{H}_{3}, & j_{3} = i_{3}-1, & i_{3} \in {V_{1}^{N}}\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{A}_{12}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{H}_{4}, & j_{3} = i_{3}, & i_{3} \in {V_{1}^{N}} ,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\quad \bar{H_{3}}= p\mu_{1} I_{(M+1)}, \ \ \bar{H_{4}}= (1-p)\mu_{1} I_{(M+1)}. \\ \left[\bar{A}_{30}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{3}, & j_{3} = i_{3}, & i_{3} \in {V_{0}^{N}} ,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \bar{C_{3}}= \mu_{3} I_{(M+1)}. \end{array} $$
When the inventory level is more than one, we have
$$\begin{array}{@{}rcl@{}} [{\Theta}_{i_{1},i_{1}-1}]_{{i_{2}}{j_{2}}} & = & \left\{ \begin{array}{lll} \bar{A_{10}^{(1)}}, & j_{2} = 0, & i_{2} = 1,\\ \bar{A_{11}^{(1)}}, & j_{2} = i_{2}, & i_{2} = 1,\\ \bar{A_{12}^{(1)}}, & j_{2} = 2, & i_{2} = 1,\\ \bar{A_{30}^{(1)}}, & j_{2} =0, & i_{2} = 3,\\ \bar{A_{31}^{(1)}}, & j_{2} = 1, & i_{2} = 3,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \end{array} $$
We will denote \({\Theta }_{i_{1},i_{1}-1} (i_{1}=2,3,\ldots , S)\) , as \(\bar {A}_{1}\) and its sub matrices are
$$\begin{array}{@{}rcl@{}} \left[\bar{A}_{10}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{H}_{3}, & j_{3} = i_{3}-1, & i_{3} =1,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{A}_{11}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{H}_{3}, & j_{3} = i_{3}-1, & i_{3} \in {V_{2}^{N}} ,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{A}_{12}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{H}_{4}, & j_{3} = i_{3}, & i_{3} \in {V_{1}^{N}} ,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{A}_{30}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{3}, & j_{3} = 0, & i_{3} =0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{A}_{31}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{3}, & j_{3} = i_{3}, & i_{3} \in {V_{1}^{N}} ,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \end{array} $$
Finally, we consider the case \(\bar {A}_{i_{1},i_{1}}, i_{1} = 0,1, \ldots , S\) . Here due to each one of the following mutually exclusive cases, a transition results:
a high/low priority customer may occur
a retrial customer may enter into the service station
a second optional service may be completed
When the inventory level is zero, we get
$$\begin{array}{@{}rcl@{}}</p><p class="noindent">[\bar{\Theta}_{00}]_{{i_{2}}{j_{2}}} & = & \left\{ \begin{array}{lll} \bar{B_{00}^{(0)}}, & j_{2} = 0, & i_{2} = 0,\\ \bar{B_{20}^{(0)}}, & j_{2} = 0, & i_{2} = 2,\\ \bar{B_{22}^{(0)}}, & j_{2} = 2, & i_{2} = 2,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \end{array} $$
the matrix \(\bar {\Theta }_{00}\) is denoted by \(\bar {B}_{0}\) and it’s sub matrices are
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{00}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in V_{0}^{N-1},\\ \bar{D}_{0}, & j_{3} = i_{3}, & i_{3} \in V_{0}^{N-1},\\ \bar{D}, & j_{3} = i_{3}, & i_{3} = N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{20}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{2}, & j_{3} = i_{3}-1, & i_{3} \in {V_{1}^{N}},\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{22}^{(0)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in {V_{1}^{N}},\\ \bar{D}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{D}_{2}, & j_{3} = i_{3}, & i_{3} = N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{D}_{0}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q) i_{4} \theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+\beta+(1-q) i_{4} \theta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{D}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q) i_{4} \theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+\beta+(1-q) i_{4} \theta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{D}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q) i_{4} \theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q) i_{4} \theta+\beta+\mu_{2}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{D}_{2}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q) i_{4} \theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{3}2+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right. \end{array} $$
$$\begin{array}{@{}rcl@{}} \bar{C_{2}}= \mu_{2} I_{(M+1)}, \ \bar{C_{1}}= \lambda_{1} I_{(M+1)} \end{array} $$
For i
1 = 1,2,…,s ,
$$\begin{array}{@{}rcl@{}} [\bar{\Theta}_{i_{1},i_{1}}]_{{i_{2}}{j_{2}}} & = & \left\{ \begin{array}{lll} \bar{B_{00}^{(1)}}, & j_{2} = 0, & i_{2} = 0,\\ \bar{B_{01}^{(1)}}, & j_{2} = 1, & i_{2} =0,\\ \bar{B_{11}^{(1)}}, & j_{2} =1, & i_{2} = 1,\\ \bar{B_{03}^{(1)}}, & j_{2} =3, & i_{2} = 0,\\ \bar{B_{20}^{(1)}}, & j_{2} =0, & i_{2} = 2,\\ \bar{B_{21}^{(1)}}, & j_{2} =1, & i_{2} =2,\\ \bar{B_{22}^{(1)}}, & j_{2} = 2, & i_{2} =2,\\ \bar{B_{31}^{(1)}}, & j_{2} =1, & i_{2} = 3,\\ \bar{B_{33}^{(1)}}, & j_{2} =3, & i_{2} = 3,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \end{array} $$
with
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{00}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{G}, & j_{3} = 0, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{G}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} -(\lambda_{1}+\lambda_{2}+\beta+i_{4} \theta), & j_{4} = 0, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{01}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = 1, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{03}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{F}, & j_{3} = i_{3}, & i_{3}=0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{F}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{11}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in V_{1}^{N-1},\\ \bar{G}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{G}_{2}, & j_{3} = i_{3}, & i_{3} =N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{G}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{1}+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{G}_{2}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{1}+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{20}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{2}, & j_{3} = 0, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{21}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{2} & j_{3} = i_{3}-1, & i_{3} \in {V_{2}^{N}},\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{22}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in V_{1}^{N-1},\\ \bar{G}_{3}, & j_{3} = i_{3}, & i_{3}\in V_{1}^{N-1},\\ \bar{G}_{4}, & j_{3} = i_{3}, & i_{3}=N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{G}_{3}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{2}+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{G}_{4}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{2}+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{31}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{F}_{1}, & j_{3} = 1, & i_{3}=0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{F}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} r\lambda_{1}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-2},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{33}^{(1)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{F}_{0}, & j_{3} = 1, & i_{3}=0,\\ \bar{C}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{H}, & j_{3} = i_{3}, & i_{3} = 0,\\ \bar{H}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{H}_{2}, & j_{3} = i_{3}, & i_{3} = N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{F}_{0}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} (1-r) \lambda_{1}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{H}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}+(1-q)i_{4}\theta+\mu_{3}+\beta), & j_{4} = i_{4}, & i_{4} \in V_{0}^{M-1},\\ -((1-r)\lambda_{1}+(1-q)M\theta+\mu_{3}+\beta), & j_{4} = i_{4}, & i_{4} =M,\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{H}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{3}+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{H}_{2}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{3}+\beta), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
For \(i_{1} = \in V_{s+1}^{S},\)
$$\begin{array}{@{}rcl@{}} \left[\bar{\Theta}_{i_{1},i_{1}}\right]_{{i_{2}}{j_{2}}} & = & \left\{ \begin{array}{lll} \bar{B_{00}^{(2)}}, & j_{2} = 0, & i_{2} = 0,\\ \bar{B_{01}^{(2)}}, & j_{2} = 1, & i_{2} =0,\\ \bar{B_{11}^{(2)}}, & j_{2} =1, & i_{2} = 1,\\ \bar{B_{03}^{(2)}}, & j_{2} =3, & i_{2} = 0,\\ \bar{B_{20}^{(2)}}, & j_{2} =0, & i_{2} = 2,\\ \bar{B_{21}^{(2)}}, & j_{2} = 1, & i_{2} = 2,\\ \bar{B_{22}^{(2)}}, & j_{2} = 2, & i_{2} =2,\\ \bar{B_{31}^{(2)}}, & j_{2} =1, & i_{2} = 3,\\ \bar{B_{33}^{(2)}}, & j_{2} =3, & i_{2} = 3,\\ \textbf{0}, & \text{otherwise}, \end{array}\right. \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{00}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{L}, & j_{3} = i_{3}, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{L}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} -(\lambda_{1}+\lambda_{2}+i_{4}\theta, & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{01}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = 1, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{03}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{F}, & j_{3} = 0, & i_{3} = 0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{11}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in V_{1}^{N-1},\\ \bar{L}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{L}_{2}, & j_{3} = i_{3}, & i_{3} =N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{L}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-2},\\ (1-q)i_{4}\theta ,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{1}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{L}_{2}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta,& j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{1}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{20}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{2}, & j_{3} = 0, & i_{3} =1,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{21}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{2}, & j_{3} = i_{3}-1, & i_{3} \in {V_{2}^{N}},\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{B}_{22}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in V_{1}^{N-1},\\ \bar{K}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{K}_{2}, & j_{3} = i_{3}, & i_{3} =N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{K}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{2}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{K}_{2}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1,& i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{2}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{31}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{F}_{1}, & j_{3} = 1, & i_{3}=0,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \left[\bar{B}_{33}^{(2)}\right]_{{i_{3}}{j_{3}}} & = & \left\{ \begin{array}{lll} \bar{F}_{0}, & j_{3} = 1, & i_{3} =0,\\ \bar{C}_{1}, & j_{3} = i_{3}+1, & i_{3} \in V_{1}^{N-1},\\ \bar{U}, & j_{3} = i_{3}, & i_{3} =0,\\ \bar{U}_{1}, & j_{3} = i_{3}, & i_{3} \in V_{1}^{N-1},\\ \bar{V}, & j_{3} = i_{3}, & i_{3} =N,\\ \textbf{0}, & \text{otherwise}, \end{array}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left[\bar{U}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1, & i_{4} \in V_{1}^{M-1},\\ (1-q)i_{4}\theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}+(1-q)i_{4}\theta+\mu_{3}), & j_{4} = i_{4}, & i_{4} \in V_{0}^{M-1},\\ -((1-r)\lambda_{1}+(1-q)M \theta+\mu_{3}), & j_{4} = i_{4}, & i_{4} =M,\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{U}_{1}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1,& i_{4} \in V_{1}^{M-1},\\ (1-q)i_{4}\theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{1}+\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{3}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right.\\ \left[\bar{V}\right]_{{i_{4}}{j_{4}}} & = & \left\{ \begin{array}{lll} \lambda_{2}, & j_{4} = i_{4}+1,& i_{4} \in V_{0}^{M-1},\\ (1-q)i_{4}\theta, & j_{4} = i_{4}-1, & i_{4} \in {V_{1}^{M}},\\ -(\lambda_{2}\bar{\delta}_{i_{4} M}+(1-q)i_{4}\theta+\mu_{3}), & j_{4} = i_{4}, & i_{4} \in {V_{0}^{M}},\\ 0, & \text{otherwise}, \end{array}\right. \end{array} $$
We denote \(\bar {\Theta }_{i_{1},i_{1}}, i_{1} = 1, 2, \ldots ,s\) as \(\bar {B}_{1}\) and \(\bar {\Theta }_{i_{1},i_{1}}, i_{1} = s + 1, s + 2, \ldots , S\) as \(\bar {B}_{2}\) . Hence the matrix \(\bar {\Theta }\) can be written in the following form
$$\begin{array}{@{}rcl@{}} \left[\bar{\Theta}\right]_{i_{1}j_{1}} & = & \left\{ \begin{array}{lll} \bar{A}_{0}, & j_{1} = i_{1}- 1, & i_{1} = 1, \\ \bar{A}_{1}, & j_{1} = i_{1} - 1, & i_{1} \in {V_{2}^{S}}, \\ \bar{C}, & j_{1} = i_{1}+Q, & i_{1} \in {V_{1}^{s}}, \\ \bar{C}_{0}, & j_{1} = i_{1}+Q, & i_{1} = 0,\\ \bar{B}_{0}, & j_{1} = i_{1}, & i_{1} = 0, \\ \bar{B}_{1}, & j_{1} = i_{1}, & i_{1} \in {V_{1}^{s}}, \\ \bar{B}_{2}, & j_{1} = i_{1}, & i_{1} \in V_{s+1}^{S}, \\ \textbf{0}, & \text{otherwise}. \end{array} \right. \end{array} $$
In Table 6 , we show the dimension of the sub matrices of \(\bar {\Theta }\) .
Table 6 The sub matrices and their dimension 
