(M, MAP)/(PH, PH)/1 Queue with Non-preemptive Priority and Working Vacation Under N-Policy

Abstract

In this paper we consider two single server queueing models with non-preemptive priority and working vacation under two distinct N-policies. High priority (type I) customers are served even in vacation mode whereas low priority (type II) customers are served only when the server comes to normal mode of service. Type I customers have only a limited waiting space L whereas type II customers have unlimited capacity. The two distinct N-policies are as described below: In model I, while service of type I customers are in progress in vacation mode (working vacation), if the number of such customers present in the system hits N ($\le L$) or the vacation timer (clock) expires, whichever occurs first, the server is switched on to normal mode. In model II, switching the server to normal mode from vacation mode occurs as soon as the accumulated number (those served out plus those present in the system) of type I customers during that working vacation hits N or the vacation timer expires, whichever occurs first. Type I customers arrive according to a Poisson process whereas type II customer’s arrival is governed by Markovian Arrival Process. Service time of type I and type II customers follow distinct phase type distributions. At a service completion epoch, finding the system empty, server takes an exponentially distributed working vacation. During working vacation, type I customers are served at a reduced rate. On vacation expiration, the service of the type I customer already in service, will start from the beginning in the normal mode of service. We analyze these models in steady state to compute the distribution of duration of service time continuously in slow mode, expected number of returns to 0 type I customer state, starting from 0 type I customer state during vacation mode of service before the arrival of a type II customer, the distribution of a p-cycle in normal mode, LSTs of busy cycle, busy period of type I customers generated during the service time of a type II customer and LSTs of waiting time distributions of type I and type II customers. We compare these models in steady state by numerical experiments to identify the superior model.

Appendices

Appendix 1

1.1 The Entries in the Block Matrices of the Infinitesimal Generator of the QBD in Model I

Define the entries of $B_{0_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}$ as transition submatrices which contains transitions of the form $(0,i_1,j_1,k_1,l_1) \rightarrow (0,i_2,j_2,k_2,l_2).$ Since none or one event alone could take place in a short interval of time with positive probability, in general, a transition such as $(i_1,i_2,j,k,l)\rightarrow (i_1',i_2',j',k',l')$ has positive rate only for exactly one of $i_1',i_2',j',k',l'$ different from $i_1,i_2,j,k,l$.

$$ \begin{aligned} B_{0_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}=\left\{ \begin{array}{ll} \lambda ({\varvec{\alpha} }\ \otimes I_n)&\quad i_1=0,i_2=1;j_1=j_2=0;1\le k_2\le m, \\ &\quad 1\le l_1,l_2 \le n\\ \lambda I_{mn}&\quad 1 \le i_1 \le N-2, i_2=i_1+1;j_1=j_2=0;\\ &\quad 1\le k_1,k_2\le m;1\le l_1,l_2 \le n\\ \lambda {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &\quad i_1=N-1,i_2=N;j_1=0,j_2=1; 1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2\le n\\ \lambda I_{mn} &\quad 1\le i_1\le L-1, i_2=i_1+1;j_1=j_2=1;\\ &\quad 1 \le k_1,k_2 \le m;1\le l_1,l_2\le n\\ \theta {\textbf{T}}^0 \otimes I_n &\quad i_1=1,i_2=0;j_1=0,j_2=0;1 \le k_1\le m;\\ &\quad 1\le l_1,l_2\le n\\ {\textbf{T}}^0 \otimes I_n &\quad i_1=1,i_2=0;j_1=1,j_2=0;1 \le k_1\le m; \\ &\quad 1\le l_1,l_2\le n\\ \theta {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n&\quad 2\le i_1\le N-1, i_2=i_1-1;j_1=0,j_2=0;\\ &\quad 1 \le k_1,k_2 \le m;1\le l_1,l_2 \le n\\ {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n &\quad 2\le i_1 \le L,i_2=i_1-1 ;j_1=j_2=1;1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2\le n\\ \eta {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &\quad 1\le i_1\le N-1,i_2=i_1;j_1=0,j_2=1; \\ &\quad 1\le k_1,k_2\le m; 1\le l_1,l_2\le n\\ D_0-\lambda I_n &\quad i_1=i_2=0;j_1=j_2=0; 1\le l_1,l_2\le n\\ \theta T \oplus D_0-(\lambda +\eta )I_{mn}&\quad 1\le i_1\le N-1, i_2=i_1;j_1=j_2=0;1 \le k_1,k_2 \le m; \\ &\quad 1\le l_1,l_2 \le n\\ T \oplus D_0-\lambda I_{mn} &\quad 1\le i_1\le L-1,i_2=i_1; j_1=j_2=1;1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2\le n\\ T\oplus D_0&\quad i_1=i_2=L;j_1=j_2=1;1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2 \le n\\ \end{array} \right. \end{aligned}$$

Define the entries of $C_{0_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}$ as transition submatrices which contains transitions of the form $(0,i_1,j_1,k_1,l_1) \rightarrow (1,i_2,j_2,k_2,l_2)$

$$ \begin{aligned} C_{0_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}=\left\{ \begin{array}{ll} D_1 &{}\quad i_1=0,i_2=0; j_1=j_2=0;1\le l_1,l_2\le n\\ I_m \otimes D_1&{}\quad 1\le i_1\le N-1,i_2=i_1;j_1=j_2=0;1 \le k_1,k_2,\le m; 1\le l_1,l_2 \le n\\ I_m \otimes D_1&{}\quad 1\le i_1\le L, i_2=i_1;j_1=j_2=1;1 \le k_1,k_2 \le m;1\le l_1,l_2 \le n \end{array} \right. \end{aligned}$$

Define the entries of $B_{1_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}$ as transition submatrices which contains transitions of the form $(1,i_1,j_1,k_1,l_1) \rightarrow (0,i_2,j_2,k_2,l_2)$.

$$ \begin{aligned} B_{1_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}=\left\{ \begin{array}{ll} {\textbf{T}'}^0\otimes I_n&{}\quad i_1=i_2=0 ; j_1=2,j_2=0;1 \le k_1\le m'; 1\le l_1,l_2\le n\\ {\textbf{T}'}^0 {\varvec{\alpha} } \otimes I_n&{}\quad 1\le i_1\le L,i_2=i_1;j_1=2,j_2=1;1 \le k_1 \le m', 1 \le k_2 \le m; \\ &{}\quad 1\le l_1,l_2\le n \end{array} \right. \end{aligned}$$

Define the entries of $A_{2_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}$ as transition submatrices which contains transitions of the form $(h,i_1,j_1,k_1,l_1) \rightarrow (h-1,i_2,j_2,k_2,l_2)$, where $h> 1$.

$$ \begin{aligned} A_{2_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}=\left\{ \begin{array}{ll} {\textbf{T}'}^0 {\varvec{\alpha} '} \otimes I_n&{}\quad i_1=i_2=0;j_1=j_2=2;1 \le k_1,k_2\le m';1\le l_1,l_2 \le n\\ {\textbf{T}'}^0 {\varvec{\alpha} } \otimes I_n &{}\quad 1\le i_1\le L,i_2=i_1;j_1=2,j_2=1; 1 \le k_1\le m', 1\le k_2\le m;\\ {} &{}\quad 1\le l_1,l_2\le n \end{array} \right. \end{aligned}$$

Define the entries of $A_{1_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}$ as transition submatrices which contains transitions of the form $(h,i_1,j_1,k_1,l_1) \rightarrow (h,i_2,j_2,k_2,l_2)$, where $h\ge 1$.

$$ \begin{aligned} A_{1_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}=\left\{ \begin{array}{ll} \lambda ({\varvec{\alpha} }\ \otimes I_n)&\quad i_1=0,i_2=1;\,j_1=j_2=0;\,1\le k_2\le m;\,\, 1\le l_1,l_2\le n\\ \lambda I_{mn}&\quad i_1=0,i_2=1;\,j_1=j_2=2;\,1\le k_1,k_2\le m';\\ &\quad 1\le l_1,l_2 \le n\\ \lambda I_{mn},\,&\quad 1\le i_1 \le N-2;\,i_2=i_1+1;\,j_1=j_2=0;\,1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2 \le n\\ \lambda {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &\quad i_1=N-1,i_2=N;\,j_1=0,j_2=1;\,1 \le k_1,k_2\le m;\\ &\quad 1\le l_1,l_2 \le n\\ \lambda I_{mn} &\quad 1\le i_1 \le L-1,i_2=i_1+1;\,j_1=j_2=1;\,1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2\le n\\ \lambda I_{mn} &\quad 1\le i_1\le L-1,i_2=i_1+1;\,j_1=j_2=2;\,1 \le k_1,k_2\le m'; \\ &\quad 1\le l_1,l_2\le n\\ \theta {\textbf{T}}^0 \otimes I_n&\quad i_1=1,i_2=0;\, j_1=j_2=0;\,1\le k_1\le m;\\ &\quad 1\le l_1,l_2\le n\\ {\textbf{T}}^0 {\varvec{\alpha} '} \otimes I_n&\quad i_1=1,i_2=0;\,j_1=1,j_2=2;\, 1\le k_1\le m, 1\le k_2\le m'; \\ &\quad 1\le l_1,l_2\le n\\ \theta {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n &\quad 2\le i_1 \le N-1,i_2=i_1-1;\,j_1=j_2=0;\, 1 \le k_1,k_2\le m; \\ &\quad 1\le l_1,l_2 \le n\\ {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n &\quad 2\le i_1 \le L, i_2=i_1-1;\,j_1=j_2=1;\,1 \le k_1,k_2\le m;\\ &\quad 1\le l_1,l_2 \le n\\ \eta ({\varvec{\alpha} '} \otimes I_n)&\quad i_1=i_2=0;\,j_1=0,j_2=2;\,1\le k_2\le m'; 1\le l_1,l_2\le n\\ \eta {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &\quad 1\le i_1\le N-1,i_2=i_1;\,j_1=0,j_2=1;\,1\le k_1,k_2\le m;\\ &\quad 1\le l_1,l_2\le n\\ D_0-(\lambda +\eta ) I_n &\quad i_1=i_2=0;\,j_1=j_2=0;\,1\le l_1,l_2\le n\\ T' \oplus D_0-\lambda I_{mn}&\quad i_1=i_2=0;j_1=j_2=2;\,1\le k_1,k_2\le m';\, 1\le l_1,l_2\le n\\ \theta T \oplus D_0-(\lambda +\eta )I_{mn}&\quad 1\le i_1\le N-1,i_2=i_1;\,j_1=j_2=0;\, 1 \le k_1,k_2 \le m;\\ &\quad 1\le l_1,l_2\le n\\ T \oplus D_0-\lambda I_{mn}&\quad 1\le i_1\le L-1, i_2=i_1;\,j_1=j_2=1;\,1 \le k_1,k_2\le m;\\ &\quad 1\le l_1,l_2 \le n\\ T' \oplus D_0-\lambda I_{mn}&\quad 1\le i_1\le L-1, i_2=i_1;\,j_1=j_2=2;\,1 \le k_1,k_2 \le m';\\ &\quad 1\le l_1,l_2\le n\\ T\oplus D_0&\quad i_1=i_2=L;\,j_1=j_2=1;\,1 \le k_1,k_2\le m; 1\le l_1,l_2 \le n\\ T'\oplus D_0&\quad i_1=i_2=L;\,j_1=j_2=2;\,1 \le k_1,k_2 \le m'; 1\le l_1,l_2\le n \end{array} \right. \end{aligned}$$

Define the entries of $A_{0_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}$ as transition submatrices which contains transitions of the form $(h,i_1,j_1,k_1,l_1) \rightarrow (h+1,i_2,j_2,k_2,l_2)$, where $h\ge 1$.

$$ \begin{aligned} A_{0_{(i_1,j_1,k_1,l_1)}}^{(i_2,j_2,k_2,l_2)}=\left\{ \begin{array}{ll} D_1 &{}\quad i_1=i_2=0;\,j_1=j_2=0;\,1\le l_1,l_2\le n\\ I_m \otimes D_1 &{}\quad i_1=i_2=0;\,j_1=j_2=2;\,1 \le k_1,k_2\le m';\, 1\le l_1,l_2 \le n\\ I_m \otimes D_1 &{}\quad 1\le i_1\le N-1,i_2=i_1;\,j_1=j_2=0;\,1 \le k_1,k_2\le m; \\ &{}\quad 1\le l_1,l_2\le n\\ I_m \otimes D_1 &{}\quad 1\le i_1\le L,i_2=i_1;\,j_1=j_2=1;\,1 \le k_1,k_2 \le m;\\ &{}\quad 1\le l_1,l_2\le n\\ I_m \otimes D_1&{}\quad 1\le i_1\le L,i_2=i_1;\,j_1=j_2=2;\,1 \le k_1,k_2 \le m';\,\\ &{}\quad 1\le l_1,l_2 \le n \end{array} \right. \end{aligned}$$

1.2 Proof of Theorem 3.1

Proof

Let $B_{c_L}$ denote the length of the busy cycle generated by type I customers arriving during the service time of a type II customer , $\hat{B_{c_L}}(s)$ the LST of the length of the busy cycle and l the number of type I customers that arrive during service time of type II customer.

Then $B_{c_L}=X+B_L^1+\cdots B_L^l$ where X denote the service time of the type II customer in service, $B_L^j$ the busy period generated by jth type I customer that arrive during X, where $1\le j\le l$.

$$ \begin{aligned} \begin{array}{lcl} \displaystyle \hat{B_{c_L}}(s)&{}=\displaystyle &{}E(e^{-sB_{c_L}}) \\ \displaystyle &=&\int _{x=0}^{\infty } E(e^{-sB_{c_L}}/X=x)P(x\le X<x+dx)\\ \displaystyle &=& \int _{x=0}^{\infty }\sum _{p=0}^{\infty }E(e^{-sB_{c_L}}/X=x,l=p)P(l=p/X=x)P(x\le X <x+dx)\\ \displaystyle &=& \int _{x=0}^{\infty }\sum _{p=0}^{\infty }E(e^{-sB_{c_L}}/X=x,l=p)\frac{e^{-\lambda x}(\lambda x)^p}{p!}{\varvec{\alpha} '} e^{T'x}{\textbf{T}'}^0 dx\\ \displaystyle &=& \int _{x=0}^{\infty }e^{-(s+\lambda )x} {\varvec{\alpha} '} e^{T'x}{\textbf{T}'}^0dx+\int _{x=0}^{\infty }\sum _{p=1}^{L-1} e^{-sx}{\varvec{\gamma} _p}(sI-T_1)^{-1}{\textbf{T}_1}^0\frac{e^{-\lambda x}(\lambda x)^p}{p!}\\ \displaystyle &{} &{} \times \, {\varvec{\alpha} '} e^{T'x}{\textbf{T}'}^0 dx+\int _{x=0}^{\infty }\sum _{p=L}^{\infty } e^{-sx}{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0\frac{e^{-\lambda x}(\lambda x)^p}{p!}{\varvec{\alpha} '} e^{T'x}{\textbf{T}'}^0 dx\\ &=& \alpha '[(s+\lambda )I-T']^{-1}{\textbf{T}'}^0+\sum _{p=1}^{L-1} {\varvec{\gamma} _p}(sI-T_1)^{-1}{\textbf{T}_1}^0\frac{\lambda ^p{\varvec{\alpha} '}}{p!}\\ &{}&{} \displaystyle \times \,\int _{x=0}^{\infty }x^pe^{-[(s+\lambda )I-T']x}{\textbf{T}'}^0dx\\ \displaystyle &{} &{} +\,\sum _{p=L}^{\infty }{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0 \frac{\lambda ^p}{p!}{\varvec{\alpha} '} \int _{x=0}^{\infty }x^p e^{[-(s+\lambda )I-T'])x}{\textbf{T}'}^0dx \end{array} \end{aligned}$$

(25)

We have,

$$\begin{aligned} \displaystyle \int _{x=0}^{\infty }x^pe^{-[(s+\lambda )I-T']x}dx=\frac{p!}{[(s+\lambda )I-T']^{p+1}} \end{aligned}$$

(26)

Substituting (26) in (25) , its third term

$$ \begin{aligned} \begin{array}{lcl} &=&\sum _{p=L}^{\infty }{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0\lambda ^p {\varvec{\alpha} '} [(s+\lambda )I-T']^{-(p+1)}{\textbf{T}'}^0\\ &=&{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0{\varvec{\alpha} '}\sum _{p=L}^{\infty } \left[ \lambda ^{-1}[(s+\lambda )I-T']\right] ^{-p}[(s+\lambda )I-T']^{-1}{\textbf{T}'}^0\\ &=&{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0{\varvec{\alpha} '} \left[ \lambda ^{-1}[(s+\lambda )I-T']\right] ^{-L}\sum _{q=0}^{\infty }\left[ \lambda ^{-1}[(s+\lambda )I-T']\right] ^{-q}[(s +\lambda )I-T']^{-1}{\textbf{T}'}^0\\ &=&{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0{\varvec{\alpha} '} \left[ \lambda ^{-1}[(s+\lambda )I-T']\right] ^{-L}[I-\lambda [(s+\lambda )I-T']^{-1}]^{-1}[(s+\lambda )I-T']^{-1}{\textbf{T}'}^0 \end{array} \end{aligned}$$

(27)

Substituting (27) in (25) gives

$$ \begin{aligned} \hat{B_{c_L}}(s)&={\varvec{\alpha} '}[(s+\lambda )I-T']^{-1}{\textbf{T}'}^0 +\sum _{p=1}^{L-1}{\varvec{\gamma} _p}(sI-T_1)^{-1}{\textbf{T}_1}^0\lambda ^p{\varvec{\alpha} '} [(s+\lambda )I-T']^{-(p+1)}{\textbf{T}'}^0 \nonumber \\&\quad+{\varvec{\gamma} _L}(sI-T_1)^{-1}{\textbf{T}_1}^0 {\varvec{\alpha} '}[\lambda ^{-1}[(s+\lambda )I-T']^{-L} \bigl [I-\lambda [(s+\lambda )I-T']^{-1}\bigr ]^{-1}\nonumber \\&\quad \times[(s+\lambda )I-T']^{-1}{\textbf{T}'}^0 \end{aligned}$$

(28)

$\square$

1.3 Proof of Theorem 3.2

Proof

Let $B_L$ denote the length of the busy period generated by type I customers arriving during the service time of a type II customer , $\hat{B_L}(s)$ the LST of the length of the busy period and l the number of type I customers that arrive during service time of type II customer.

Then $B_L=B_L^1+\cdots B_L^l$ ,where $B_L^j$ denote the busy period generated by jth type I customer that arrive during X, where $1\le j\le l$. Proceeding as in the above proof, we get the required result.

$\square$

Appendix 2

1.1 The Entries in the Block Matrices of the Infinitesimal Generator of the QBD in Model II

Define the entries of $G_{0_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}$ as transition submatrices which contains transitions of the form $(0,i_1,j_1,k_1,l_1,m_1) \rightarrow (0,i_2,j_2,k_2,l_2,m_2).$ Since none or one event alone could take place in a short interval of time with positive probability, in general, a transition such as $(i_1,j_1,k_1,l_1,m_1,n_1)\rightarrow (i_2,j_2,k_2,l_2,m_2,n_2)$ has positive rate only for exactly one of $i_2,j_2,k_2,l_2,m_2,n_2$ different from $i_1,j_1,k_1,l_1,m_1,n_1$.

$$ \begin{aligned} G_{0_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}=\left\{ \begin{array}{ll} \lambda ({\varvec{\alpha} }\ \otimes I_n)& i_1=0,i_2=1;j_1=j_2=0; k_2=1;\\ &\quad 1\le l_2\le m, 1\le m_1,m_2 \le n\\ \lambda I_{mn}&\quad 1 \le i_1 \le N-2, i_2=i_1+1;j_1=j_2=0;i_1\le k_1\le N-2,\\ &\quad k_2=k_1+1;1\le l_1,l_2 \le m;1\le m_1,m_2\le n\\ \lambda {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &\quad 1\le i_1 \le N-1,i_2=i_1+1; j_1=0,j_2=1;\\ &\quad k_1=N-1;1\le l_1,l_2 \le m;1\le m_1,m_2\le n\\ \lambda I_{mn}&\quad 1 \le i_1 \le L-1,i_2=i_1+1; j_1=j_2=1;\\ &\quad 1\le l_1,l_2 \le m;1\le m_1,m_2\le n\\ \eta {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n)&\quad 1\le i_1\le N-1;j_1=0,j_2=1; \\ &\quad i_1\le k_1\le N-1;1\le l_1,l_2\le m;1\le m_1,m_2\le n\\ \theta {\textbf{T}}^0 \otimes I_n &\quad i_1=1,i_2=0;j_1=0,j_2=0;1\le k_1\le N-1,\\ &\quad 1 \le l_1\le m; 1\le m_1,m_2\le n\\ {\textbf{T}}^0 \otimes I_n &\quad i_1=1,i_2=0;j_1=1,j_2=0;1 \le l_1\le m; \\ &\quad 1\le m_1,m_2\le n\\ \theta {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n&\quad 2\le i_1\le N-1, i_2=i_1-1;j_1=0,j_2=0; i_1\le k_1\le \\ &\quad N-1, k_2=k_1; 1 \le l_1,l_2 \le m;1\le m_1,m_2 \le n\\ {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n &\quad 2\le i_1 \le L,i_2=i_1-1 ;j_1=j_2=1;\\ &\quad 1 \le l_1,l_2 \le m; 1\le m_1,m_2\le n\\ D_0-\lambda I_n &\quad i_1=i_2=0;j_1=j_2=0; 1\le m_1,m_2\le n\\ \theta T \oplus D_0-(\lambda +\eta )I_{mn}&\quad 1\le i_1\le N-1, i_2=i_1;j_1=j_2=0;i_1\le k_1\le N-1,\\ &\quad k_2=k_1;1 \le l_1,l_2 \le m; 1\le m_1,m_2 \le n\\ T \oplus D_0-\lambda I_{mn} &\quad 1\le i_1\le L-1,i_2=i_1; j_1=j_2=1;1 \le l_1,l_2 \le m;\\ &\quad 1\le m_1,m_2\le n\\ T\oplus D_0&\quad i_1=i_2=L;j_1=j_2=1;1 \le l_1,l_2 \le m;\\ &\quad 1\le m_1,m_2 \le n\\ \end{array} \right. \end{aligned}$$

Define the entries of $H_{0_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_1)}$ as transition submatrices which contains transitions of the form $(0,i_1,j_1,k_1,l_1,m_1) \rightarrow (1,i_2,j_2,k_2,l_2,m_2)$

$$ \begin{aligned} H_{0_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}=\left\{ \begin{array}{ll} D_1 &\quad i_1=i_2=0; j_1=j_2=0; 1\le m_1,m_2\le n\\ I_m \otimes D_1&\quad 1\le i_1\le N-1,i_2=i_1;j_1=j_2=0;i_1\le k_1\le N-1,k_2=k_1;\\ &\quad 1 \le l_1,l_2\le m; 1\le m_1,m_2 \le n\\ I_m \otimes D_1&\quad 1\le i_1\le L, i_2=i_1;j_1=j_2=1;1 \le l_1,l_2 \le m;\\ &\quad 1\le m_1,m_2 \le n \end{array} \right. \end{aligned}$$

Define the entries of $H_{1_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}$ as transition submatrices which contains transitions of the form $(1,i_1,j_1,k_1,l_1,m_1) \rightarrow (0,i_2,j_2,k_2,l_2,m_2)$.

$$ \begin{aligned} H_{1_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}=\left\{ \begin{array}{ll} {\textbf{T}'}^0 \otimes I_n&\quad i_1=i_2=0 ; j_1=2,j_2=0;1 \le l_1\le m'; \\ &\quad 1\le m_1,m_2\le n\\ {\textbf{T}'}^0 {\varvec{\alpha} } \otimes I_n&\quad 1\le i_1\le L,i_2=i_1;j_1=2,j_2=1;1 \le l_1 \le m',1\le l_2\le m;\\ &\quad 1\le m_1,m_2\le n \end{array} \right. \end{aligned}$$

Define the entries of $A_{2_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}$ as transition submatrices which contains transitions of the form $(h,i_1,j_1,k_1,l_1,m_1) \rightarrow (h-1,i_2,j_2,k_2,l_2,m_2)$, where $h\ge 1$.

$$ \begin{aligned} A_{2_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}=\left\{ \begin{array}{ll} {\textbf{T}'}^0 {\varvec{\alpha} '} \otimes I_n&{}\quad i_1=i_2=0;j_1=j_2=2;1 \le l_1,l_2\le m';\\ {} &{}\quad 1\le m_1,m_2 \le n\\ {\textbf{T}'}^0 {\varvec{\alpha} } \otimes I_n &{}\quad 1\le i_1\le L,i_2=i_1;j_1=2,j_2=1; 1 \le l_1\le m',1\le l_2\le m; \\ &{}\quad 1 \le m_1,m_2\le n \end{array} \right. \end{aligned}$$

Define the entries of $A_{1_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}$ as transition submatrices which contains transitions of the form $(h,i_1,j_1,k_1,l_1,m_1) \rightarrow (h,i_2,j_2,k_2,l_2,m_2)$, where $h\ge 1$.

$$ \begin{aligned} A_{1_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}=\left\{ \begin{array}{ll} \lambda ({\varvec{\alpha} }\ \otimes I_n)&{}\quad i_1=0,i_2=1;j_1=j_2=0; k_2=1;\\ {} &{}\quad 1\le l_2\le m;\, 1\le m_1,m_2\le n\\ \lambda I_{mn},\,&{}\quad 1\le i_1 \le N-2;i_2=i_1+1;j_1=j_2=0;i_1\le k_1\le N-2,\\ {} &{}\quad k_2=k_1+1;1 \le l_1,l_2 \le m; 1\le m_1,m_2 \le n\\ \lambda {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &{}\quad 1\le i_1\le N-1;j_1=0,j_2=1;k_1=N-1;1 \le l_1,l_2\le m;\\ &{}\quad 1\le m_1,m_2 \le n\\ \lambda I_{mn} &{}\quad 1\le i_1 \le L-1,i_2=i_1+1;j_1=j_2=1;1 \le l_1,l_2 \le m;\\ &{}\quad 1\le m_1,m_2\le n\\ \lambda I_{mn} &{}\quad 0\le i_1\le L-1,i_2=i_1+1;j_1=j_2=2;1 \le l_1,l_2\le m'; \\ &{}\quad 1\le m_1,m_2\le n\\ \eta ({\varvec{\alpha} '} \otimes I_n) &{}\quad i_1=i_2=0;j_1=0,j_2=2; 1\le l_2\le m';\\ {} &{}\quad 1\le m_1,m_2\le n\\ \eta {\textbf{e}}(m)\otimes ({\varvec{\alpha} } \otimes I_n) &{}\quad 1\le i_1\le N-1,i_2=i_1;j_1=0,j_2=1;i_1\le k_1\le N-1;\\ &{}\quad 1\le l_1,l_2\le m; 1\le m_1,m_2\le n\\ \theta {\textbf{T}}^0 \otimes I_n&{}\quad i_1=1,i_2=0;j_1=j_2=0;1\le k_1\le N-1;\\ &{}\quad 1\le l_1\le m;1\le m_1,m_2\le n\\ {\textbf{T}}^0 {\varvec{\alpha} '} \otimes I_n&{}\quad i_1=1,i_2=0;j_1=1,j_2=2; 1\le l_1\le m,1\le l_2\le m'; \\ &{}\quad 1\le m_1,m_2\le n\\ \theta {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n &{}\quad 2\le i_1 \le N-1,i_2=i_1-1;j_1=j_2=0; i_1\le k_1\le N-1, \\ &{}\quad k_2=k_1;1 \le l_1,l_2\le m; 1\le m_1,m_2 \le n\\ {\textbf{T}}^0 {\varvec{\alpha} } \otimes I_n &{}\quad 2\le i_1 \le L, i_2=i_1-1;j_1=j_2=1;1 \le l_1,l_2\le m;\\ &{}\quad 1\le m_1,m_2 \le n\\ D_0-(\lambda +\eta ) I_n &{}\quad i_1=i_2=0;j_1=j_2=0;1\le m_1,m_2\le n\\ T' \oplus D_0-\lambda I_{mn}&{}\quad i_1=i_2=0;j_1=j_2=2;1\le l_1,l_2\le m'; 1\le m_1,m_2\le n\\ \theta T \oplus D_0-(\lambda +\eta )I_{mn}&{}\quad 1\le i_1\le N-1,i_2=i_1;j_1=j_2=0; i_1 \le k_1 \le N-1,\\ {} &{}\quad k_2=k_1; 1\le l_1,l_2\le m;1\le m_1,m_2\le n\\ T \oplus D_0-\lambda I_{mn}&{}\quad 1\le i_1\le L-1, i_2=i_1;j_1=j_2=1;1 \le l_1,l_2\le m;\\ &{}\quad 1\le m_1,m_2 \le n\\ T' \oplus D_0-\lambda I_{mn}&{}\quad 1\le i_1\le L-1, i_2=i_1;j_1=j_2=2;1 \le l_1,l_2 \le m';\\ &{}\quad 1\le m_1,m_2\le n\\ T\oplus D_0&{}\quad i_1=i_2=L;j_1=j_2=1;1 \le l_1,l_2\le m; 1\le m_1,m_2 \le n\\ T'\oplus D_0&{}\quad i_1=i_2=L;j_1=j_2=2;1 \le l_1,l_2 \le m'; 1\le m_1,m_2\le n \end{array} \right. \end{aligned}$$

Define the entries of $A_{0_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}$ as transition submatrices which contains transitions of the form $(h,i_1,j_1,k_1,l_1,m_1) \rightarrow (h+1,i_2,j_2,k_2,l_2,m_2)$, where $h\ge 1$.

$$ \begin{aligned} A_{0_{(i_1,j_1,k_1,l_1,m_1)}}^{(i_2,j_2,k_2,l_2,m_2)}=\left\{ \begin{array}{ll} D_1 &{}\quad i_1=i_2=0;j_1=j_2=0;1\le m_1,m_2\le n\\ I_{m'} \otimes D_1 &{}\quad i_1=i_2=0;j_1=j_2=2;1 \le l_1,l_2\le m'; 1\le m_1,m_2 \le n\\ I_m \otimes D_1 &{}\quad 1\le i_1\le N-1,i_2=i_1;j_1=j_2=0;i_1\le k_1\le N-1,k_2=k_1; \\ &{}\quad 1 \le l_1,l_2\le m;1\le m_1,m_2\le n\\ I_m \otimes D_1 &{}\quad 1\le i_1\le L,i_2=i_1;j_1=j_2=1;1 \le l_1,l_2 \le m;\\ &{}\quad 1\le m_1,m_2\le n\\ I_{m'} \otimes D_1&{}\quad 1\le i_1\le L,i_2=i_1;j_1=j_2=2;1 \le l_1,l_2 \le m'; 1\le m_1,m_2 \le n \end{array} \right. \end{aligned}$$