A perishable inventory system with service facility and feedback customers

Abstract

In this article, we consider a continuous review perishable inventory system with service facility consisting of finite waiting hall and a single server. The primary customers arrive according to a Markovian arrival process. An arriving customer, who finds the waiting hall is full, is considered to be lost. The individual customer’s unit demand is satisfied after a random time of service which is distributed as exponential. The life time of each item is assumed to be exponentially distributed. The items are replenished based on variable ordering policy. The lead time is assumed to have phase type distribution. After the service completion, the primary customer may decide either to join the secondary (feedback) queue, which is of infinite size, or leave the system according to a Bernoulli trial and the server decides to serve either for primary or feedback customer according to a Bernoulli trail. The primary and secondary service are at different counters. The service time for feedback customers is assumed to be independent exponential distribution. After the service completion for feedback customer, the server starts immediately for primary customer’s service, whenever the inventory level and the primary customer level is positive, otherwise the server becomes idle for an exponential duration. If the primary customer level and inventory level becomes positive during the server idle period then he starts service for primary customer immediately. After completing his idle period, the server goes to secondary counter to serve for feedback customer, if any. The joint probability distribution of the system is obtained in the steady state. Important system performance measures are derived and the long-run total expected cost rate is also calculated. The results are illustrated numerically.

Appendix

$\text {For }i\in E^{2}_{s},$

$$\begin{aligned}&\left[ \tilde{C}_{i}\right] _{kl} = \left\{ \begin{array}{ll} \tilde{C}_{1}^{(i)}, &{} \quad l=k-1; \ k\in E^{2}_{N}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ \tilde{C}_{1}^{(i)}\right] _{kl} = \left\{ \begin{array}{ll} qp_1 \mu _1 I_{m_1 m_2}, &{}\quad l=k; \ k\in E^{1}_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\&\tilde{C}_{s+1}=I_{N}\otimes (qp_1 \mu _1 I_{m_1}\otimes \alpha ); \qquad \tilde{C}_{s+2}=I_{N}\otimes (qp_1\mu _1 I_{m_1});\\&\left[ B_{12}^{(0)}\right] _{ij} = \left\{ \begin{array}{ll} \tilde{L}_{i},&{}\quad j=i-1; \ i\in E^{1}_{s+1}\\ \tilde{L}_{s+2},&{}\quad j=i-1; \ i \in E^{s+2}_{S}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\&\left[ \tilde{L}_{1}\right] _{kl} = \left\{ \begin{array}{ll} \tilde{L}_{1}^{(1)},&{}\quad l=k-1; \ k\in E^{1}_{N}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. , \quad \left[ \tilde{L}_{1}^{(1)}\right] _{kl} = \left\{ \begin{array}{ll} p_1\mu _1 I_{m_1m_2},&{}\quad l=k+1; \ k\in E^{1}_{s}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{2}_{s}$

$\text {For }j\in E^{Q}_{S},$

$$\begin{aligned} \left[ G_{j} \right] _{kl}= & {} \left\{ \begin{array}{ll} G_1^{(j)},&{} \quad l=k; \ k\in E^{1}_N\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad G_1^{(j)}= {\mathbf {e}}_{(s+1)}{(j-Q+1)}\otimes \left( I_{m_1}\otimes T^0\right) . \end{aligned}$$

$\text {For }i\in E^{1}_{s}$,

$$\begin{aligned} G_i= & {} {\mathbf {e}}_{N}^{'}(1)\otimes G_1^{(i)}; \quad G_1^{(i)}=I_{(s+1-i)}\otimes (D_1\otimes I_{m_2}); \quad G_{s+1}= {\mathbf {e}}_{N}^{'}(1)\otimes D_1;\\ \left[ B_{00}^{(1)} \right] _{ij}= & {} \left\{ \begin{array}{ll} F_{i}, &{} \quad j=i; \ i\in E^{0}_{S}\\ F_{ii-1}, &{}\quad j=i-1; \ i\in E^{1}_{S}\\ F_{ij}, &{}\quad j\in E^{Q+i}_{S}; \ i\in E^{0}_{s}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }j\in E^{Q}_{S},$

$$\begin{aligned} F_1^{(0j)}= & {} {\mathbf {e}}_{(s+1)}{(j-Q+1)}\otimes \left( I_{m_1}\otimes T^0\right) , \quad F_{0j} = I_{(N+1)}\otimes F_1^{(0j)} \end{aligned}$$

$\text {For } i\in E^{1}_{s}, \quad j\in E^{Q+i}_{S},$

$$\begin{aligned} F_{ij}= & {} {\mathbf {e}}_{(s+1-i)}{(j-Q+1)}\otimes \left( I_{m_1}\otimes T^{0}\right) ,\quad F_{10}={\mathbf {e}}_{(N+1)}^{'}(1)\otimes F_{0}^{(10)}\\ \left[ F_{0}^{(10)} \right] _{ij}= & {} \left\{ \begin{array}{ll} \gamma I_{m_1m_2}, &{} \quad j=i; \ i\in E^{1}_{s}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{2}_{s}$,

$$\begin{aligned} \left[ F_{ii-1} \right] _{kl}= & {} \left\{ \begin{array}{ll} i\gamma I_{m_1m_2}, &{} \quad l=k; \ k\in E^{1}_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad F_{s+1, s} = (s+1)\gamma I_{m_1}\otimes \alpha \end{aligned}$$

$\text {For }i\in E^{s+2}_{S}, \qquad F_{ii-1}=i\gamma I_{m_1}$

$$\begin{aligned} \left[ F_0\right] _{ij}= & {} \left\{ \begin{array}{ll} F_1^{(0)}, &{} \quad j=i; \ i\in E^{0}_{N-1}\\ F_{2}^{(0)}, &{}\quad j=i; \ i=N\\ F_3^{(0)}, &{}\quad j=i+1; \ i\in E^{0}_{N-1}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad F_1^{(0)}=I_{(s+1)}\otimes (D_0\oplus T); \\ F_2^{(0)}= & {} I_{(s+1)}\otimes (D\oplus T); \qquad F_3^{(0)} = I_{(s+1)}\otimes (D_1\otimes I_{m_2}). \end{aligned}$$

$\text {For }i\in E^{1}_{s}, \qquad F_{i}=I_{(s+1-i)}\otimes (D_0\oplus T-i\gamma I_{m_1m_2})$

$\text {For }i\in E^{s+1}_{S}, \qquad F_i=D_0-i\gamma I_{m_1}$

$$\begin{aligned} \left[ B_{10}^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{ll} H_i, &{} \quad j=i; \ i\in E^{1}_{s+1}\\ H_{s+2}, &{}\quad j=i; \ i\in E^{s+2}_{S}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ H_{1}\right] _{ij}=\left\{ \begin{array}{ll} H_1^{(1)}, &{}\quad j=i-1; \ i\in E^{1}_{N}\\ H_2^{(1)}, &{} \quad j=i; \ i\in E^{1}_{N}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ H_{1}^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{ll} \bar{p}_1\mu _1 I_{m_1m_2}, &{}\quad j=i; \ i\in E^{1}_{s}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ H_{2}^{(1)}\right] _{ij}=\left\{ \begin{array}{ll} \gamma I_{m_1m_2}, &{} \quad j=i; \ i\in E^{1}_{s}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{2}_{s}$,

$$\begin{aligned} H_i= & {} {\mathbf {e}}_{N}(1)\otimes H_1^{(i)}, \quad \left[ H_{1}^{(i)}\right] _{ij}=\left\{ \begin{array}{ll} \bar{p}_1\mu _1 I_{m_1m_2}, &{}\quad j=i; \ i\in E^{1}_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ H_{s+1}= & {} {\mathbf {e}}_{N}(1)\otimes (\bar{p}_1 \mu _1 I_{m_1}\otimes \alpha ); \quad H_{s+2}={\mathbf {e}}_{N}(1)\otimes (\bar{p}_1 \mu _1 I_{m_1});\\ \left[ B_{11}^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{ll} K_i, &{} \quad j=i; \ i\in E^1_S\\ K_{ii-1}, &{}\quad j=i-1; \ i\in E^2_S\\ K_{ij}, &{} \quad j\in E^{Q+i}_S; \ i\in E^1_s\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For } i\in E^1_s, \qquad j\in E^{Q+i}_{S};$

$$\begin{aligned} K_1^{(ij)}= & {} {\mathbf {e}}_{(s+1-i)}{(j-Q+1)} \otimes \left( I_{m_1}\otimes T^{0}\right) ; \quad K_{ij}= I_{(N)} \otimes K_1^{(ij)} \end{aligned}$$

$\text {For }i\in E^2_s,$

$$\begin{aligned} \left[ K_{ii-1}\right] _{kl}= & {} \left\{ \begin{array}{lll} K_2^{(i,i-1)}, &{}\quad l=k; &{} k\in E^1_N\\ K_1^{(i,i-1)}, &{}\quad l=k-1; &{} k\in E^2_N\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ K_{1}^{(i,i-1)}\right] _{kl}= & {} \left\{ \begin{array}{lll} \bar{p}_1\mu _1 I_{m_1m_2}, &{} \quad l=k; &{} k\in E^1_{s+1-i}\\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \\ \left[ K_{2}^{(i,i-1)}\right] _{kl}= & {} \left\{ \begin{array}{lll} i\gamma I_{m_1m_2}, &{}\quad l=k; &{} k\in E^1_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ K_{s+1s}\right] _{kl}= & {} \left\{ \begin{array}{lll} (s+1)\gamma I_{m_1}\otimes \alpha , &{}\quad l=k; &{} k\in E^{1}_{N}\\ \bar{p}_1\mu _1 I_{m_1}\otimes \alpha , &{}\quad l=k-1; &{} k\in E^{2}_{N}\\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{s+2}_{S},$

$$\begin{aligned} \left[ K_{ii-1}\right] _{kl}= & {} \left\{ \begin{array}{lll} i\gamma I_{m_1}, &{}\quad l=k; &{} k\in E^{1}_{N}\\ \bar{p}_1\mu _1 I_{m_1}, &{}\quad l=k-1; &{} k\in E^{2}_{N}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^1_s,$

$$\begin{aligned} \left[ K_{i}\right] _{kl}= & {} \left\{ \begin{array}{lll} K_1^{(i)}, &{}\quad l=k; &{}\quad k\in E^1_{N-1}\\ K_2^{(i)}, &{}\quad l=k; &{}\quad k=N\\ K_3^{(i)}, &{}\quad l=k+1; &{} \quad k\in E^1_{N-1}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad K_3^{(i)}= I_{(s+1-i)}\otimes (D_1\otimes I_{m_2})\\ K_1^{(i)}= & {} I_{(s+1-i)}\otimes \left( D_0\oplus T-(\mu _1+i\gamma )I_{m_1m_2}\right) ; \\ K_2^{(i)}= & {} I_{(s+1-i)}\otimes \left( D\oplus T-(\mu _1+i\gamma )I_{m_1m_2}\right) \end{aligned}$$

$\text {For }i\in E^{s+1}_{S}$,

$\text {For }i\in E^{2}_{s},$

$$\begin{aligned} \left[ C_1^{(i)}\right] _{kl}= & {} \left\{ \begin{array}{lll} p_2\mu _2 I_{m_1m_2}, &{} \quad l=k-1; &{}\quad k\in E^2_{s+2-i}\\ p_2\mu _2 I_{m_1}\otimes \alpha , &{}\quad l=s+1-i; &{}\quad k=1\\ {\mathbf {0}},&{} \quad \text {otherwise}. \\ \end{array} \right. ,\quad C_i={\mathbf {e}}_{(N+1)}(1) \otimes C_1^{(i)}\\ C_{s+1}= & {} {\mathbf {e}}_{(N+1)}(1) \otimes (p_2\mu _2 I_{m_1}); \quad \left[ A_{21}^{(0)}\right] _{ij}=\left\{ \begin{array}{lll} L_{i-1}, &{}\quad j=i; &{}\quad i\in E^{1}_{s+1}\\ L_{s+1}, &{} \quad j=i; &{} \quad i\in E^{s+1}_{S}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{1}_{s},$

$\text {For }i\in E^{1}_s,$

$$\begin{aligned} \left[ M_1^{(i)}\right] _{kl}= & {} \left\{ \begin{array}{lll} \bar{p}_2\mu _2 I_{m_1m_2}, &{}\quad l=k-1; &{} \quad k\in E^2_{s+2-i}\\ \bar{p}_2\mu _2 I_{m_1}\otimes \alpha , &{}\quad l=s+1-i; &{}\quad k=1\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad M_i= {\mathbf {e}}_{(N+1)}(1) \otimes M_1^{(i)}\\ M_{s+1}= & {} {\mathbf {e}}_{(N+1)}(1)\otimes \left( \bar{p}_2\mu _2 I_{m_1}\right) ; \quad \left[ B_{21}^{(2)}\right] _{ij}=\left\{ \begin{array}{lll} P_{i-1}, &{}\quad j=i; &{}\quad i\in E^{1}_{s}\\ P_{s+1}, &{}\quad j=i-1; &{}\quad i\in E^{s+1}_{S}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^1_s,$

$\text {For }i\in E^1_s,$

$$\begin{aligned} \left[ R_1^{(i)}\right] _{ij}= & {} \left\{ \begin{array}{ll} \theta I_{m_1m_2}, &{}\quad j=i+1; \ i\in E^1_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad R_i={\mathbf {e}}_{(N+1)}^{'}(1)\otimes R_1^{(i)}\\ R_{s+1}= & {} {\mathbf {e}}_{(N+1)}^{'}(1)\otimes (\theta I_{m_1});\\ \left[ A_{00}^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{ll} U_i, &{} \quad j=i; \ i\in E^0_S\\ F_{ii-1}, &{}\quad j=i-1; \ i\in E^1_S\\ F_{ij}&{}\quad j\in E^{Q+i}_S; \ i\in E^0_s\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ U_0\right] _{ij}=\left\{ \begin{array}{ll} U_1^{(0)}, &{}\quad j=i; \ i\in E^0_{N-1}\\ U_2^{(0)}, &{}\quad j=i; \ i=N\\ F_3^{(0)}, &{} \quad j=i+1; \ i\in E^0_{N-1}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ U_1^{(0)}= & {} I_{(s+1)}\otimes \left( D_0\oplus T-\theta I_{m_1m_2}\right) ; \qquad U_2^{(0)} = I_{(s+1)}\otimes \left( D\oplus T-\theta I_{m_1m_2}\right) ; \end{aligned}$$

$\text {For }i\in E^1_s, \qquad U_{i}= I_{(s+1-i)}\otimes (D_0\oplus T-(\theta +i\gamma )I_{m_1m_2});$

$\text {For }i\in E^{s+1}_{S}, \qquad U_{i}=D_0-(\theta +i\gamma )I_{m_1};$

$$\begin{aligned} \left[ A_{12}^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{ll} V_i, &{}\quad j=i; \ \ i\in E^1_{s+1}\\ V_{s+1}, &{}\quad j=i; \ \ i\in E^{s+2}_{S}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ V_1\right] _{ij}=\left\{ \begin{array}{ll} V_1^{(1)}, &{}\quad j=i-1; \ \ i\in E^1_{N}\\ V_{2}^{(1)}, &{} \quad j=i; \ \ i\in E^{1}_{N}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ V_1^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{ll} \bar{p}_1\mu _1 I_{m_1m_2}, &{}\quad j=i+1; \ \ i\in E^1_s\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ V_2^{(1)}\right] _{ij}=\left\{ \begin{array}{ll} \gamma I_{m_1m_2}, &{}\quad j=i; \ \ i\in E^1_s\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^2_s,$

$$\begin{aligned}&\left[ V_i\right] _{ij}=\left\{ \begin{array}{ll} V_1^{(i)}, &{}\quad j=i-1; \ i=1\\ V_2^{(i)}, &{} \quad j=i-1; \ i\in E^2_N\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ V_1^{(i)}\right] _{kl}\!=\!\left\{ \begin{array}{ll} \bar{p}_1\mu _1 I_{m_1m_2}, &{}\quad l\!=\!k\!+\!1; \ \ k\in E^1_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\&\left[ V_2^{(i)}\right] _{kl}=\left\{ \begin{array}{ll} \bar{q}\bar{p}_1 \mu _1 I_{m_1m_2}, &{}\quad l=k+1; \ \ l\in E^1_{s+1-i}\\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \\&\left[ V_{s+1}\right] _{ij}=\left\{ \begin{array}{ll} V_1^{(s+1)}, &{} \quad j=i-1; \ \ i=1\\ V_2^{(s+1)}, &{}\quad j=i-1; \ \ i\in E^2_N\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad V_1^{(s+1)}={\mathbf {e}}_{2}^{'}(2)\otimes \left( \bar{p}_1\mu _1 I_{m_1}\otimes \alpha \right) ; \\&V_2^{(s+1)}={\mathbf {e}}_{2}^{'}(2)\otimes \left( \bar{q}\bar{p}_1\mu _1 I_{m_1}\otimes \alpha \right) ; \quad \left[ V_{s+2}\right] _{ij}=\left\{ \begin{array}{ll} \bar{p}_1\mu _1 I_{m_1}, &{} \quad j=i-1; \ \ i=1\\ \bar{q}\bar{p}_1\mu _1 I_{m_1}, &{}\quad j=i-1; \ \ i\in E^2_N\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\&\left[ A_{11}^{(1)}\right] _{ij}=\left\{ \begin{array}{lll} K_i, &{} \quad j=i; &{} \quad i\in E^1_S\\ W_{ii-1}, &{} \quad j=i-1; &{}\quad i\in E^2_S\\ K_{ij} &{} \quad j \in E^{Q+i}_S; &{}\quad i\in E^1_s\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^2_s,$

$$\begin{aligned} \left[ W_{ii-1}\right] _{kl}= & {} \left\{ \begin{array}{lll} K_2^{(i,i-1)}, &{}\quad l=k; &{} \quad k\in E^1_N\\ W_1^{(i,i-1)}, &{}\quad l=k-1; &{}\quad k\in E^2_N\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ W_1^{(i,i-1)}\right] _{kl}= & {} \left\{ \begin{array}{lll} q\bar{p}_1 \mu _1 I_{m_1m_2}, &{} \quad l=k; &{}\quad k\in E^1_{s+1-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ W_{s+1s}\right] _{kl}= & {} \left\{ \begin{array}{lll} (s+1)\gamma I_{m_1}\otimes \alpha , &{} \quad l=k; &{} \quad k\in E^1_N\\ q\bar{p}_1 \mu _1 I_{m_1}\otimes \alpha , &{}\quad l=k-1; &{} \quad k\in E^2_{N}\\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{s+2}_S,$

$$\begin{aligned} \left[ W_{ii-1}\right] _{kl}= & {} \left\{ \begin{array}{lll} i\gamma I_{m_1}, &{} \quad l=k; &{}\quad k\in E^1_N\\ q\bar{p}_1 \mu _1 I_{m_1}, &{}\quad l=k-1; &{}\quad k\in E^2_{N}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\ \left[ A_{22}^{(1)}\right] _{ij}= & {} \left\{ \begin{array}{lll} Z_i, &{} \quad j=i; &{}\quad i\in E^0_S\\ Z_{ii-1}, &{}\quad j=i-1; &{} \quad i\in E^1_S\\ Z_{ij}, &{}\quad j \in E^{Q+i}_S; &{} \quad i\in E^0_s\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }j\in E^Q_S$

$$\begin{aligned} Z_{0j}= & {} I_{(N+1)}\otimes Z_1^{(0j)}; \quad Z_1^{(0j)}={\mathbf {e}}_{(s+2)}(i-Q+2) \otimes \left( I_{m_1}\otimes T^0\right) ; \end{aligned}$$

$\text {For }i\in E^1_s; \qquad j\in E^{Q+i}_S;$

$$\begin{aligned} Z_{ij}= & {} I_{(N+1)}\otimes Z_1^{(ij)}; \quad Z_1^{(ij)}={\mathbf {e}}_{(s+2-i)} (j-Q+2) \otimes \left( I_{m_1}\otimes T^0\right) ; \end{aligned}$$

$\text {For }i\in E^1_s;$

$$\begin{aligned} \left[ Z_1^{(i,i-1)}\right] _{kl}= & {} \left\{ \begin{array}{ll} i\gamma I_{m_1}, &{}\quad l=k; \ \ k=1\\ i\gamma I_{m_1m_2}, &{}\quad l=k; \ \ k\in E^2_{s+2-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad Z_{ii-1}=I_{(N+1)}\otimes Z_1^{(i,i-1)}\\ Z_{s+1s}= & {} I_{(N+1)}\otimes \left( {\mathbf {e}}_{2}^{'}(1)\otimes (s+1)\gamma I_{m_1}\right) \\ \end{aligned}$$

$\text {For }i\in E^{s+2}_{S}; \qquad Z_{ii-1}= I_{(N+1)}\otimes i\gamma I_{m_1}$;

$$\begin{aligned}&\left[ Z_0\right] _{ij}=\left\{ \begin{array}{ll} Z_1^{(0)}, &{}\quad j=i; \ \ i\in E^{0}_{N-1}\\ Z_2^{(0)}, &{}\quad j=i; \ \ i=N\\ Z_3^{(0)}, &{}\quad j=i+1; \ \ i\in E^{0}_{N-1} \\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ Z_1^{(0)}\right] _{ij}=\left\{ \begin{array}{ll} D_0-\mu _2 I_{m_1}, &{}\quad j=i; \ \ i=1\\ D_0\oplus T-\mu _2 I_{m_1m_2}, &{}\quad j=i; \ \ i\in E^2_{s+2}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\&\left[ Z_3^{(0)}\right] _{ij}=\left\{ \begin{array}{ll} D_1, &{} \quad j=i; \ \ i=1\\ D_1\otimes I_{m_2}, &{}\quad j=i; \ \ i\in E^2_{s+2}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ Z_2^{(0)}\right] _{ij}=\left\{ \begin{array}{ll} D-\mu _2 I_{m_1}, &{} \quad j=i; \ \ i=1\\ D\oplus T-\mu _2 I_{m_1m_2}, &{}\quad j=i; \ \ i\in E^2_{s+2}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^1_s,$

$$\begin{aligned}&\left[ Z_i\right] _{kl}=\left\{ \begin{array}{ll} Z_1^{(i)}, &{}\quad l=k; \ \ k \in E^{0}_{N-1} \\ Z_2^{(i)}, &{} \quad l=k; \ \ k=N \\ Z_3^{(i)}, &{} \quad l=k+1; \ \ k\in E^{0}_{N-1} \\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. ,\quad \left[ Z_3^{(i)}\right] _{kl}=\left\{ \begin{array}{ll} D_1, &{}\quad l=k; \ \ k=1\\ D_1\otimes I_{m_2}, &{}\quad l\!=\!k; \ \ k\!\in \! E^2_{s+2-i}\\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \\&\left[ Z_1^{(i)}\right] _{kl}=\left\{ \begin{array}{lll} D_0-(\mu _2+i\gamma ) I_{m_1}, &{} \quad l=k; &{}\quad k=1\\ D_0\oplus T-(\mu _2+i\gamma ) I_{m_1m_2}, &{}\quad l=k; &{}\quad k\in E^2_{s+2-i}\\ {\mathbf {0}}, &{}\quad \text {otherwise}. \\ \end{array} \right. \\&\left[ Z_2^{(i)}\right] _{kl}=\left\{ \begin{array}{lll} D-(\mu _2+i\gamma ) I_{m_1}, &{} \quad l=k; &{} \quad k=1 \\ D\oplus T-(\mu _2+i\gamma ) I_{m_1m_2},&{}\quad l=k; &{}\quad k\in E^2_{s+2-i}\\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

$\text {For }i\in E^{s+1}_{S},$

$$\begin{aligned} \left[ Z_i\right] _{kl}= & {} \left\{ \begin{array}{lll} D_0-(\mu _2+i\gamma )I_{m_1}, &{} \quad l=k; &{}\quad k\in E^{0}_{N-1}\\ D-(\mu _2+i\gamma )I_{m_1}, &{} \quad l=k; &{}\quad k=N \\ D_1, &{}\quad l=k+1; &{} \quad k\in E^{0}_{N-1} \\ {\mathbf {0}}, &{} \quad \text {otherwise}. \\ \end{array} \right. \end{aligned}$$

See Tables 14 and 15.

Table 14 The submatrices and their dimensions

Table 15 The submatrices and their dimensions