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A retrial inventory system with priority customers and second optional service

Abstract

In this paper, we investigate a single server (s, Q) perishable inventory model consisting of two priority customers, say, type-1 and type-2. The customers arrival flows are independent Poisson processes, and the service times of the type 1 and type 2 customers are exponentially distributed. The server offers two different types of services - first with ordinary service (essential service) and the second with optional service. The idle server first gives ordinary service to the arriving customers (type 1/type 2). Upon first essential service completion, then the server gives additional service (second optional) only to the type 1 customers. We assume that the type 1 customers have both types of priorities (non-preemptive priority and preemptive priority) over the type 2 customers. We discussed retrial concepts only for type-2 customers. The stationary probability distribution of the inventory level, status of the server, number of customer in the orbit and number of customers in the waiting line are obtained by matrix methods and some numerical illustrations are provided.

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References

  1. 1.

    Berman, O., Kim, E.: Stochastic models for inventory management at service facility. Stoch. Model. 15(4), 695–718 (1999)

    Article  Google Scholar 

  2. 2.

    Berman, O., Sapna, K.P.: Inventory management at service facility for systems with arbitrarily distributed service times. Commun. Stat. Stoch. Model. 16 (3), 343–360 (2000)

    Article  Google Scholar 

  3. 3.

    Çakanyildirim, M., Bookbinder, J.H., Gerchak, Y.: Continuous review inventory models where random lead time depends on lot size and reserved capacity. Int. J. Prod. Econ. 68(3), 217–228 (2000)

    Article  Google Scholar 

  4. 4.

    Durán, A., Gutiérrez, G., Zequeira, R.I.: A continuous review inventory model with order expediting. Int. J. Prod. Econ. 87(2), 157–169 (2004)

    Article  Google Scholar 

  5. 5.

    Elango, C., Arivarignan, G.: A continuous review perishable inventory systems with poisson demand and partial backlogging. In: Balakrishnan, N., Kannan, N., Srinivasan, M.R. (eds.) Statistical Methods and Practice: Recent Advances. Narosa Publishing House, New Delhi (2003)

    Google Scholar 

  6. 6.

    Gaver, D.P., Jacobs, P.A., Latouche, G.: Finite birth-and-death models in randomly changing environments. Adv. Appl. Probab. 16, 715–731 (1984)

    Article  Google Scholar 

  7. 7.

    Goyal, S.K., Giri, B.C.: Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 34(1), 1–16 (2001)

    Article  Google Scholar 

  8. 8.

    Jeganathan, K., Anbazhagan, N., Kathiresan, J.: A retrial inventory system with non-preemptive priority service. Int. J. Inf. Manag. Sci. 24(1), 57–77 (2013)

    Google Scholar 

  9. 9.

    Jeganathan, K.: Linear retrial inventory system with second optional service under mixed priority service. TWMS J. Appl. Eng. Math. 5(2), 249–268 (2015)

    Google Scholar 

  10. 10.

    Ke, J.-C., Wu, C.-H., Pearn, W.L.: Multi-server retrial queue with second optional service: algorithmic computation and optimisation. Int. J. Syst. Sci. 42(10), 1755–1769 (2011)

    Article  Google Scholar 

  11. 11.

    Jinting, W., Qing Zhao, J.: A discrete-time Geo/G/1 retrial queue with starting failures and second optional service. Comput. Math. Appl. 53(1), 115–127 (2007)

    Article  Google Scholar 

  12. 12.

    Kalpakam, S., Arivarignan, G.: Inventory system with random supply quantity. OR Spektrum 12, 139–145 (1990)

    Article  Google Scholar 

  13. 13.

    Kalpakam, S., Arivarignan, G.: A coordinated multicommodity (s, S) inventory system. Mathl. Comput. Modell. 18, 69–73 (1993)

    Article  Google Scholar 

  14. 14.

    Karthik, T., Sivakumar, B., Arivarignan, G.: An inventory system with two types of customers and retrial demands. Int. J. Syst. Sci. Oper. Logist. 2(2), 90–112 (2015)

    Google Scholar 

  15. 15.

    Krishnamoorthy, A., Anbazhagan, N.: Perishable inventory system at service facilities with N policy. Stoch. Anal. Appl. 26, 120–135 (2008)

    Article  Google Scholar 

  16. 16.

    Liu, L., Yang, T.: An (s, s) random lifetime inventory model with a positive lead time. Eur. J. Oper. Res. 113, 52–63 (1999)

    Article  Google Scholar 

  17. 17.

    Nahmias, S.: Perishable inventory theory: a review. Oper. Res. 30, 680–708 (1982)

    Article  Google Scholar 

  18. 18.

    Ning, Z., Zhaotong, L.: A queueing-inventory system with two classes of customers. Int. J. Prod. Econ. 129, 225–231 (2011)

    Article  Google Scholar 

  19. 19.

    Raafat, F.: A survey of literature on continuously deteriorating inventory models. J. Oper. Res. Soc. 42, 27–37 (1991)

    Article  Google Scholar 

  20. 20.

    Yadavalli, V.S.S., Schoor, C.V., Strashein, J.J., Udayabakaran, S.: A single product perishing inventory model with demand interaction. ORiON 20(2), 109–124 (2004)

    Article  Google Scholar 

  21. 21.

    Yadavalli, V.S.S., Anbazhagan, N.: A Retrial Inventory System with Impatient Customers. Appl. Math. Inf. Sci. Int. J. 9(2), 637–650 (2015)

    Google Scholar 

Download references

Acknowledgments

The authors would like to express their appreciation to the anonymous referee for significant remarks, which really improved the value of this paper. The work of K. Jeganathan is supported by UGC-BSR Research Start-Up-Grant F.30-82/2014(BSR), India. The work of N. Anbazhagan is supported by UGC- Research Award for the year 2014-16, India.

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Correspondence to K. Jeganathan.

Appendix

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

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Jeganathan, K., Kathiresan, J. & Anbazhagan, N. A retrial inventory system with priority customers and second optional service. OPSEARCH 53, 808–834 (2016). https://doi.org/10.1007/s12597-016-0261-x

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Keywords

  • Mixed Priority
  • (s, Q) policy
  • Markov process
  • Service facility
  • Ordinary and additional service