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On a queueing-inventory with reservation, cancellation, common life time and retrial

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Abstract

In this paper we model a queueing-inventory system that has applications in railway and airline reservation systems. Maximum items in the inventory is \(S\) which have a random common life time; this includes those that are sold in particular cycle. A customer, on arrival to an idle server with at least one item in inventory, is immediately taken for service; or else he joins the buffer of maximum size \(S\) depending on number of items in the inventory (the buffer capacity varies and is, at any time, equal to the number of items in the inventory). The arrival of customers constitutes a Poisson process, demanding exactly one item each from the inventory. If there is no item in the inventory, the arriving customer first queue up in a finite waiting space of capacity \(K\). When it overflows an arrival goes to an orbit of infinite capacity with probability \(p\) or is lost forever with probability \(1-p\). From the orbit he retries for service according to an exponentially distributed inter-occurrence time. The service time follows an exponential distribution. Cancellation of sold items before its expiry is permitted. Inventory gets added through cancellation of purchased items, until the expiry time. Cancellation time is assumed to be negligible. We analyze this system. Several performance characteristics are computed; expected sojourn time of the system in a cycle with “no inventory” and also “maximum inventory” are computed. Some illustrative numerical examples are presented. An optimization problem is numerically analyzed.

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Acknowledgments

The authors thank the referee(s) for their critical comments which helped in improving the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Cochin, 682022, India

    A. Krishnamoorthy, Dhanya Shajin & B. Lakshmy

Authors
  1. A. Krishnamoorthy
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  2. Dhanya Shajin
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  3. B. Lakshmy
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Corresponding author

Correspondence to A. Krishnamoorthy.

Additional information

Dedicated to the memory of Prof. J. R. Artalejo. First and second author’s research supported by Kerala State Council for Science, Technology & Environment (No. 001/KESS/2013/CSTE).

Appendices

Appendix 1

Sub-matrices are

$$\begin{aligned} H_0= & {} \left[ {\begin{array}{*{20}c} {h_0^0 } &{}\quad {h_{01}^0 } &{}\quad {} &{}\quad {} &{}\quad {} {} \\ {h_{10}^0 } &{}\quad {h_1^0 } &{}\quad {h_{12}^0 } &{}\quad {} &{}\quad {} {} \\ {} &{}\quad {} \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad {} \\ {} &{}\quad {} &{}\quad {h_{S - 1S - 2}^0 } &{}\quad {h_{S - 1}^0 } &{}\quad {h_{S - 1S}^0 } \\ {} &{}\quad {} {} &{}\quad {} &{}\quad {h_{SS - 1}^0 } &{}\quad {h_S^0 } \\ \end{array}} \right] ,\\ H'_0= & {} \left[ {\begin{array}{*{20}c} {h_0^0 } &{}\quad {h_{01}^0 } &{}\quad {} &{}\quad {} &{}\quad {} {} \\ {h_{10}^0 } &{}\quad {h_1^{'0} } &{}\quad {h_{12}^0 } &{}\quad {} &{}\quad {} {} \\ {} &{}\quad {} \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad {} \\ {} &{}\quad {} &{}\quad {h_{S - 1S - 2}^0 } &{}\quad {h_{S - 1}^{'0} } &{}\quad {h_{S - 1S}^0 } \\ {} &{}\quad {} &{}\quad {} &{}\quad {h_{SS - 1}^0 } &{}\quad {h_S^{'0} } \\ \end{array}} \right] , \end{aligned}$$

\(N =\ diag\ (0,n_1,\ldots ,n_S)\) where \(h_{0}^{0}=-(\lambda +\alpha +S\beta ),\ h_{01}^{0}=[S\beta \ \ 0 \ \ 0],\ h_{10}^0 = [0 \ \ \mu \ \ \mu ]^T,\)

$$\begin{aligned}&\left( {h_i^0 } \right) _{jk} = \left\{ \begin{array}{ll} - (\lambda + \alpha + (S - i)\beta ),&{}\quad k = j = 1 \\ - (\lambda + \alpha + \mu + (S - i)\beta ),&{}\quad k = j = 2,\ldots ,i + 2 \\ \lambda ,&{}\quad k = j + 1,j = 1,\ldots ,i + 1 \\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. ,1 \le i \le S \\&\left( {h_{i\ i - 1}^0 } \right) _{jk} = \left\{ \begin{array}{ll} \mu ,&{}\quad k = 1,j = 2,3 \\ \mu , &{}\quad k = j - 1,j = 4,\ldots ,i + 2 \\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. ,2 \le i \le S \\&\left( {h_{i\ i + 1}^0 } \right) _{jk} = \left\{ \begin{array}{ll} (S - i)\beta ,&{}\quad k = j = 1,2,\ldots ,i + 2 \\ 0,&{}\quad \text {otherwise}, \\ \end{array} \right. ,1 \le i \le S - 1 \\&\left( {h_i^{'0} } \right) _{jk} = \left\{ \begin{array}{ll} - (\lambda + \alpha + \eta + (S - i)\beta ),&{}\quad k = j = 1 \\ - (\lambda + \alpha + \mu + (S - i)\beta ),&{}\quad k = j = 2,\ldots ,i + 2 \\ \lambda ,&{}\quad k = j + 1,j = 1,\ldots ,i + 1 \\ 0, &{}\quad \text {otherwise}\\ \end{array} \right. ,1 \le i \le S, \\&\left( {n_{i} } \right) _{jk} = \left\{ \begin{array}{ll} \eta ,&{}\quad k =2, j = 1 \\ 0,&{}\quad \text {otherwise}, \\ \end{array} \right. ,1 \le i \le S. \end{aligned}$$

The dimensions of the matrices \(h^0_{i\ i-1},\ h^0_{i\ i+1}\) are, respectively, \((i+2) \times (i+1),\ (i+2) \times (i+3)\). The matrices \(h_i^0,\ h_i^{'0},\ n_i\) are square matrices of order \((i+2),\ 1 \le i \le S\).

$$\begin{aligned}&\left( L_0\right) _{jk} = \left\{ \begin{array}{ll} \lambda ,&{}\quad k = j= 1, \\ \lambda , &{}\quad 2 \le k \le S+1, j=\sum _{i=1}^k i+(k-1)\\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. \!, \\&\left( M_0\right) _{jk} = \left\{ \begin{array}{ll} (S-j+1)\beta , &{}\quad 1 \le j \le S, k=\sum _{i=1}^{j+1} i + j\\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. \!\!, \\&\left( L\right) _{jk} = \left\{ \begin{array}{ll} \lambda , &{}\quad 1 \le j \le S+1, k=j\\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. \!\!, \\&\left( L_1\right) _{jk} = \left\{ \begin{array}{ll} p\lambda , &{}\quad 1 \le j \le S+1, k=j\\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. \!\!, \\&\left( M\right) _{jk} = \left\{ \begin{array}{ll} (S-j+1)\beta , &{}\quad 1 \le j \le S, k=j+1\\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. \!, \\&\left( H\right) _{jk} = \left\{ \begin{array}{ll} -(\lambda +S\beta +\alpha ) , &{}\quad k=j=1,\\ -(\lambda +(S-j+1)\beta +\mu +\alpha ) , &{}\quad 2 \le j \le S+1, k=j \\ \mu , &{}\quad 2 \le j \le S+1, k=j-1,\\ 0, &{}\quad \text {otherwise},\\ \end{array} \right. \!, \\&\left( H_1\right) _{jk} = \left\{ \begin{array}{ll} -(p\lambda +S\beta +\alpha ) , &{}\quad k=j=1,\\ -(p\lambda +(S-j+1)\beta +\mu +\alpha ) , &{}\quad 2 \le j \le S+1, k=j \\ \mu , &{}\quad 2 \le j \le S+1, k=j-1,\\ 0, &{}\quad \text {otherwise}.\\ \end{array} \right. \end{aligned}$$

Appendix 2

The following matrices give transition rates from the state \((i,n_3,k_1)\rightarrow (j,m_3,k_2)\) where \(i(j)\) represents the number of items in the inventory; \(n_3(m_3)\), the number of customers in the buffer and \(k_l, \text {for } l=1,2,\) are status of the server.

$$\begin{aligned}&\check{H}_{00(i,j)}^{(k_1 ,k_2 )} (n_3 ,m_3 ) \\&\quad = \left\{ \begin{array}{llll} S\beta ,&{}\quad j = i + 1,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,i = 0, \\ (S - i)\beta ,&{}\quad j = i + 1,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,1 \le i \le S - 2, \\ &{}\quad j = i + 1,&{}\quad m_3 = n_3 ,&{}\quad k_2 = k_1 = 1,1 \le i \le S - 2,0 \le n_3 \le i, \\ \mu ,&{}\quad j = i - 1,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 - 1,1 \le i \le S,k_1 = 1, \\ &{}\quad j = i - 1,&{}\quad m_3 = n_3 - 1,&{}\quad k_2 = k_1 - 1,1 \le i \le S,k_1 = 1,n_3 = 1, \\ &{}\quad j = i - 1,&{}\quad m_3 = n_3 - 1,&{}\quad k_2 = k_1 = 1,2 \le i \le S,2 \le n_3 \le i, \\ \lambda ,&{}\quad j = i,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 + 1,1 \le i \le S,k_1 = 0, \\ &{}\quad j = i,&{}\quad m_3 = n_3 + 1,&{}\quad k_2 = k_1 = 1,1 \le i \le S,0 \le n_3 \le i - 1, \\ - (\lambda + \alpha + S\beta ),&{}\quad j = i,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,i = 0, \\ - (\lambda + \alpha + (S - i)\beta ),&{}\quad j = i,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,1 \le i \le S, \\ - (\lambda + \mu + \alpha + (S - i)\beta ),&{}\quad j = i,&{}\quad m_3 = n_3 ,&{}\quad k_2 = k_1 = 1,1 \le i \le S,0 \le n_3 \le i, \\ 0,&{}\quad \text {otherwise}, \\ \end{array} \right. \\&\check{H}_{0(i,j)}^{(k_1 ,k_2 )} (n_3 ,m_3 )\\&\quad = \left\{ \begin{array}{llll} S\beta ,&{}\quad j = i + 1,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,i = 0, \\ (S - i)\beta ,&{}\quad j = i + 1,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,1 \le i \le S - 2, \\ &{}\quad j = i + 1,&{}\quad m_3 = n_3 ,&{}\quad k_2 = k_1 = 1,1 \le i \le S - 2,0 \le n_3 \le i, \\ \mu ,&{}\quad j = i - 1,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 - 1,1 \le i \le S,k_1 = 1, \\ &{}\quad j = i - 1,&{}\quad m_3 = n_3 - 1,&{}\quad k_2 = k_1 - 1,1 \le i \le S,k_1 = 1,n_3 = 1, \\ &{}\quad j = i - 1,&{}\quad m_3 = n_3 - 1,&{}\quad k_2 = k_1 = 1,2 \le i \le S,2 \le n_3 \le i, \\ \lambda ,&{}\quad j = i,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 + 1,1 \le i \le S,k_1 = 0, \\ &{}\quad j = i,&{}\quad m_3 = n_3 + 1,&{}\quad k_2 = k_1 = 1,1 \le i \le S,0 \le n_3 \le i - 1, \\ - (\lambda + \alpha + S\beta ),&{}\quad j = i,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,i = 0, \\ - (\lambda + \alpha + \eta + (S - i)\beta ),&{}\quad j = i,&{}\quad m_3 = n_3 = 0,&{}\quad k_2 = k_1 = 0,1 \le i \le S, \\ - (\lambda + \mu + \alpha + (S - i)\beta ),&{}\quad j = i,&{}\quad m_3 = n_3 ,&{}\quad k_2 = k_1 = 1,1 \le i \le S,0 \le n_3 \le i, \\ 0,&{}\quad \text {otherwise}, \\ \end{array} \right. \end{aligned}$$

\(\check{H}_{00},\ \check{H}_0,\ \check{B}_0\) are square matrices of order \(U_1\) and dimension of the matrices \(\check{B}_1,\ \check{M}_0\) are \((S+1)\times U_1\). \(\check{B}\) is a square matrix of order \(S+1\).

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Krishnamoorthy, A., Shajin, D. & Lakshmy, B. On a queueing-inventory with reservation, cancellation, common life time and retrial. Ann Oper Res 247, 365–389 (2016). https://doi.org/10.1007/s10479-015-1849-x

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