# Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

#### Literature Review

## 2. Model Description

**Cancellation policy:**

## 3. Analysis

- Transition due to ordinary customers arrival
- (a)
- $(u,v)\to (u,v-1)$: rate ${E}_{1}\otimes {I}_{{k}_{2}}$, $u=0,1,\cdots ,N;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}v=1,2,\cdots ,S$.
- (b)
- $(u,0)\to (u+1,0)$: rate ${r}_{1}{E}_{1}\otimes {I}_{{k}_{2}}$, $u=0,1,\cdots ,N-1$.

- Transition due to impulse customers arrival$(u,v)\to (u,v-1)$: rate ${I}_{{k}_{1}}\otimes p{F}_{1}$, $u=0,1,\cdots ,N;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}v=1,2,\cdots ,S$.
- Transition due to cancellation$(u,v)\to (u,v+1)$: rate $(S-v)\beta {I}_{{k}_{1}}\otimes {I}_{{k}_{2}}$, $u=0,1,\cdots ,N;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}v=0,1,\cdots ,S-1.$
- Transition due to approach from pooled customers$(u,v)\to (u-1,v-1)$: rate $\theta {I}_{{k}_{1}}\otimes {I}_{{k}_{2}}$, $u=1,2,\cdots ,N;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}v=1,2,\cdots ,S.$
- Transition due to replenishment$(u,v)\to (u,v+Q)$: rate $\mu {I}_{{k}_{1}}\otimes {I}_{{k}_{2}}$, $u=0,1,\cdots ,N;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}v=0,1,\cdots ,s,$

#### 3.1. Steady State Probability Vector

- ${\mathrm{\Theta}}_{{u}_{1}}=({\mathrm{\Theta}}_{({u}_{1},0)},{\mathrm{\Theta}}_{({u}_{1},1)},{\mathrm{\Theta}}_{({u}_{1},2)},\cdots ,{\mathrm{\Theta}}_{({u}_{1},S)}),{u}_{1}=0,1,\cdots ,N$
- ${\mathrm{\Theta}}_{({u}_{1},{u}_{2})}=({\mathrm{\Theta}}_{({u}_{1},{u}_{2},1)},{\mathrm{\Theta}}_{({u}_{1},{u}_{2},2)},\cdots ,{\mathrm{\Theta}}_{({u}_{1},{u}_{2},{k}_{1})}),{u}_{1}=0,1,\cdots ,N;\phantom{\rule{3.33333pt}{0ex}}{u}_{2}=0,1,\cdots ,S$
- ${\mathrm{\Theta}}_{({u}_{1},{u}_{2},{u}_{3})}=({\mathrm{\Theta}}_{({u}_{1},{u}_{2},{u}_{3},1)},{\mathrm{\Theta}}_{({u}_{1},{u}_{2},{u}_{3},2)},\cdots ,{\mathrm{\Theta}}_{({u}_{1},{u}_{2},{u}_{3},{k}_{2})}),{u}_{1}=0,1,\cdots ,N;{u}_{2}=0,1,\cdots ,S;\phantom{\rule{3.33333pt}{0ex}}{u}_{3}=1,2,\cdots ,{k}_{1}$.

**Gaver Algorithm**:

- Determine the matrices ${Z}_{n}$ recursively by initializing,${Z}_{0}={A}_{00}$${Z}_{n}={A}_{11}+{A}_{10}(-{Z}_{n-1}^{-1}){A}_{01},1\le n\le N-1$${Z}_{N}={A}_{22}+{A}_{10}(-{Z}_{N-1}^{-1}){A}_{01}.$
- Compute the limiting probability vectors ${\mathrm{\Theta}}_{n}$ using,${\mathrm{\Theta}}_{n}={\mathrm{\Theta}}_{(n+1)}{A}_{10}(-{Z}_{n}^{-1})$, for $n=0,\cdots ,N-1.$
- Determine the system of equations${\mathrm{\Theta}}_{N}{Z}_{N}=\mathbf{0}$;$\sum _{n=0}^{N}{\mathrm{\Theta}}_{n}\mathbf{e}=1.$

#### 3.2. Few Significant of the System Peculiarities

- Mean inventory levelLet ${\eta}_{I}$ is mean inventory level in the steady state. Since ${\mathrm{\Theta}}_{({i}_{1},{i}_{2})}$ denote the limiting probability vector with the inventory level represents as ${i}_{1}$ and the number of customers in the pool represents as ${i}_{2}$. This is given by${\eta}_{I}=\sum _{{i}_{1}=0}^{N}\sum _{{i}_{2}=1}^{S}{i}_{2}{\mathrm{\Theta}}_{({i}_{1},{i}_{2})}\mathbf{e}.$
- Mean reorder rateLet ${\eta}_{R}$ denote the mean reorder rate in the steady-state. When the inventory level reduces to s from $s+1$ due to any of the following situations, a reorder is triggered:
- (a)
- The purchase of an ordinary customer.
- (b)
- Any one of pooled customers approaches.
- (c)
- The purchase of an impulse customer.

This is lead to${\eta}_{R}=\sum _{{i}_{1}=0}^{N}{\mathrm{\Theta}}_{({i}_{1},s+1)}({E}_{1}\otimes {I}_{{k}_{2}})\mathbf{e}+\sum _{{i}_{1}=1}^{N}{\mathrm{\Theta}}_{({i}_{1},s+1)}\theta I\mathbf{e}+\sum _{{i}_{1}=0}^{N}{\mathrm{\Theta}}_{({i}_{1},s+1)}({I}_{{k}_{1}}\otimes p{F}_{1})\mathbf{e}.$ - Mean number of customers in the poolLet ${\eta}_{PC}$ denote the mean number of customers in the pool. Since ${\mathrm{\Theta}}_{{i}_{1}}$ denote the stationary probability vector with the inventory level ${i}_{1}$. Hence, the mean number of customers in the pool is given by${\eta}_{PC}=\sum _{{i}_{1}=1}^{N}{i}_{1}{\mathrm{\Theta}}_{{i}_{1}}\mathbf{e}.$ Mean rate of arrival of impulse customersLet ${\eta}_{IC}$ denote the mean rate of arrival of impulse customers in the steady state. Then, ${\eta}_{IC}$ is given by${\eta}_{IC}=\sum _{{i}_{1}=0}^{N}\sum _{{i}_{2}=0}^{S}{\mathrm{\Theta}}_{({i}_{1},{i}_{2})}({I}_{{k}_{1}}\otimes {F}_{1})\mathbf{e}.$
- Mean number of lost customers in the systemLet ${\eta}_{L}$ denote the mean number of lost customers in the system. This is given by${\eta}_{L}=\sum _{{i}_{1}=0}^{N-1}{\mathrm{\Theta}}_{({i}_{1},0)}({r}_{2}{E}_{1}\otimes {I}_{{k}_{2}})\mathbf{e}+\sum _{{i}_{2}=0}^{S}{\mathrm{\Theta}}_{(N,{i}_{2})}({E}_{1}\otimes {I}_{{k}_{2}})\mathbf{e}+\sum _{{i}_{1}=0}^{N}{\mathrm{\Theta}}_{({i}_{1},0)}({I}_{{k}_{1}}\otimes {F}_{1})\mathbf{e}+\sum _{{i}_{1}=0}^{N}\sum _{{i}_{2}=1}^{S}{\mathrm{\Theta}}_{({i}_{1},{i}_{2})}({I}_{{k}_{1}}\otimes q{F}_{1})\mathbf{e}.$
- Mean cancellation rate of return productLet ${\eta}_{C}$ denote the mean cancellation rate of return product in the steady state. Then, ${\eta}_{C}$ is given by${\eta}_{C}=\sum _{{i}_{1}=0}^{N}\sum _{{i}_{2}=0}^{S-1}{\mathrm{\Theta}}_{({i}_{1},{i}_{2})}(S-{i}_{2})\beta \mathbf{e}.$

#### 3.3. Construction of the Cost Feature

## 4. Numerical Illustration

**Hyper-exponential (HEX):**${E}_{0}={F}_{0}=\left[\begin{array}{cc}-15& 0\\ 0& -5\end{array}\right];{E}_{1}={F}_{1}=\left[\begin{array}{cc}13.5& 1.5\\ 4.5& 0.5\end{array}\right]$**Erlang (ER):**${E}_{0}={F}_{0}=\left[\begin{array}{ccc}-3& 3& 0\\ 0& -3& 3\\ 0& 0& -3\end{array}\right];{E}_{1}={F}_{1}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 3& 0& 0\end{array}\right]$**Negative Correlation (NC):**${E}_{0}={F}_{0}=\left[\begin{array}{ccc}-2.35& 2.35& 0\\ 0& -2.35& 0\\ 0& 0& -3.5\end{array}\right]$; ${E}_{1}={F}_{1}=\left[\begin{array}{ccc}0& 0& 0\\ 0.0235& 0& 2.3265\\ 3.465& 0& 0.035\end{array}\right]$**Positive Correlation (PC):**${E}_{0}={F}_{0}=\left[\begin{array}{ccc}-2.35& 2.35& 0\\ 0& -2.35& 0\\ 0& 0& -3.5\end{array}\right];{E}_{1}={F}_{1}=\left[\begin{array}{ccc}0& 0& 0\\ 2.3265& 0& 0.0235\\ 0.035& 0& 3.465\end{array}\right]$

**bold**in each column indicate the minimum cost rate whereas, the least cost rate is specified in each row by underlining the values. Thus a value (bold and underlined) spectacles the local minimum of the function $C(S,s)$. The optimal cost value

**${C}^{*}(S,s)$ = 34.3798**is achieve at

**${S}^{*}$ = 23**,

**${s}^{*}$ = 7**with the values ${r}_{1}=0.6,\phantom{\rule{3.33333pt}{0ex}}{r}_{2}=1-{r}_{1},\phantom{\rule{3.33333pt}{0ex}}p=0.6,\phantom{\rule{3.33333pt}{0ex}}q=1-p,\phantom{\rule{3.33333pt}{0ex}}\theta =6.5,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{1}=12.5,\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{2}=12.5,\phantom{\rule{3.33333pt}{0ex}}\mu =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =1,\phantom{\rule{3.33333pt}{0ex}}N=7,\phantom{\rule{3.33333pt}{0ex}}{c}_{h}=0.9,\phantom{\rule{3.33333pt}{0ex}}{c}_{w}=3,\phantom{\rule{3.33333pt}{0ex}}{c}_{s}=10$, ${c}_{cl}=0.1,\phantom{\rule{3.33333pt}{0ex}}{c}_{i}=0.3$. Table 1 and Figure 1 shows that the function $C(S,s)$ is convex.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\left[A\right]}_{ij}$ | The element submatrix at $(i,j)$ the position of A |

$\mathbf{0}$ | Zero matrix |

$\mathbf{e}$ | A column vector of 1’s appropriate dimension |

S | Maximum inventory level |

$A\otimes B$ | Kronecker product of matrices A and B |

$A\oplus B$ | Kronecker sum of matrices A and B |

MAP | Markovian Arrival Process |

PH | Phase-type |

PHF | Phase-type with Failure. |

## References

- Jeganathan, K.; Harikrishnan, T.; Selvakumar, S.; Anbazhagan, N.; Amutha, S.; Acharya, S.; Dhakal, R.; Joshi, G.P. Analysis of Interconnected Arrivals on Queueing-Inventory System with Two Multi-Server Service Channels and One Retrial Facility. Electronics
**2021**, 9, 576. [Google Scholar] [CrossRef] - Sigman, K.; Simchi-Levi, D. Light traffic heuristic for an M/G/1 queue with limited inventory. Ann. Oper. Res.
**1992**, 40, 371–380. [Google Scholar] [CrossRef] - Melikov, A.Z.; Molchanov, A.A. Stock optimization in transportation/storage systems. Cybern. Syst. Anal.
**1992**, 28, 484–487. [Google Scholar] [CrossRef] - Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach; Courier Corporation: North Chelmsford, MA, USA, 1994. [Google Scholar]
- Berman, O.; Kaplan, E.H.; Shevishak, D.G. Deterministic approximations for inventory management at service facilities. IIE Trans.
**1993**, 25, 98–104. [Google Scholar] [CrossRef] - Sivakumar, B.; Arivarignan, G. A stochastic inventory system with postponed demands. Perform. Eval.
**2009**, 66, 47–58. [Google Scholar] [CrossRef] - Manuel, P.; Sivakumar, B.; Arivarignan, G. A perishable inventory system with service facilities, MAP arrivals and PH Service times. J. Syst. Sci. Syst. Eng.
**2007**, 16, 62–73. [Google Scholar] [CrossRef] - Krishnamoorthy, A.; Vladimir, V.; Manjunath, A.S.; Dhanya, S. Single server with several services. Reliab. Theory Appl.
**2017**, 12, 14–30. [Google Scholar] - Shajin, D.; Krishnamoorthy, A. On a queueing-inventory system with impatient customers, advanced reservation, cancellation, overbooking and common life time. Oper. Res.
**2021**, 21, 1229–1253. [Google Scholar] [CrossRef] - Shajin, D.; Krishnamoorthy, A.; Dudin, A.; Joshua, V.; Jacob, V. On a queueing-inventory system with advanced reservation and cancellation for the next K time frames ahead: The case of overbooking. Queueing Syst.
**2020**, 94, 3–37. [Google Scholar] [CrossRef] - Ko, S.-S. A Nonhomogeneous Quasi-birth-death process approach for an (s,S) policy for a perishable inventory system with retrial demands. J. Ind. Manag. Optim.
**2020**, 37, 1415–1433. [Google Scholar] - Chakravarthy, S.R. Analysis of MAP/PH
_{1},PH_{2}/1 queue with vacations and optional secondary services. Appl. Math. Model.**2013**, 37, 8886–8902. [Google Scholar] [CrossRef] - Krishnamoorthy, A.; Shajin, D.; Lakshmy, B. GI/M/1 type queueing-inventory systems with postponed work, reservation, cancellation and common life time. Indian J. Pure Appl. Math.
**2016**, 47, 357–388. [Google Scholar] [CrossRef] - Nair, S.S.; Jose, K.P. A PH Distributed Production Inventory Model with Different Modes of Service and MAP Arrivals. In Applied Probability and Stochastic Processes; Infosys Science Foundation Series; Springer: Singapore, 2020. [Google Scholar]
- Punalal, J.; Babu, S. The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer Induced Interruption and Retrial of Customers. In Applied Probability and Stochastic Processes; Infosys Science Foundation Series; Springer: Singapore, 2020. [Google Scholar]
- Ayyappan, G.; Gowthami, R. Analysis of MAP, PH2OA/PH1I, PH2O/1 retrial queue with vacation, feedback, two-way communication and impatient customers. Soft Comput.
**2020**, 25, 9811–9838. [Google Scholar] [CrossRef] - Lee, S.; Dudin, S.; Dudina, O.; Kim, C.; Klimenok, V. A Priority Queue with Many Customer Types, Correlated Arrivals and Changing Priorities. Mathematics
**2020**, 8, 1292. [Google Scholar] [CrossRef] - Klimenok, V.; Dudin, A.; Dudina, O.; Kochetkova, I. Queuing System with Two Types of Customers and Dynamic Change of a Priority. Mathematics
**2020**, 8, 824. [Google Scholar] [CrossRef] - Dudin, A.; Dudin, S. Analysis of a Priority Queue with Phase-Type Service and Failures. Int. J. Stoch. Anal.
**2016**, 2016, 9152701. [Google Scholar] [CrossRef] [Green Version] - He, Q.-M. The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue. Queueing Syst.
**2001**, 38, 397–418. [Google Scholar] [CrossRef] - Salini, S.; Nair, K.P.; Jose, A. MAP/PH/1 Production Inventory Model with Varying Service Rates. Int. J. Pure Appl. Math.
**2017**, 117, 373–381. [Google Scholar] - Radhamani, V.; Devi, P.C.; Sivakumar, B. A Stochastic Inventory System with Postponed Demands and Infinite Pool in Discrete-Time Setup. J. Oper. Res. Soc. China
**2014**, 2, 455–480. [Google Scholar] [CrossRef] [Green Version] - Barron, Y.; Hermel, D. Shortage decision policies for a fluid production model with MAP arrivals. Int. J. Prod. Res.
**2017**, 55, 3946–3969. [Google Scholar] [CrossRef] - Gaver, D.P.; Jacobs, P.A.; Latouche, G. Finite Birth- And- Death Models in Randomly Changing Environments. Adv. Appl. Probab.
**1984**, 16, 715–731. [Google Scholar] [CrossRef]

$\mathit{S}/\mathit{s}$ | 5 | 6 | 7 | 8 |
---|---|---|---|---|

21 | 73.9092 | 50.1316 | 35.9508 | 51.6445 |

22 | 70.1498 | 48.7175 | 35.1863 | 50.1225 |

23 | 66.6530 | 47.3281 | 34.3798 | 48.6287 |

24 | 64.2888 | 46.8634 | 34.4369 | 48.0564 |

25 | 64.3960 | 46.9908 | 35.0336 | 48.0747 |

**Table 2.**Ramification of the rate of lead time $\left(\mu \right)$ and approach from the pool customer $\left(\theta \right)$ on the optimal values.

$\mathit{\mu}/\mathit{\theta}$ | 0.5 | 1.0 | 1.5 | 1.5 | 2.0 | |||||
---|---|---|---|---|---|---|---|---|---|---|

6 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 |

35.2053 | 34.9025 | 34.5938 | 34.2810 | 33.9654 | ||||||

8 | 24 | 7 | 24 | 7 | 24 | 7 | 23 | 7 | 23 | 7 |

32.0167 | 31.7418 | 31.4634 | 31.1825 | 30.8967 | ||||||

10 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

29.1054 | 28.8532 | 28.5981 | 28.3413 | 28.0834 | ||||||

12 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

26.5210 | 26.2909 | 26.0586 | 25.8250 | 25.5906 | ||||||

14 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

24.2303 | 24.0214 | 23.8108 | 23.5992 | 23.3871 |

**Table 3.**Ramification of the cost rate of a lost customer during the stock out period $\left({c}_{cl}\right)$ and cost rate of waiting customer in the pool $\left({c}_{w}\right)$ on the optimal values.

${\mathit{c}}_{\mathbf{cl}}/{\mathit{c}}_{\mathit{w}}$ | 1 | 2 | 3 | 4 | 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

0.1 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

25.3237 | 26.3775 | 27.4312 | 28.4850 | 29.5388 | ||||||

0.2 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 |

20.6208 | 21.7063 | 22.7918 | 23.8773 | 24.9628 | ||||||

0.3 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 |

15.8716 | 16.9572 | 18.0427 | 19.1282 | 20.2137 | ||||||

0.4 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 |

11.1225 | 12.2080 | 13.2935 | 14.3790 | 15.4645 | ||||||

0.5 | 22 | 7 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 |

6.3611 | 7.4589 | 8.5444 | 9.6299 | 10.7154 |

**Table 4.**Ramification of the cost of setup cost $\left({c}_{s}\right)$ and holding cost $\left({c}_{h}\right)$ on the optimal values.

${\mathit{c}}_{\mathit{s}}/{\mathit{c}}_{\mathit{h}}$ | 0.7 | 0.8 | 0.9 | 1.0 | 1.1 | |||||
---|---|---|---|---|---|---|---|---|---|---|

8 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 | 23 | 7 |

24.8072 | 25.1731 | 25.5389 | 25.9047 | 26.2705 | ||||||

9 | 24 | 7 | 24 | 7 | 24 | 7 | 23 | 7 | 23 | 7 |

25.7475 | 26.1340 | 26.5205 | 26.9057 | 27.2716 | ||||||

10 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

26.6582 | 27.0447 | 27.4312 | 27.8178 | 28.2043 | ||||||

11 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

27.5689 | 27.9555 | 28.3420 | 28.7285 | 29.1150 | ||||||

12 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 | 24 | 7 |

28.4797 | 28.8662 | 29.2527 | 29.6392 | 30.0258 |

Customer Arrivals | $\mathit{\theta}$ | ${\mathit{\eta}}_{\mathit{I}}$ | ${\mathit{\eta}}_{\mathit{R}}$ | ${\mathit{\eta}}_{\mathit{PC}}$ | ${\mathit{\eta}}_{\mathit{IC}}$ | ${\mathit{\eta}}_{\mathit{L}}$ | ${\mathit{\eta}}_{\mathit{C}}$ |
---|---|---|---|---|---|---|---|

MAP with HEX | 6 | 7.1489 | 5.5542 | 0.9492 | 0.1955 | 0.9655 | 4.9701 |

8 | 6.1845 | 5.2418 | 1.8282 | 0.1841 | 0.7165 | 4.8983 | |

10 | 5.5445 | 5.1297 | 2.1221 | 0.1755 | 0.598 | 4.8031 | |

12 | 5.2971 | 5.1185 | 2.3209 | 0.1587 | 0.2655 | 4.7455 | |

14 | 4.1842 | 5.0395 | 2.7986 | 0.1484 | 0.0495 | 4.2965 | |

MAP with ER | 6 | 10.4716 | 1.5605 | 4.7701 | 0.1907 | 0.2718 | 3.9203 |

8 | 10.3639 | 1.1016 | 4.8031 | 0.1618 | 0.1711 | 3.7462 | |

10 | 10.2159 | 0.7828 | 4.8355 | 0.1421 | 0.1071 | 3.3194 | |

12 | 9.9523 | 0.5579 | 4.8672 | 0.1271 | 0.0669 | 3.2674 | |

14 | 9.6010 | 0.3977 | 4.8983 | 0.1148 | 0.0419 | 3.0355 | |

MAP with NC | 6 | 11.9361 | 0.8136 | 3.6126 | 0.5072 | 0.1869 | 6.5235 |

8 | 11.8047 | 0.6569 | 3.6502 | 0.4694 | 0.1415 | 6.1452 | |

10 | 11.7142 | 0.5300 | 3.6872 | 0.4375 | 0.1076 | 5.4512 | |

12 | 10.1372 | 0.4278 | 3.7235 | 0.4099 | 0.0821 | 5.2201 | |

14 | 9.2649 | 0.3455 | 3.7592 | 0.3857 | 0.0629 | 5.1258 | |

MAP with PC | 6 | 11.8179 | 0.0055 | 3.0061 | 0.4916 | 0.1555 | 8.2835 |

8 | 11.9699 | 0.0063 | 3.0113 | 0.4738 | 0.1655 | 8.1521 | |

10 | 11.9704 | 0.0069 | 3.0160 | 0.4590 | 0.1731 | 7.9203 | |

12 | 11.4244 | 0.0076 | 3.0203 | 0.4464 | 0.1790 | 7.5603 | |

14 | 10.1432 | 0.0082 | 3.0244 | 0.4352 | 0.1835 | 7.3595 |

Customer Arrivals | $\mathit{\mu}$ | ${\mathit{\eta}}_{\mathit{I}}$ | ${\mathit{\eta}}_{\mathit{R}}$ | ${\mathit{\eta}}_{\mathit{PC}}$ | ${\mathit{\eta}}_{\mathit{IC}}$ | ${\mathit{\eta}}_{\mathit{L}}$ | ${\mathit{\eta}}_{\mathit{C}}$ |
---|---|---|---|---|---|---|---|

MAP with HEX | 0.5 | 9.2511 | 1.2543 | 2.3533 | 0.7530 | 0.9532 | 6.2472 |

1.0 | 5.4524 | 1.0216 | 2.3751 | 0.4652 | 0.7592 | 6.1552 | |

1.5 | 5.2455 | 1.0178 | 2.3751 | 0.3728 | 0.5313 | 5.2428 | |

2.0 | 3.9425 | 1.0149 | 2.3751 | 0.2086 | 0.3097 | 5.1588 | |

2.5 | 3.7058 | 1.0126 | 2.3751 | 0.0532 | 0.0927 | 5.0214 | |

MAP with ER | 0.5 | 9.5325 | 1.2029 | 3.8355 | 0.6417 | 0.8013 | 7.2472 |

1.0 | 9.9282 | 1.1225 | 3.8355 | 0.5124 | 0.7032 | 6.8435 | |

1.5 | 8.8140 | 1.0212 | 3.8356 | 0.4532 | 0.6233 | 6.5472 | |

2.0 | 8.2287 | 1.0129 | 3.8356 | 0.3681 | 0.5237 | 6.3758 | |

2.5 | 8.0266 | 1.0025 | 3.8357 | 0.2522 | 0.4251 | 6.2457 | |

MAP with NC | 0.5 | 6.2459 | 2.0054 | 3.6872 | 0.7490 | 0.1595 | 7.8643 |

1.0 | 6.1205 | 2.0056 | 3.6873 | 0.6378 | 0.1645 | 7.5184 | |

1.5 | 13.0476 | 2.0055 | 3.6873 | 0.5431 | 0.1635 | 7.4259 | |

2.0 | 6.0102 | 2.0053 | 3.6874 | 0.4629 | 0.1585 | 7.3445 | |

2.5 | 5.3139 | 2.0051 | 3.6874 | 0.3952 | 0.1510 | 6.2454 | |

MAP with PC | 0.5 | 5.8184 | 2.0051 | 4.5212 | 1.7060 | 0.9654 | 6.9637 |

1.0 | 3.9954 | 2.0051 | 4.6122 | 1.5971 | 0.7652 | 6.7235 | |

1.5 | 2.3729 | 2.0049 | 4.6222 | 1.5089 | 0.6449 | 6.6282 | |

2.0 | 2.9522 | 2.0046 | 4.7522 | 1.4371 | 0.5372 | 6.4645 | |

2.5 | 1.7177 | 2.0043 | 4.7622 | 1.3782 | 0.3286 | 6.1278 |

Customer Arrivals | $\mathit{\beta}$ | ${\mathit{\eta}}_{\mathit{I}}$ | ${\mathit{\eta}}_{\mathit{R}}$ | ${\mathit{\eta}}_{\mathit{PC}}$ | ${\mathit{\eta}}_{\mathit{IC}}$ | ${\mathit{\eta}}_{\mathit{L}}$ | ${\mathit{\eta}}_{\mathit{C}}$ |
---|---|---|---|---|---|---|---|

MAP with HEX | 0.5 | 4.3915 | 4.7884 | 1.0554 | 38.8344 | 9.9514 | 4.2575 |

1.0 | 4.3805 | 4.4750 | 1.0320 | 30.7307 | 8.9913 | 4.1702 | |

1.5 | 4.2681 | 4.3167 | 1.1123 | 30.3223 | 8.9913 | 4.1434 | |

2.0 | 4.1868 | 4.0668 | 1.9921 | 30.2715 | 2.0151 | 1.5565 | |

2.5 | 3.6345 | 3.2879 | 2.0475 | 30.1281 | 1.9440 | 1.3230 | |

MAP with ER | 0.5 | 6.5733 | 2.5566 | 1.0214 | 1.1602 | 5.1051 | 0.7440 |

1.0 | 2.0654 | 1.4214 | 1.2613 | 1.0315 | 4.0583 | 0.6516 | |

1.5 | 0.8832 | 1.3771 | 1.3315 | 0.7274 | 1.4230 | 0.5629 | |

2.0 | 0.4074 | 0.3074 | 1.3701 | 0.3122 | 0.5364 | 0.2604 | |

2.5 | 0.1988 | 0.0065 | 1.4128 | 0.1682 | 0.2350 | 0.1353 | |

MAP with NC | 0.5 | 7.7680 | 4.1544 | 0.9291 | 0.0302 | 0.1130 | 0.0098 |

1.0 | 2.0747 | 3.5491 | 0.9314 | 0.0139 | 0.0139 | 0.0024 | |

1.5 | 1.5672 | 1.5672 | 0.9437 | 0.0002 | 0.0006 | 0.0001 | |

2.0 | 1.7586 | 0.5786 | 0.9560 | 0.0001 | 0.0005 | 0.0001 | |

2.5 | 0.2107 | 0.3754 | 0.9684 | 0.0001 | 0.0004 | 0.0001 | |

MAP with PC | 0.5 | 0.3234 | 0.1616 | 1.1871 | 2.2619 | 3.4010 | 1.0797 |

1.0 | 0.3160 | 0.0788 | 1.1248 | 1.0117 | 1.7219 | 0.5441 | |

1.5 | 0.0590 | 0.0206 | 1.2137 | 0.1521 | 0.1198 | 0.1073 | |

2.0 | 0.0087 | 0.0019 | 1.3521 | 0.0194 | 0.0161 | 0.0158 | |

2.5 | 0.0020 | 0.0003 | 1.4287 | 0.0040 | 0.0034 | 0.0035 |

Ordinary/Impulse Arrivals | MAP with ER | MAP with NC | MAP with PC | |||
---|---|---|---|---|---|---|

MAP with ER | 23 | 7 | 23 | 8 | 24 | 7 |

23.1112 | 21.6972 | 12.0351 | ||||

MAP with NC | 23 | 7 | 23 | 8 | 24 | 7 |

22.8082 | 21.4556 | 21.1303 | ||||

MAP with PC | 23 | 7 | 23 | 8 | 24 | 7 |

21.5698 | 23.8045 | 22.7606 |

Impulse Customer Arrivals | ${\mathit{\eta}}_{\mathit{I}}$ | ${\mathit{\eta}}_{\mathit{R}}$ | ${\mathit{\eta}}_{\mathit{PC}}$ | ${\mathit{\eta}}_{\mathit{IC}}$ | ${\mathit{\eta}}_{\mathit{L}}$ | ${\mathit{\eta}}_{\mathit{C}}$ |
---|---|---|---|---|---|---|

MAP of ordinary arrival with ER | ||||||

MAP with ER | 4.0749 | 0.3074 | 1.0370 | 0.3122 | 0.5364 | 0.2604 |

MAP with NC | 2.8386 | 0.2935 | 0.9444 | 0.2783 | 0.0878 | 0.0417 |

MAP with PC | 1.8369 | 0.0038 | 1.0198 | 0.0189 | 0.0562 | 0.0154 |

MAP of ordinary arrival with NC | ||||||

MAP with ER | 1.7586 | 0.0055 | 0.5786 | 0.5431 | 0.1635 | 0.1720 |

MAP with NC | 0.6741 | 0.0161 | 0.0016 | 0.9280 | 0.4787 | 0.0128 |

MAP with PC | 0.0164 | 0.0086 | 0.0029 | 0.8591 | 0.2563 | 0.0031 |

MAP of ordinary arrival with PC | ||||||

MAP with ER | 8.0759 | 0.0019 | 0.0194 | 0.0161 | 0.0158 | 2.1521 |

MAP with NC | 2.2589 | 0.0171 | 0.0015 | 0.8594 | 0.5084 | 1.2315 |

MAP with PC | 0.0909 | 0.8807 | 0.2235 | 0.1300 | 0.1486 | 1.1763 |

${\mathit{\lambda}}_{1}$ | Ordinary/Impulse Arrivals | MAP with ER | MAP with NC | MAP with PC | |||
---|---|---|---|---|---|---|---|

1.0 | MAP with ER | 30 | 9 | 30 | 8 | 31 | 9 |

3.2312 | 2.8333 | 1.9895 | |||||

MAP with NC | 33 | 7 | 33 | 7 | 31 | 9 | |

2.7514 | 2.8149 | 2.8366 | |||||

MAP with PC | 31 | 8 | 34 | 8 | 33 | 7 | |

3.0706 | 2.8275 | 2.8560 | |||||

1.5 | MAP with ER | 33 | 7 | 35 | 7 | 35 | 8 |

2.9048 | 2.9277 | 2.8302 | |||||

MAP with NC | 35 | 9 | 33 | 8 | 33 | 8 | |

3.7805 | 2.8542 | 2.8954 | |||||

MAP with PC | 31 | 8 | 34 | 8 | 33 | 7 | |

24.7267 | 2.3890 | 20.5588 | |||||

2.0 | MAP with ER | 33 | 7 | 35 | 7 | 35 | 8 |

2.8922 | 2.8921 | 2.8967 | |||||

MAP with NC | 35 | 9 | 33 | 8 | 33 | 8 | |

2.9265 | 2.8177 | 2.8953 | |||||

MAP with PC | 31 | 8 | 34 | 8 | 33 | 7 | |

2.8855 | 2.8851 | 2.8920 |

${\mathit{\lambda}}_{2}$ | Impulse/Ordinary Arrivals | MAP with ER | MAP with NC | MAP with PC | |||
---|---|---|---|---|---|---|---|

0.2 | MAP with ER | 30 | 9 | 30 | 8 | 31 | 9 |

4.2172 | 1.3851 | 0.1314 | |||||

MAP with NC | 33 | 7 | 33 | 7 | 31 | 9 | |

1.3100 | 1.0523 | 1.0050 | |||||

MAP with PC | 31 | 8 | 34 | 8 | 33 | 7 | |

1.1253 | 0.5222 | 7.0213 | |||||

0.4 | MAP with ER | 38 | 3 | 38 | 3 | 38 | 3 |

3.0233 | 2.2223 | 2.3503 | |||||

MAP with NC | 35 | 9 | 33 | 8 | 33 | 8 | |

1.0123 | 2.2211 | 2.4125 | |||||

MAP with PC | 31 | 8 | 34 | 8 | 33 | 7 | |

3.4561 | 2.1847 | 2.4875 | |||||

0.6 | MAP with ER | 35 | 9 | 33 | 8 | 33 | 8 |

2.5654 | 3.4842 | 1.1551 | |||||

MAP with NC | 33 | 9 | 31 | 11 | 35 | 8 | |

3.2484 | 1.5422 | 0.4412 | |||||

MAP with PC | 31 | 8 | 34 | 8 | 33 | 7 | |

3.1654 | 2.1685 | 3.1517 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vinitha, V.; Anbazhagan, N.; Amutha, S.; Jeganathan, K.; Joshi, G.P.; Cho, W.; Seo, S.
Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System. *Processes* **2021**, *9*, 2146.
https://doi.org/10.3390/pr9122146

**AMA Style**

Vinitha V, Anbazhagan N, Amutha S, Jeganathan K, Joshi GP, Cho W, Seo S.
Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System. *Processes*. 2021; 9(12):2146.
https://doi.org/10.3390/pr9122146

**Chicago/Turabian Style**

Vinitha, V., N. Anbazhagan, S. Amutha, K. Jeganathan, Gyanendra Prasad Joshi, Woong Cho, and Suseok Seo.
2021. "Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System" *Processes* 9, no. 12: 2146.
https://doi.org/10.3390/pr9122146