# Queueing-Inventory Systems with Catastrophes under Various Replenishment Policies

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

- The c-customers (consumer customers) arrive in the system according to the Markovian arrival process ($MAP$) with representation ${({\mathit{D}}_{0},{\mathit{D}}_{1})}_{m}$. The underlying Markov chain of the $MAP$ is governed by the matrix $\mathit{D}$$(={\mathit{D}}_{0}+{\mathit{D}}_{1})$. Such that, the entries of matrix ${\mathit{D}}_{0}$ denote the transition rates without arrival while the entries of matrix ${\mathit{D}}_{1}$ denote the transition rates with arrival. So, the arrival rate of c-customers is given by ${\lambda}^{+}=\mathit{\delta}{\mathit{D}}_{1}\mathit{e}$ where $\mathit{\delta}$ is the stationary probability vector of the generator matrix $\mathit{D}$ and it is satisfied$$\mathit{\delta}\mathit{D}=\mathbf{0},\phantom{\rule{3.33333pt}{0ex}}\mathit{\delta}\mathit{e}=1.$$
- The service times of the c-customers follow $PH$-distribution with representation ${(\mathit{\beta},T)}_{n}$ where $\mathit{\beta}$ is the initial probability vector, $\mathit{\beta}\mathit{e}=1$, and $\mathit{T}$ is a sub-generator matrix. The matrix $\mathit{T}$ holds the transition rates among the n transient states and ${\mathit{T}}^{0}$ is a column vector containing the absorption rates into state 0 from the transient states. It is clear that $\mathit{T}\mathit{e}+{\mathit{T}}^{0}=\mathbf{0}$. The phase-type distribution has the service rate $\mu =1/\left[\mathit{\beta}{(-\mathit{T})}^{-1}\mathit{e}\right]$.
- The system also receives n-customers (negative customers) that the arrivals occur according to the Poisson process with rate ${\lambda}^{-}$. When an n-customer arrives in the system, there are three possible cases; (i) there is at least one c-customer in the queue $(QL>0)$, and only the c-customer is pushed out from the queue (i.e., the servicing of the c-customer in the server continues), (ii) the queue has no c-customer $(QL=0)$ and the server is busy with a c-customer, then the c-customer in the server is forced out of the system. However, in this case, the inventory level does not change, since stocks are released after the completion of servicing a c-customer is assumed, and (iii) there are no c-customers in the system. The arrived n-customer has no effect on the operation of the system.
- A hybrid sales scheme is used in the system. When a c-customer arrives in the system, if the inventory level is zero $(IL=0)$, then the c-customer either joins the queue of infinite capacity with probability ${\theta}_{1}$ (called backorder sale scheme), or leaves the system unserved with probability ${\theta}_{2}$ (called lost sale scheme). Note that ${\theta}_{1}+{\theta}_{2}=1$. If the inventory level occurs to be zero with the completion servicing of a c-customer, the c-customer in the queue (if any) waits for a replenishment.
- In the warehouse part of the system, catastrophic events can occur according to the Poisson process with parameter $\kappa $. When a catastrophic event occurs, all items, even the item that is at the status of release to the c-customer in the inventory are instantly destroyed. If the c-customer’s service is interrupted due to a catastrophe, then he returns to the queue. In other words, the catastrophic event only destroys the items in the inventory and does not cause c-customers out of the system. Hence, if the number of items in the inventory is zero, then the disaster has no effect on the operation of the system.
- Two inventory replenishment policies are considered in this study. That is an $(s,S)$-type policy for Model-1 and an $(s,Q)$-type policy for Model-2. The lead time of order follows an exponential distribution with parameter $\eta $ for both replenishment policies. In an $(s,S)$-type policy (sometimes this policy is called “Up to S”), when the inventory level drops to the reorder point s, $0\le s<S$, an order is placed for replenishment and upon replenishment the inventory level becomes S. This policy states that the replenishment quantity varies in order to fill the maximum capacity of the inventory when the reorder is placed. In an $(s,Q)$-type policy, when the inventory level drops to the reorder point s, $s<\frac{S}{2}$, an order quantity of a $Q=S-s$ is placed for replenishment and upon replenishment the inventory level becomes a sum of the current items in the inventory and order quantity. This policy states that the replenishment quantity is always fixed.

## 3. The Steady-State Analysis

#### 3.1. Model-1 with $(s,S)$-Type Replenishment Policy

#### 3.1.1. Stability Condition

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Note:**In the paper [37], the authors studied the queueing-inventory system which we have discussed here by considering Poisson arrival and exponentially distributed service times. They obtained the closed-form solution of the probabilities for the special case. We suggest the paper in [37] to see the stability condition of the system under Poisson arrival and exponential service.

#### 3.1.2. The Steady-State Probability Vector of the Matrix $\mathit{G}$

**G**in (2). That is, $\mathit{x}$ satisfies

#### 3.2. Model-2 with $(s,Q)$-Type Replenishment Policy

#### 3.2.1. Stability Condition

**Theorem**

**2.**

#### 3.2.2. The Steady-State Probability Vector of the Matrix $\tilde{\mathit{G}}$

## 4. Performance Measures of Model-1 and Model-2

- The probability that there is no c-customer in the system$${P}_{idle}=\mathit{x}\left(0\right)\mathit{e}.$$
- The mean number of c-customers in the system$$E\left(N\right)=\sum _{k=1}^{\infty}k\phantom{\rule{3.33333pt}{0ex}}\mathit{x}\left(k\right)\mathit{e}=\mathit{x}\left(1\right){(\mathit{I}-\mathit{R})}^{-2}\mathit{e}.$$
- The mean loss rate of c-customers because of no inventory$${E}_{I}\left(LR\right)={\theta}_{2}\left[\mathit{x}(0,0){\mathit{D}}_{1}{\mathit{e}}_{m}+\sum _{k=1}^{\infty}\mathit{x}(k,0)({I}_{n}\otimes {\mathit{D}}_{1}){\mathit{e}}_{mn}\right].$$
- The mean loss rate of c-customers because of n-customer$${E}_{N}\left(LR\right)={\lambda}^{-}\left[1-\mathit{x}\left(0\right)\mathit{e}\right].$$
- The mean loss rate of c-customers$$E\left(LR\right)={E}_{I}\left(LR\right)+{E}_{N}\left(LR\right).$$
- The mean number of items in the inventory$$E\left(I\right)=\sum _{i=1}^{S}i\phantom{\rule{3.33333pt}{0ex}}x(0,i){\mathit{e}}_{m}+\sum _{k=1}^{\infty}\sum _{i=1}^{S}i\phantom{\rule{3.33333pt}{0ex}}x(k,i){\mathit{e}}_{mn}.$$
- The mean reorder rate$$E\left(RR\right)=\sum _{k=1}^{\infty}i\phantom{\rule{3.33333pt}{0ex}}x(k,s+1)({\mathit{T}}^{0}\otimes {I}_{m}){\mathit{e}}_{m}+\kappa \left[\sum _{i=1}^{S}x(0,i){\mathit{e}}_{m}+\sum _{k=1}^{\infty}\sum _{i=1}^{S}x(k,i){\mathit{e}}_{mn}\right].$$
- The mean order size$${E}_{1}\left(OS\right)=\sum _{i=S-s}^{S}i\phantom{\rule{3.33333pt}{0ex}}x(0,S-i){\mathit{e}}_{m}+\sum _{k=1}^{\infty}\sum _{i=S-s}^{S}i\phantom{\rule{3.33333pt}{0ex}}x(k,S-i){\mathit{e}}_{mn}.$$$${E}_{2}\left(OS\right)=Q\left[\sum _{i=0}^{s}\tilde{x}(0,i){\mathit{e}}_{m}+\sum _{k=1}^{\infty}\sum _{i=0}^{s}\tilde{x}(k,i){\mathit{e}}_{mn}\right].$$

## 5. Numerical Study

#### 5.1. The Effect of Parameters on Performance Measures

#### 5.2. Optimization

- ${c}_{k}:$
- the fixed cost of one order,
- ${c}_{r}:$
- the unit cost of the order size,
- ${c}_{h}:$
- the holding cost per item in the inventory per unit of time,
- ${c}_{ps}:$
- the damaging cost per item in the inventory,
- ${c}_{l}:$
- the cost incured due to the loss of a c-customer,
- ${c}_{w}:$
- the waiting cost of a c-customer in the system.

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QS | Queueing System |

QIS | Queueing Inventory System |

ICS | Inventory Control System |

MAP | Markovian Arrival Process |

PH | Phase-type distribution |

IL | Inventory Level |

QL | Queue Length |

CTMC | Continuous Time Markov Chain |

QBD | Quasi-birth-and-death process |

ETC | Expected Total Cost |

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As It Is Varied | It Is Fixed |
---|---|

the arrival rate of c-customers: ${\lambda}^{+}$ | ${\lambda}^{-}=1$, $\phantom{\rule{3.33333pt}{0ex}}\mu =8$, $\phantom{\rule{3.33333pt}{0ex}}\eta =1$, $\phantom{\rule{3.33333pt}{0ex}}\kappa =1$, $\phantom{\rule{3.33333pt}{0ex}}{\theta}_{1}=0.6$ |

the arrival rate of n-customers: ${\lambda}^{-}$ | ${\lambda}^{+}=5$, $\phantom{\rule{3.33333pt}{0ex}}\mu =8$, $\phantom{\rule{3.33333pt}{0ex}}\eta =1$, $\phantom{\rule{3.33333pt}{0ex}}\kappa =1$, $\phantom{\rule{3.33333pt}{0ex}}{\theta}_{1}=0.6$ |

the service rate of c-customers: $\mu $ | ${\lambda}^{+}=5$, $\phantom{\rule{3.33333pt}{0ex}}{\lambda}^{-}=1$, $\phantom{\rule{3.33333pt}{0ex}}\eta =1$, $\phantom{\rule{3.33333pt}{0ex}}\kappa =1$, $\phantom{\rule{3.33333pt}{0ex}}{\theta}_{1}=0.6$ |

the rate of the catastrophic events: $\kappa $ | ${\lambda}^{+}=5$, $\phantom{\rule{3.33333pt}{0ex}}{\lambda}^{-}=1$, $\phantom{\rule{3.33333pt}{0ex}}\mu =8$, $\phantom{\rule{3.33333pt}{0ex}}\eta =1$, $\phantom{\rule{3.33333pt}{0ex}}{\theta}_{1}=0.6$ |

the probability that c-customer joins the queue when the inventory level is zero: ${\theta}_{1}$ | ${\lambda}^{+}=4$, $\phantom{\rule{3.33333pt}{0ex}}{\lambda}^{-}=1$, $\phantom{\rule{3.33333pt}{0ex}}\mu =8$, $\phantom{\rule{3.33333pt}{0ex}}\eta =1$, $\phantom{\rule{3.33333pt}{0ex}}\kappa =1$ |

ERLA | HEXA | ||||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | EXPS | HEXS | ERLS | EXPS | HEXS | |

4.2 | 3.239 | 3.490 | 5.611 | 7.730 | 8.133 | 10.894 | |

4.4 | 3.848 | 4.179 | 6.994 | 9.530 | 10.046 | 13.654 | |

${\lambda}^{+}$ | 4.6 | 4.663 | 5.106 | 8.925 | 11.967 | 12.646 | 17.501 |

4.8 | 5.811 | 6.426 | 11.789 | 15.438 | 16.373 | 23.198 | |

5 | 7.559 | 8.458 | 16.444 | 20.759 | 22.140 | 32.449 | |

0.4 | 3.401 | 3.707 | 6.344 | 9.298 | 9.772 | 13.120 | |

0.6 | 4.384 | 4.808 | 8.496 | 11.889 | 12.534 | 17.199 | |

$\kappa $ | 0.8 | 5.686 | 6.291 | 11.589 | 15.463 | 16.380 | 23.117 |

1 | 7.559 | 8.458 | 16.444 | 20.759 | 22.140 | 32.449 | |

1.2 | 10.577 | 12.023 | 25.194 | 29.468 | 31.767 | 49.303 | |

7.6 | 9.620 | 10.940 | 22.927 | 27.554 | 29.633 | 45.447 | |

8 | 7.559 | 8.458 | 16.444 | 20.759 | 22.140 | 32.449 | |

$\mu $ | 8.4 | 6.323 | 6.989 | 12.837 | 16.701 | 17.717 | 25.201 |

8.8 | 5.499 | 6.018 | 10.549 | 14.009 | 14.802 | 20.592 | |

9.2 | 4.909 | 5.329 | 8.975 | 12.095 | 12.741 | 17.411 | |

1 | 7.559 | 8.458 | 16.444 | 20.759 | 22.140 | 32.449 | |

1.4 | 4.317 | 4.701 | 7.931 | 11.502 | 12.095 | 16.254 | |

${\lambda}^{-}$ | 1.8 | 2.957 | 3.159 | 4.778 | 7.644 | 7.979 | 10.175 |

2.2 | 2.216 | 2.331 | 3.200 | 5.555 | 5.767 | 7.059 | |

2.6 | 1.753 | 1.822 | 2.296 | 4.262 | 4.405 | 5.205 |

ERLA | HEXA | ||||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | EXPS | HEXS | ERLS | EXPS | HEXS | |

4.2 | 3.701 | 4.001 | 6.579 | 9.563 | 10.081 | 13.596 | |

4.4 | 4.560 | 4.976 | 8.584 | 12.213 | 12.924 | 17.831 | |

${\lambda}^{+}$ | 4.6 | 5.811 | 6.412 | 11.701 | 16.100 | 17.133 | 24.402 |

4.8 | 7.803 | 8.737 | 17.165 | 22.329 | 23.979 | 35.903 | |

5 | 11.486 | 13.156 | 29.116 | 33.888 | 37.021 | 61.022 | |

0.4 | 4.462 | 4.861 | 8.427 | 13.026 | 13.702 | 18.572 | |

0.6 | 5.900 | 6.499 | 11.895 | 17.145 | 18.173 | 25.651 | |

$\kappa $ | 0.8 | 7.997 | 8.947 | 17.641 | 23.348 | 25.032 | 37.437 |

1 | 11.486 | 13.156 | 29.116 | 33.888 | 37.021 | 61.022 | |

1.2 | 18.705 | 22.381 | 63.549 | 55.978 | 63.556 | 131.820 | |

7.6 | 16.591 | 19.688 | 52.949 | 50.813 | 57.091 | 111.116 | |

8 | 11.486 | 13.156 | 29.116 | 33.888 | 37.021 | 61.022 | |

$\mu $ | 8.4 | 8.971 | 10.066 | 20.110 | 25.573 | 27.542 | 42.060 |

8.8 | 7.472 | 8.265 | 15.396 | 20.636 | 22.028 | 32.114 | |

9.2 | 6.477 | 7.086 | 12.507 | 17.370 | 18.426 | 26.003 | |

1 | 11.486 | 13.156 | 29.116 | 33.888 | 37.021 | 61.022 | |

1.4 | 5.187 | 5.675 | 9.862 | 14.842 | 15.683 | 21.456 | |

${\lambda}^{-}$ | 1.8 | 3.270 | 3.498 | 5.346 | 9.048 | 9.451 | 12.058 |

2.2 | 2.354 | 2.476 | 3.412 | 6.281 | 6.516 | 7.939 | |

2.6 | 1.822 | 1.892 | 2.386 | 4.682 | 4.833 | 5.677 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 3.266 | 3.324 | 3.345 | 3.408 | 3.334 | 3.397 | |

4.2 | 3.209 | 3.275 | 3.280 | 3.350 | 3.268 | 3.338 | |

${\lambda}^{+}$ | 4.4 | 3.154 | 3.228 | 3.217 | 3.294 | 3.204 | 3.281 |

4.6 | 3.099 | 3.182 | 3.154 | 3.238 | 3.141 | 3.226 | |

4.8 | 3.046 | 3.138 | 3.092 | 3.184 | 3.080 | 3.172 | |

0.2 | 4.000 | 4.088 | 4.140 | 4.227 | 4.054 | 4.147 | |

0.4 | 3.696 | 3.797 | 3.807 | 3.907 | 3.747 | 3.851 | |

$\kappa $ | 0.6 | 3.431 | 3.537 | 3.513 | 3.616 | 3.475 | 3.582 |

0.8 | 3.199 | 3.303 | 3.255 | 3.358 | 3.234 | 3.339 | |

1 | 2.994 | 3.094 | 3.030 | 3.130 | 3.020 | 3.120 | |

0.1 | 3.655 | 3.665 | 3.774 | 3.795 | 3.767 | 3.796 | |

0.3 | 3.500 | 3.526 | 3.606 | 3.643 | 3.598 | 3.639 | |

${\theta}_{1}$ | 0.5 | 3.343 | 3.390 | 3.432 | 3.487 | 3.422 | 3.478 |

0.7 | 3.191 | 3.259 | 3.256 | 3.328 | 3.245 | 3.316 | |

0.9 | 3.039 | 3.127 | 3.077 | 3.165 | 3.068 | 3.155 | |

1 | 2.994 | 3.094 | 3.030 | 3.130 | 3.020 | 3.120 | |

1.4 | 3.108 | 3.184 | 3.159 | 3.242 | 3.150 | 3.231 | |

${\lambda}^{-}$ | 1.8 | 3.212 | 3.260 | 3.270 | 3.336 | 3.266 | 3.328 |

2.2 | 3.306 | 3.325 | 3.368 | 3.416 | 3.368 | 3.412 | |

2.6 | 3.391 | 3.380 | 3.453 | 3.483 | 3.459 | 3.486 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 2.266 | 2.289 | 2.275 | 2.303 | 2.250 | 2.277 | |

4.2 | 2.214 | 2.240 | 2.221 | 2.252 | 2.200 | 2.231 | |

${\lambda}^{+}$ | 4.4 | 2.162 | 2.192 | 2.167 | 2.201 | 2.150 | 2.184 |

4.6 | 2.109 | 2.143 | 2.113 | 2.150 | 2.101 | 2.138 | |

4.8 | 2.057 | 2.095 | 2.060 | 2.100 | 2.051 | 2.091 | |

0.2 | 2.949 | 2.984 | 2.976 | 3.015 | 2.960 | 3.000 | |

0.4 | 2.634 | 2.671 | 2.648 | 2.689 | 2.633 | 2.675 | |

$\kappa $ | 0.6 | 2.382 | 2.421 | 2.390 | 2.432 | 2.377 | 2.420 |

0.8 | 2.176 | 2.217 | 2.180 | 2.223 | 2.171 | 2.215 | |

1 | 2.005 | 2.047 | 2.007 | 2.050 | 2.001 | 2.045 | |

0.1 | 2.559 | 2.563 | 2.624 | 2.635 | 2.581 | 2.594 | |

0.3 | 2.456 | 2.467 | 2.496 | 2.515 | 2.454 | 2.473 | |

${\theta}_{1}$ | 0.5 | 2.335 | 2.354 | 2.351 | 2.377 | 2.320 | 2.345 |

0.7 | 2.193 | 2.219 | 2.195 | 2.225 | 2.177 | 2.207 | |

0.9 | 2.030 | 2.059 | 2.027 | 2.059 | 2.020 | 2.053 | |

1 | 2.005 | 2.047 | 2.007 | 2.050 | 2.001 | 2.045 | |

1.4 | 2.121 | 2.152 | 2.124 | 2.161 | 2.112 | 2.148 | |

${\lambda}^{-}$ | 1.8 | 2.218 | 2.236 | 2.222 | 2.252 | 2.205 | 2.233 |

2.2 | 2.301 | 2.303 | 2.306 | 2.327 | 2.285 | 2.303 | |

2.6 | 2.371 | 2.355 | 2.378 | 2.389 | 2.353 | 2.362 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 0.635 | 0.637 | 0.640 | 0.639 | 0.628 | 0.626 | |

4.2 | 0.645 | 0.646 | 0.648 | 0.647 | 0.637 | 0.635 | |

${\lambda}^{+}$ | 4.4 | 0.655 | 0.655 | 0.656 | 0.654 | 0.647 | 0.644 |

4.6 | 0.664 | 0.663 | 0.665 | 0.662 | 0.657 | 0.653 | |

4.8 | 0.673 | 0.671 | 0.673 | 0.669 | 0.666 | 0.662 | |

0.2 | 0.504 | 0.497 | 0.493 | 0.486 | 0.492 | 0.485 | |

0.4 | 0.564 | 0.557 | 0.556 | 0.549 | 0.552 | 0.545 | |

$\kappa $ | 0.6 | 0.612 | 0.606 | 0.607 | 0.600 | 0.601 | 0.594 |

0.8 | 0.650 | 0.645 | 0.648 | 0.642 | 0.642 | 0.636 | |

1 | 0.682 | 0.678 | 0.681 | 0.677 | 0.676 | 0.671 | |

0.1 | 0.582 | 0.584 | 0.589 | 0.590 | 0.577 | 0.576 | |

0.3 | 0.599 | 0.602 | 0.608 | 0.608 | 0.595 | 0.593 | |

${\theta}_{1}$ | 0.5 | 0.622 | 0.625 | 0.629 | 0.628 | 0.616 | 0.614 |

0.7 | 0.648 | 0.649 | 0.651 | 0.649 | 0.640 | 0.638 | |

0.9 | 0.674 | 0.672 | 0.674 | 0.672 | 0.668 | 0.665 | |

1 | 0.682 | 0.678 | 0.681 | 0.677 | 0.676 | 0.671 | |

1.4 | 0.663 | 0.663 | 0.664 | 0.661 | 0.656 | 0.652 | |

${\lambda}^{-}$ | 1.8 | 0.646 | 0.648 | 0.649 | 0.648 | 0.639 | 0.637 |

2.2 | 0.630 | 0.636 | 0.637 | 0.637 | 0.626 | 0.624 | |

2.6 | 0.617 | 0.624 | 0.625 | 0.628 | 0.614 | 0.613 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 0.764 | 0.754 | 0.751 | 0.739 | 0.741 | 0.729 | |

${\lambda}^{+}$ | 4.2 | 0.774 | 0.762 | 0.762 | 0.749 | 0.754 | 0.740 |

4.4 | 0.784 | 0.770 | 0.773 | 0.758 | 0.767 | 0.751 | |

0.2 | 0.615 | 0.606 | 0.604 | 0.594 | 0.605 | 0.595 | |

0.4 | 0.683 | 0.670 | 0.672 | 0.659 | 0.672 | 0.659 | |

$\kappa $ | 0.6 | 0.735 | 0.719 | 0.726 | 0.709 | 0.725 | 0.708 |

0.8 | 0.777 | 0.758 | 0.769 | 0.750 | 0.768 | 0.748 | |

1 | 0.811 | 0.789 | 0.805 | 0.783 | 0.803 | 0.782 | |

0.1 | 0.686 | 0.686 | 0.675 | 0.672 | 0.658 | 0.652 | |

0.3 | 0.716 | 0.713 | 0.705 | 0.699 | 0.689 | 0.680 | |

0.5 | 0.748 | 0.741 | 0.735 | 0.726 | 0.723 | 0.712 | |

${\theta}_{1}$ | 4.6 | 0.793 | 0.777 | 0.784 | 0.766 | 0.779 | 0.762 |

4.8 | 0.802 | 0.783 | 0.794 | 0.775 | 0.791 | 0.772 | |

0.7 | 0.778 | 0.765 | 0.768 | 0.753 | 0.760 | 0.745 | |

0.9 | 0.806 | 0.787 | 0.801 | 0.782 | 0.798 | 0.779 | |

1 | 0.811 | 0.789 | 0.805 | 0.783 | 0.803 | 0.782 | |

1.4 | 0.791 | 0.776 | 0.782 | 0.765 | 0.777 | 0.760 | |

${\lambda}^{-}$ | 1.8 | 0.773 | 0.764 | 0.762 | 0.749 | 0.754 | 0.740 |

2.2 | 0.755 | 0.753 | 0.745 | 0.735 | 0.733 | 0.723 | |

2.6 | 0.738 | 0.742 | 0.730 | 0.724 | 0.716 | 0.709 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 5.891 | 5.928 | 5.953 | 5.983 | 5.896 | 5.927 | |

4.2 | 5.960 | 5.998 | 6.012 | 6.043 | 5.960 | 5.993 | |

${\lambda}^{+}$ | 4.4 | 6.028 | 6.066 | 6.071 | 6.103 | 6.025 | 6.058 |

4.6 | 6.095 | 6.133 | 6.130 | 6.163 | 6.090 | 6.125 | |

4.8 | 6.161 | 6.200 | 6.188 | 6.222 | 6.155 | 6.191 | |

0.2 | 4.852 | 4.828 | 4.797 | 4.772 | 4.795 | 4.767 | |

0.4 | 5.267 | 5.257 | 5.247 | 5.234 | 5.222 | 5.210 | |

$\kappa $ | 0.6 | 5.629 | 5.636 | 5.632 | 5.635 | 5.599 | 5.604 |

0.8 | 5.947 | 5.970 | 5.962 | 5.982 | 5.930 | 5.952 | |

1 | 6.227 | 6.265 | 6.247 | 6.281 | 6.220 | 6.257 | |

0.1 | 5.545 | 5.567 | 5.607 | 5.625 | 5.547 | 5.562 | |

0.3 | 5.654 | 5.682 | 5.732 | 5.754 | 5.671 | 5.691 | |

${\theta}_{1}$ | 0.5 | 5.806 | 5.840 | 5.876 | 5.903 | 5.816 | 5.843 |

0.7 | 5.980 | 6.020 | 6.032 | 6.066 | 5.982 | 6.017 | |

0.9 | 6.167 | 6.215 | 6.199 | 6.241 | 6.166 | 6.212 | |

1 | 6.227 | 6.265 | 6.247 | 6.281 | 6.220 | 6.257 | |

1.4 | 6.087 | 6.127 | 6.125 | 6.158 | 6.085 | 6.118 | |

${\lambda}^{-}$ | 1.8 | 5.966 | 6.010 | 6.021 | 6.055 | 5.973 | 6.005 |

2.2 | 5.861 | 5.912 | 5.931 | 5.969 | 5.880 | 5.912 | |

2.6 | 5.770 | 5.831 | 5.853 | 5.896 | 5.802 | 5.835 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 4.605 | 4.611 | 4.613 | 4.614 | 4.573 | 4.574 | |

4.2 | 4.666 | 4.669 | 4.671 | 4.670 | 4.637 | 4.637 | |

${\lambda}^{+}$ | 4.4 | 4.725 | 4.726 | 4.728 | 4.724 | 4.700 | 4.698 |

4.6 | 4.784 | 4.781 | 4.784 | 4.778 | 4.763 | 4.759 | |

4.8 | 4.841 | 4.835 | 4.840 | 4.832 | 4.825 | 4.818 | |

0.2 | 4.036 | 4.006 | 3.993 | 3.959 | 3.994 | 3.959 | |

0.4 | 4.319 | 4.293 | 4.294 | 4.265 | 4.288 | 4.259 | |

$\kappa $ | 0.6 | 4.548 | 4.527 | 4.535 | 4.511 | 4.524 | 4.501 |

0.8 | 4.738 | 4.722 | 4.732 | 4.714 | 4.720 | 4.704 | |

1 | 4.897 | 4.888 | 4.896 | 4.885 | 4.886 | 4.876 | |

0.1 | 4.236 | 4.248 | 4.241 | 4.247 | 4.181 | 4.182 | |

0.3 | 4.365 | 4.375 | 4.379 | 4.382 | 4.322 | 4.323 | |

${\theta}_{1}$ | 0.5 | 4.521 | 4.529 | 4.532 | 4.534 | 4.485 | 4.486 |

0.7 | 4.691 | 4.696 | 4.698 | 4.698 | 4.665 | 4.668 | |

0.9 | 4.872 | 4.875 | 4.875 | 4.876 | 4.861 | 4.865 | |

1 | 4.897 | 4.888 | 4.896 | 4.885 | 4.886 | 4.876 | |

1.4 | 4.771 | 4.770 | 4.773 | 4.766 | 4.750 | 4.745 | |

${\lambda}^{-}$ | 1.8 | 4.659 | 4.671 | 4.668 | 4.668 | 4.634 | 4.634 |

2.2 | 4.562 | 4.589 | 4.579 | 4.586 | 4.536 | 4.542 | |

2.6 | 4.476 | 4.519 | 4.501 | 4.518 | 4.452 | 4.464 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 0.837 | 0.848 | 0.869 | 0.884 | 0.861 | 0.879 | |

4.2 | 0.886 | 0.899 | 0.918 | 0.935 | 0.909 | 0.929 | |

${\lambda}^{+}$ | 4.4 | 0.936 | 0.952 | 0.967 | 0.987 | 0.958 | 0.979 |

4.6 | 0.988 | 1.007 | 1.016 | 1.039 | 1.007 | 1.031 | |

4.8 | 1.040 | 1.063 | 1.066 | 1.091 | 1.058 | 1.084 | |

0.2 | 0.642 | 0.664 | 0.723 | 0.750 | 0.677 | 0.712 | |

0.4 | 0.788 | 0.811 | 0.849 | 0.876 | 0.818 | 0.851 | |

$\kappa $ | 0.6 | 0.908 | 0.933 | 0.953 | 0.981 | 0.933 | 0.964 |

0.8 | 1.009 | 1.035 | 1.041 | 1.069 | 1.029 | 1.058 | |

1 | 1.094 | 1.120 | 1.117 | 1.144 | 1.109 | 1.137 | |

0.1 | 1.832 | 1.833 | 1.918 | 1.939 | 1.883 | 1.930 | |

0.3 | 1.435 | 1.442 | 1.499 | 1.518 | 1.478 | 1.507 | |

${\theta}_{1}$ | 0.5 | 1.037 | 1.048 | 1.081 | 1.097 | 1.069 | 1.090 |

0.7 | 0.635 | 0.645 | 0.656 | 0.669 | 0.651 | 0.665 | |

0.9 | 0.217 | 0.222 | 0.222 | 0.227 | 0.221 | 0.226 | |

1 | 1.094 | 1.120 | 1.117 | 1.144 | 1.109 | 1.137 | |

1.4 | 1.073 | 1.092 | 1.104 | 1.128 | 1.095 | 1.119 | |

${\lambda}^{-}$ | 1.8 | 1.057 | 1.071 | 1.093 | 1.115 | 1.083 | 1.106 |

2.2 | 1.045 | 1.056 | 1.083 | 1.103 | 1.074 | 1.095 | |

2.6 | 1.036 | 1.045 | 1.075 | 1.094 | 1.067 | 1.086 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 0.882 | 0.899 | 0.926 | 0.948 | 0.907 | 0.935 | |

4.2 | 0.937 | 0.958 | 0.979 | 1.005 | 0.960 | 0.990 | |

${\lambda}^{+}$ | 4.4 | 0.995 | 1.019 | 1.033 | 1.061 | 1.015 | 1.047 |

4.6 | 1.054 | 1.082 | 1.087 | 1.118 | 1.071 | 1.105 | |

4.8 | 1.114 | 1.147 | 1.142 | 1.176 | 1.128 | 1.165 | |

0.2 | 0.768 | 0.801 | 0.857 | 0.896 | 0.804 | 0.852 | |

0.4 | 0.903 | 0.938 | 0.968 | 1.007 | 0.931 | 0.977 | |

$\kappa $ | 0.6 | 1.012 | 1.048 | 1.059 | 1.097 | 1.033 | 1.076 |

0.8 | 1.102 | 1.139 | 1.134 | 1.171 | 1.117 | 1.158 | |

1 | 1.177 | 1.214 | 1.197 | 1.234 | 1.187 | 1.226 | |

0.1 | 1.873 | 1.873 | 2.032 | 2.065 | 1.953 | 2.064 | |

0.3 | 1.481 | 1.492 | 1.589 | 1.619 | 1.538 | 1.592 | |

${\theta}_{1}$ | 0.5 | 1.084 | 1.101 | 1.149 | 1.174 | 1.120 | 1.155 |

0.7 | 0.673 | 0.690 | 0.700 | 0.719 | 0.689 | 0.711 | |

0.9 | 0.234 | 0.242 | 0.238 | 0.246 | 0.237 | 0.245 | |

1 | 1.177 | 1.214 | 1.197 | 1.234 | 1.187 | 1.226 | |

1.4 | 1.142 | 1.171 | 1.180 | 1.213 | 1.162 | 1.198 | |

${\lambda}^{-}$ | 1.8 | 1.116 | 1.138 | 1.164 | 1.196 | 1.143 | 1.176 |

2.2 | 1.095 | 1.113 | 1.150 | 1.180 | 1.127 | 1.159 | |

2.6 | 1.079 | 1.094 | 1.137 | 1.167 | 1.115 | 1.145 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 0.759 | 0.752 | 0.724 | 0.727 | 0.745 | 0.745 | |

4.2 | 0.787 | 0.782 | 0.756 | 0.761 | 0.776 | 0.777 | |

${\lambda}^{+}$ | 4.4 | 0.815 | 0.813 | 0.788 | 0.794 | 0.806 | 0.809 |

4.6 | 0.843 | 0.843 | 0.819 | 0.827 | 0.836 | 0.840 | |

4.8 | 0.871 | 0.873 | 0.851 | 0.860 | 0.865 | 0.871 | |

0.2 | 0.743 | 0.736 | 0.747 | 0.748 | 0.756 | 0.755 | |

0.4 | 0.790 | 0.787 | 0.783 | 0.788 | 0.796 | 0.798 | |

$\kappa $ | 0.6 | 0.830 | 0.830 | 0.818 | 0.825 | 0.831 | 0.835 |

0.8 | 0.866 | 0.869 | 0.851 | 0.860 | 0.864 | 0.870 | |

1 | 0.898 | 0.903 | 0.882 | 0.893 | 0.894 | 0.902 | |

0.1 | 0.438 | 0.417 | 0.446 | 0.434 | 0.459 | 0.446 | |

0.3 | 0.601 | 0.583 | 0.572 | 0.567 | 0.586 | 0.578 | |

${\theta}_{1}$ | 0.5 | 0.713 | 0.702 | 0.675 | 0.676 | 0.696 | 0.693 |

0.7 | 0.802 | 0.799 | 0.771 | 0.778 | 0.791 | 0.795 | |

0.9 | 0.882 | 0.889 | 0.864 | 0.878 | 0.878 | 0.889 | |

1 | 0.898 | 0.903 | 0.882 | 0.893 | 0.894 | 0.902 | |

1.4 | 1.173 | 1.166 | 1.142 | 1.148 | 1.161 | 1.163 | |

${\lambda}^{-}$ | 1.8 | 1.410 | 1.384 | 1.363 | 1.359 | 1.387 | 1.379 |

2.2 | 1.615 | 1.562 | 1.554 | 1.536 | 1.579 | 1.559 | |

2.6 | 1.790 | 1.707 | 1.720 | 1.685 | 1.744 | 1.710 |

ERLA | HEXA | MPCA | |||||
---|---|---|---|---|---|---|---|

Values of the Parameters | ERLS | HEXS | ERLS | HEXS | ERLS | HEXS | |

4 | 0.778 | 0.776 | 0.755 | 0.762 | 0.772 | 0.777 | |

4.2 | 0.809 | 0.810 | 0.788 | 0.798 | 0.805 | 0.812 | |

${\lambda}^{+}$ | 4.4 | 0.840 | 0.844 | 0.822 | 0.834 | 0.836 | 0.845 |

4.6 | 0.870 | 0.877 | 0.856 | 0.870 | 0.868 | 0.879 | |

4.8 | 0.900 | 0.910 | 0.889 | 0.905 | 0.899 | 0.912 | |

0.2 | 0.783 | 0.783 | 0.794 | 0.800 | 0.800 | 0.805 | |

0.4 | 0.828 | 0.832 | 0.829 | 0.839 | 0.837 | 0.846 | |

$\kappa $ | 0.6 | 0.867 | 0.874 | 0.862 | 0.875 | 0.871 | 0.882 |

0.8 | 0.901 | 0.911 | 0.893 | 0.909 | 0.901 | 0.915 | |

1 | 0.930 | 0.944 | 0.923 | 0.940 | 0.929 | 0.945 | |

0.1 | 0.435 | 0.415 | 0.452 | 0.439 | 0.469 | 0.459 | |

0.3 | 0.604 | 0.589 | 0.587 | 0.583 | 0.602 | 0.597 | |

${\theta}_{1}$ | 0.5 | 0.726 | 0.719 | 0.700 | 0.704 | 0.719 | 0.720 |

0.7 | 0.828 | 0.831 | 0.808 | 0.821 | 0.824 | 0.833 | |

0.9 | 0.924 | 0.942 | 0.915 | 0.938 | 0.923 | 0.943 | |

1 | 0.930 | 0.944 | 0.923 | 0.940 | 0.929 | 0.945 | |

1.4 | 1.206 | 1.208 | 1.187 | 1.200 | 1.201 | 1.212 | |

${\lambda}^{-}$ | 1.8 | 1.441 | 1.424 | 1.410 | 1.414 | 1.430 | 1.431 |

2.2 | 1.642 | 1.597 | 1.600 | 1.590 | 1.624 | 1.612 | |

2.6 | 1.813 | 1.738 | 1.764 | 1.738 | 1.789 | 1.764 |

$\mathit{s}=3$ | $\mathit{s}=5$ | $\mathit{s}=7$ | |||||
---|---|---|---|---|---|---|---|

MAP | PH | ${\mathit{S}}^{*}$ | ${\mathit{ETC}}^{*}$ | ${\mathit{S}}^{*}$ | ${\mathit{ETC}}^{*}$ | ${\mathit{S}}^{*}$ | ${\mathit{ETC}}^{*}$ |

ERLA | ERLS | 12 | 1522.323 | 12 | 1525.292 | 12 | 1537.011 |

EXPS | 12 | 1577.132 | 12 | 1579.566 | 12 | 1590.679 | |

HEXS | 14 | 2028.332 | 14 | 2028.057 | 14 | 2033.688 | |

EXPA | ERLS | 13 | 1656.619 | 13 | 1656.629 | 13 | 1664.424 |

EXPS | 13 | 1714.634 | 13 | 1714.218 | 13 | 1721.526 | |

HEXS | 15 | 2171.108 | 15 | 2169.237 | 15 | 2172.631 | |

HEXA | ERLS | 18 | 2414.265 | 17 | 2403.514 | 16 | 2398.433 |

EXPS | 18 | 2497.895 | 17 | 2487.835 | 17 | 2482.838 | |

HEXS | 20 | 3045.628 | 19 | 3036.796 | 19 | 3032.003 | |

MNCA | ERLS | 13 | 1705.272 | 13 | 1705.334 | 13 | 1713.199 |

EXPS | 13 | 1760.042 | 13 | 1759.702 | 13 | 1767.099 | |

HEXS | 15 | 2210.253 | 15 | 2208.484 | 15 | 2211.984 | |

MPCA | ERLS | 39 | 28,273.244 | 38 | 28,245.179 | 36 | 28,217.734 |

EXPS | 40 | 28,343.299 | 39 | 28,316.826 | 37 | 28,290.719 | |

HEXS | 45 | 28,862.644 | 43 | 28,840.331 | 42 | 28,818.321 |

$\mathit{s}=3$ | $\mathit{s}=5$ | $\mathit{s}=7$ | |||||
---|---|---|---|---|---|---|---|

MAP | PH | ${\mathit{S}}^{*}$ | ${\mathit{ETC}}^{*}$ | ${\mathit{S}}^{*}$ | ${\mathit{ETC}}^{*}$ | ${\mathit{S}}^{*}$ | ${\mathit{ETC}}^{*}$ |

ERLA | ERLS | 15 | 1522.217 | 17 | 1528.293 | 19 | 1546.239 |

EXPS | 15 | 1576.974 | 17 | 1582.440 | 19 | 1599.634 | |

HEXS | 17 | 2027.992 | 19 | 2029.560 | 21 | 2039.175 | |

EXPA | ERLS | 16 | 1656.341 | 18 | 1658.732 | 20 | 1671.566 |

EXPS | 16 | 1714.313 | 18 | 1716.220 | 20 | 1728.459 | |

HEXS | 18 | 2170.712 | 20 | 2170.295 | 22 | 2176.910 | |

HEXA | ERLS | 21 | 2413.769 | 22 | 2403.575 | 23 | 2400.146 |

EXPS | 21 | 2497.376 | 22 | 2487.828 | 24 | 2484.688 | |

HEXS | 23 | 3045.173 | 24 | 3036.695 | 25 | 3032.973 | |

MNCA | ERLS | 16 | 1705.006 | 18 | 1707.478 | 20 | 1720.441 |

EXPS | 16 | 1759.736 | 18 | 1761.756 | 20 | 1774.156 | |

HEXS | 18 | 2209.873 | 20 | 2209.594 | 22 | 2216.393 | |

MPCA | ERLS | 42 | 28,273.102 | 43 | 28,244.961 | 43 | 28,217.343 |

EXPS | 43 | 28,343.165 | 44 | 28,316.619 | 44 | 28,290.345 | |

HEXS | 48 | 28,862.522 | 48 | 28,840.099 | 49 | 28,817.971 |

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## Share and Cite

**MDPI and ACS Style**

Ozkar, S.; Melikov, A.; Sztrik, J.
Queueing-Inventory Systems with Catastrophes under Various Replenishment Policies. *Mathematics* **2023**, *11*, 4854.
https://doi.org/10.3390/math11234854

**AMA Style**

Ozkar S, Melikov A, Sztrik J.
Queueing-Inventory Systems with Catastrophes under Various Replenishment Policies. *Mathematics*. 2023; 11(23):4854.
https://doi.org/10.3390/math11234854

**Chicago/Turabian Style**

Ozkar, Serife, Agassi Melikov, and Janos Sztrik.
2023. "Queueing-Inventory Systems with Catastrophes under Various Replenishment Policies" *Mathematics* 11, no. 23: 4854.
https://doi.org/10.3390/math11234854