# Analysis of M/M/1/N Stochastic Queueing—Inventory System with Discretionary Priority Service and Retrial Facility

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## Abstract

**:**

## 1. Introduction

#### Literature Review

## 2. Model Description

#### 2.1. Definitions

#### 2.1.1. Non-Preemptive Priority (Head-of-The-Line Priority Discipline)

#### 2.1.2. Preemptive Priority Discipline

#### 2.1.3. Mixed-Priority Discipline

#### 2.1.4. Discretionary Priority Discipline

#### 2.2. Notations

#### 2.3. Explanation of the Model

- The HP customer is at service;
- The LP customer is receiving stage-II service;
- The LP customer is at stage-I service, when the inventory is greater than L;
- The HP customer decides to wait with probability $(1-r)$;
- With zero inventory level.

- When the inventory is zero;
- On interruption from an HP customer at stage-I;
- If the server is busy with HP or LP customers.

## 3. Analysis

#### 3.1. Steady-State Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.2. Computation of R-Matrix

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.3. System Performance Measures

- Expected inventory level:$${E}_{I}=\sum _{{i}_{1}=0}^{\infty}\sum _{{j}_{1}=1}^{S}{j}_{1}\left\{u({i}_{1},{j}_{1},0,0)+\sum _{{k}_{1}=1}^{3}\sum _{{l}_{1}=0}^{N}u({i}_{1},{j}_{1},{k}_{1},{l}_{1})\right\}$$
- Expected perishable rate:$${E}_{P}=\sum _{{i}_{1}=0}^{\infty}\left\{\sum _{{j}_{1}=1}^{S}{j}_{1}\gamma u({i}_{1},{j}_{1},0,0)+\sum _{{j}_{1}=1}^{S}\sum _{{k}_{1}=1}^{3}\sum _{{l}_{1}=0}^{N}({j}_{1}-1)\gamma u({i}_{1},{j}_{1},{k}_{1},{l}_{1})\right\}$$
- Expected reorder rate:$$\begin{array}{c}\hfill {E}_{R}=\sum _{{i}_{1}=0}^{\infty}(s+1)\gamma u({i}_{1},(s+1),0,0)+\sum _{{i}_{1}=0}^{\infty}\sum _{{k}_{1}=1}^{3}\sum _{{l}_{1}=0}^{N}s\gamma u({i}_{1},(s+1),{k}_{1},{l}_{1})+\\ \hfill \sum _{{i}_{1}=0}^{\infty}{\mu}_{{L}_{2}}u({i}_{1},(s+1),2,0)+\sum _{{i}_{1}=0}^{\infty}\sum _{{l}_{1}=0}^{N}{\mu}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},(s+1),3,{l}_{1}).\end{array}$$
- Expected number of HP customers in the waiting hall:$${E}_{WH}=\sum _{{i}_{1}=0}^{\infty}\sum _{{l}_{1}=0}^{N}{l}_{1}\left\{u({i}_{1},0,0,{l}_{1})+\sum _{{j}_{1}=1}^{S}\sum _{{k}_{1}=1}^{3}u({i}_{1},{j}_{1},{k}_{1},{l}_{1})\right\}$$
- Expectation of HP customer entering into the waiting hall:$${E}_{EWH}=\sum _{{i}_{1}=0}^{\infty}\left\{\sum _{{l}_{1}=0}^{N-1}{\lambda}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},0,0,{l}_{1})+\sum _{{j}_{1}=1}^{S}{\lambda}_{H}u({i}_{1},{j}_{1},0,0)+\sum _{{j}_{1}=1}^{S}\sum _{{k}_{1}=1}^{3}\sum _{{l}_{1}=0}^{N-1}{\lambda}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},{k}_{1},{l}_{1})\right\}$$
- Expected waiting time of an HP:$${E}_{WHP}={E}_{WH}/{E}_{EWH}$$
- Expected number of LP customers in the orbit:$${E}_{O}=\sum _{{i}_{1}=0}^{\infty}{i}_{1}\phantom{\rule{3.33333pt}{0ex}}u\left({i}_{1}\right)\mathbf{e}$$
- Expected number of LP customers lost:$\begin{array}{c}{E}_{LPL}=\sum _{{i}_{1}=0}^{\infty}\sum _{{l}_{1}=0}^{N}(1-p){\lambda}_{L}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},0,0,{l}_{1})+\sum _{{i}_{1}=0}^{\infty}\sum _{{1}_{1}=1}^{S}\sum _{{k}_{1}=1}^{3}\sum _{{l}_{1}=0}^{N}(1-p){\lambda}_{L}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},{k}_{1},{l}_{1})\hfill \\ +\sum _{{i}_{1}=0}^{\infty}\sum _{{j}_{1}=1}^{L}\sum _{{l}_{1}=0}^{N}r{\lambda}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},0,{l}_{1})\hfill \end{array}$
- Expected number of HP customers lost:$${E}_{HPL}=\sum _{{i}_{1}=0}^{\infty}\left\{{\lambda}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},0,0,N)+\sum _{{j}_{1}=1}^{S}\sum _{{k}_{1}=1}^{3}{\lambda}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},{k}_{1},N)\right\}$$
- Expected number of times an HP customer interrupts an LP customer:$${E}_{HPIL}=\sum _{{i}_{1}=0}^{\infty}\sum _{{j}_{1}=1}^{L}r{\lambda}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},1,0)$$
- Probability of server being idle:$${E}_{SI}=\sum _{{i}_{1}=0}^{\infty}\left\{\sum _{{l}_{1}=0}^{N}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},0,0,{l}_{1})+\sum _{{j}_{1}=1}^{S}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},0,0)\right\}$$
- Probability of server being busy:$${E}_{SB}=\sum _{{i}_{1}=0}^{\infty}\sum _{{j}_{1}=1}^{S}\sum _{{k}_{1}=1}^{3}\sum _{{l}_{1}=0}^{N}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},{k}_{1},{l}_{1})$$
- Expected number of times a server carries out the orbital search:$${E}_{OS}=\sum _{{i}_{1}=1}^{\infty}\sum _{{j}_{1}=1}^{S}\left\{q{\mu}_{{L}_{2}}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},2,0)+q{\mu}_{H}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},3,0)\right\}$$
- Overall rate of retrial:$${E}_{ORR}=\sum _{{i}_{1}=1}^{\infty}{i}_{1}{\lambda}_{r}\phantom{\rule{3.33333pt}{0ex}}u\left({i}_{1}\right)\mathbf{e}$$
- Successful rate of retrial:$${E}_{SRR}=\sum _{{i}_{1}=1}^{\infty}\sum _{{j}_{1}=1}^{S}{i}_{1}{\lambda}_{r}\phantom{\rule{3.33333pt}{0ex}}u({i}_{1},{j}_{1},0,0)$$
- Fraction of successful rate of retrial:$${E}_{FSRR}={E}_{SRR}/{E}_{ORR}$$

#### 3.4. Cost Analysis

## 4. Numerical Discussions

**Example**

**1.**

- Mixed priority: At $r=0.23$, the optimum value is obtained similarly at ${S}^{*}=30$, ${s}^{*}=4$ and $T{C}^{*}=3.074810.$ as shown in Table 2. This $T{C}^{*}$ shows the impact of interruption, based on the discretionary priority. On comparing $T{C}^{*}$ with case (i), as we expected, the discretionary priority service process gives the minimum optimal total cost rather than the usual priority service patterns.
- Preemptive priority discipline: At $r=1$, the optimum total cost is obtained at ${S}^{*}=31$, ${s}^{*}=2$, and $T{C}^{*}=2.983274$, as shown in Table 3. Even though the $T{C}^{*}$ of this case is minimum than that of the case (i), we conclude that case (ii) provides the best service discipline. This is because, when a company allows preemptive priority, it may lose its LP customers. The growth of the company obviously depends upon all types of customers. So, the company must satisfy them by providing their best service. In such a way, discretionary priority discipline is considered the best one.

**Example**

**2.**

- Mixed priority: At r = 0.5, the total cost $T{C}^{*}$ follows the same pattern of increment and decrement as in case (i), for all the parameters considered in Table 4, respectively. The optimum total cost obtained is $T{C}^{*}=2.890126$ at $\gamma =0.05$ and $\beta =1.8$. Thus, the optimum value obtained in discretionary priority is less than that of non-preemptive priority.
- Preemptive priority discipline: At $r=1$, for the same value of the parameter as in case (i), we obtain the optimum total cost to be $T{C}^{*}=2.48149$, which is less than those of non-preemptive and discretionary priority disciplines. However, depending on this $T{C}^{*}$, we cannot make the inference that preemptive priority discipline is economical. Although it seems like preemptive priority discipline is more profitable than the other priority discipline, it leads to the loss of new patrons for the business. A new LP customer who is ignored or made to wait for a long duration may leave the system before trying the product even once. Hence, discretionary priority is best suited to increase the customer base.

**Example**

**3.**

- Mixed priority: At $r=0.5$, the measures ${E}_{WHP}$ and ${E}_{WLP}$ show the same variations as in case (i) for the respective parameters, but the difference is that the waiting time of HP customers is less, and the waiting time of LP customers is higher, compared with those in case (i).
- Preemptive Priority discipline: At $r=1$, similar to case (i), the waiting times remain with the same variation with respect to the parameters considered in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, and the measure ${E}_{WHP}$ is considerably less and ${E}_{WLP}$ is much higher compared with those in case (i).

**Example**

**4.**

- Mixed priority: At $r=0.5$, the loss of HP customers encountered by following discretionary priority is less compared with non-preemptive priority. In the same way, the loss of LP customers by following discretionary priority is greater compared with non-preemptive priority.
- Preemptive Priority: As seen previously, the measure ${E}_{HPL}$ is a little less compared with case (i), and ${E}_{LPL}$ is slightly greater than mixed priority. Thus, the loss of an HP customer heightens with non-discretionary priority, and an LP customer rises with preemptive priority. However, the intensified loss of any one type of customer will affect the reputation of the organization, which in turn decreases the growth of the business. Hence, to maintain an admissible loss of any type of customer, the following discretionary priority is advisable.

**Example**

**5.**

- The ${E}_{HPIL}$ increases with ${\mu}_{H}$ despite the increment in ${\mu}_{{L}_{1}}$ on positive interruption.Since an increase in service rates allows the server to serve all customers in less time, the measure ${E}_{SB}$ decreases with an increase in service rate and increases with arrival rate.
- It is clear from the table that, as the service rate, ${\mu}_{H}$, increases, the server may complete the service to all HP customers, and the LP customers obtain a successful chance of retrial.

**Example**

**6.**

- Mixed priority: In contrast to case (i), we obtain the same increment and reduction in the case $r=0.5$ for the parameters, respectively. However, with discretionary priority, ${E}_{WH}$ decreases and ${E}_{O}$ increases.
- Preemptive priority discipline: As discussed in case (i), the changes with respect to other parameters at $r=1$ are in the same order, but ${E}_{WH}$ decreases, and ${E}_{O}$ increases slightly higher compared with case (i). With this, we conclude that the waiting hall accumulates non-preemptive priority, and the customers in orbit increase with preemptive priority. This also affects the business environment and once again shows the importance of discretionary priority discipline.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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S|s | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

28 | 3.187610 | 3.132378 | 3.114067 | 3.102593 | 3.110016 | 3.132837 | 3.164871 |

29 | 3.170042 | 3.116397 | 3.108295 | 3.093581 | 3.099043 | 3.120513 | 3.151056 |

30 | 3.157677 | 3.103865 | 3.097418 | 3.088793 | 3.092117 | 3.111897 | 3.140980 |

31 | 3.153721 | 3.101571 | 3.095904 | 3.089001 | 3.095888 | 3.114345 | 3.142301 |

32 | 3.162313 | 3.113600 | 3.111644 | 3.106273 | 3.114821 | 3.135832 | 3.163509 |

S|s | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

28 | 3.156901 | 3.107505 | 3.094919 | 3.087682 | 3.098520 | 3.119306 | 3.155629 |

29 | 3.140263 | 3.092402 | 3.089834 | 3.079195 | 3.088026 | 3.107578 | 3.142287 |

30 | 3.128573 | 3.080375 | 3.079441 | 3.074810 | 3.081410 | 3.099400 | 3.132570 |

31 | 3.125040 | 3.078352 | 3.078033 | 3.075131 | 3.085328 | 3.102140 | 3.134130 |

32 | 3.133964 | 3.090639 | 3.094001 | 3.092575 | 3.104508 | 3.124117 | 3.155706 |

S|s | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

28 | 3.027256 | 3.005787 | 3.018169 | 3.027169 | 3.050287 | 3.068396 | 3.120758 |

29 | 3.014473 | 2.994246 | 3.015881 | 3.020835 | 3.041772 | 3.058894 | 3.109163 |

30 | 3.005563 | 2.984233 | 3.007418 | 3.018146 | 3.036500 | 3.052381 | 3.100798 |

31 | 3.003744 | 2.983274 | 3.006425 | 3.018997 | 3.041181 | 3.056248 | 3.103275 |

32 | 3.013988 | 2.996584 | 3.023343 | 3.037225 | 3.061479 | 3.080069 | 3.126179 |

33 | 3.022344 | 3.005740 | 3.032852 | 3.047970 | 3.072497 | 3.092072 | 3.138950 |

**Table 4.**The effect of the parameters $\gamma $, $\beta $, ${\lambda}_{H}$, ${\lambda}_{r}$, ${\mu}_{H}$, and r.

$\mathit{\gamma}$ | ${\mathit{\lambda}}_{\mathit{H}}$ | ${\mathit{\lambda}}_{\mathit{r}}$ | ${\mathit{\mu}}_{\mathit{H}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ | |||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\beta}=\mathbf{1}.\mathbf{7}$ | $\mathbf{\beta}=\mathbf{1}.\mathbf{8}$ | $\mathbf{\beta}=\mathbf{1}.\mathbf{7}$ | $\mathbf{\beta}=\mathbf{1}.\mathbf{8}$ | $\mathbf{\beta}=\mathbf{1}.\mathbf{7}$ | $\mathbf{\beta}=\mathbf{1}.\mathbf{8}$ | ||||

0.05 | 0.85 | 0.60 | 2.75 | 2.979013 | 2.973860 | 2.942602 | 2.938585 | 2.899332 | 2.896864 |

2.80 | 2.954551 | 2.948898 | 2.918055 | 2.913559 | 2.874628 | 2.871711 | |||

2.85 | 2.931692 | 2.925533 | 2.895107 | 2.890126 | 2.851522 | 2.848149 | |||

0.70 | 2.75 | 2.990586 | 2.985467 | 2.954113 | 2.950135 | 2.910731 | 2.908310 | ||

2.80 | 2.966412 | 2.960791 | 2.929858 | 2.925397 | 2.886322 | 2.883448 | |||

2.85 | 2.943827 | 2.937697 | 2.907188 | 2.902239 | 2.863496 | 2.860163 | |||

0.95 | 0.60 | 2.75 | 2.991737 | 2.987089 | 2.957554 | 2.953994 | 2.916601 | 2.914532 | |

2.80 | 2.966064 | 2.960974 | 2.931800 | 2.927819 | 2.890698 | 2.888237 | |||

2.85 | 2.941959 | 2.936423 | 2.907617 | 2.903210 | 2.866367 | 2.863511 | |||

0.70 | 2.75 | 2.991737 | 2.996258 | 2.957554 | 2.963084 | 2.916601 | 2.923494 | ||

2.80 | 2.966064 | 2.970501 | 2.931800 | 2.937274 | 2.890698 | 2.897572 | |||

2.85 | 2.941959 | 2.946289 | 2.907617 | 2.913009 | 2.866367 | 2.873197 | |||

0.1 | 0.85 | 0.60 | 2.75 | 3.106973 | 3.099336 | 3.064676 | 3.058380 | 3.014288 | 3.009778 |

2.80 | 3.083714 | 3.075479 | 3.041333 | 3.034460 | 2.990770 | 2.985716 | |||

2.85 | 3.062043 | 3.053205 | 3.019577 | 3.012126 | 2.968839 | 2.963238 | |||

0.70 | 2.75 | 3.118530 | 3.110927 | 3.076163 | 3.069904 | 3.025633 | 3.021167 | ||

2.80 | 3.095568 | 3.087364 | 3.053121 | 3.046281 | 3.002421 | 2.997407 | |||

2.85 | 3.074181 | 3.065369 | 3.031652 | 3.024231 | 2.980781 | 2.975217 | |||

0.95 | 0.60 | 2.75 | 3.118081 | 3.111224 | 3.078620 | 3.073032 | 3.031223 | 3.027343 | |

2.80 | 3.093557 | 3.086174 | 3.054006 | 3.047916 | 3.006432 | 3.002082 | |||

2.85 | 3.070585 | 3.062675 | 3.030949 | 3.024355 | 2.983203 | 2.978382 | |||

0.70 | 2.75 | 3.127136 | 3.120334 | 3.087575 | 3.082047 | 3.040010 | 3.036197 | ||

2.80 | 3.102982 | 3.095650 | 3.063340 | 3.057305 | 3.015606 | 3.011320 | |||

2.85 | 3.080360 | 3.072498 | 3.040641 | 3.034099 | 2.992743 | 2.987982 | |||

0.15 | 0.85 | 0.60 | 2.75 | 3.227871 | 3.124522 | 3.180782 | 3.172613 | 3.124522 | 3.118335 |

2.80 | 3.205585 | 3.101958 | 3.158412 | 3.149586 | 3.101958 | 3.095151 | |||

2.85 | 3.184858 | 3.080966 | 3.137602 | 3.128126 | 3.080966 | 3.073538 | |||

0.70 | 2.75 | 3.239407 | 3.229767 | 3.239407 | 3.184112 | 3.135819 | 3.129678 | ||

2.80 | 3.217425 | 3.207099 | 3.217425 | 3.161390 | 3.113568 | 3.106804 | |||

2.85 | 3.196987 | 3.185980 | 3.196987 | 3.140222 | 3.092876 | 3.085485 | |||

0.95 | 0.60 | 2.75 | 3.237823 | 3.229153 | 3.194032 | 3.186784 | 3.141275 | 3.135916 | |

2.80 | 3.214209 | 3.204947 | 3.170322 | 3.162508 | 3.117363 | 3.111471 | |||

2.85 | 3.192134 | 3.182281 | 3.148160 | 3.139778 | 3.095006 | 3.088582 | |||

0.70 | 2.75 | 3.246812 | 3.238204 | 3.202911 | 3.195730 | 3.149959 | 3.144675 | ||

2.80 | 3.223575 | 3.214372 | 3.179589 | 3.171837 | 3.126443 | 3.120623 | |||

2.85 | 3.201857 | 3.192058 | 3.157792 | 3.149469 | 3.104462 | 3.098105 |

N | $\mathit{\beta}$ | ${\mathit{\mu}}_{\mathit{H}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ | |||
---|---|---|---|---|---|---|---|---|

${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | |||

7 | 1.6 | 2.75 | 0.003336 | 0.005413 | 0.003179 | 0.005173 | 0.002990 | 0.004882 |

2.80 | 0.003239 | 0.005245 | 0.003083 | 0.005006 | 0.002894 | 0.004717 | ||

2.85 | 0.003156 | 0.005097 | 0.002999 | 0.004859 | 0.002811 | 0.004570 | ||

1.7 | 2.75 | 0.003173 | 0.005191 | 0.003024 | 0.004963 | 0.002845 | 0.004687 | |

2.80 | 0.003068 | 0.005012 | 0.002920 | 0.004786 | 0.002742 | 0.004511 | ||

2.85 | 0.002977 | 0.004854 | 0.002829 | 0.004628 | 0.002652 | 0.004355 | ||

1.8 | 2.75 | 0.003054 | 0.005025 | 0.002911 | 0.004806 | 0.002741 | 0.004542 | |

2.80 | 0.002943 | 0.004839 | 0.002802 | 0.004622 | 0.002633 | 0.004360 | ||

2.85 | 0.002846 | 0.004673 | 0.002706 | 0.004457 | 0.002537 | 0.004197 | ||

8 | 1.6 | 2.75 | 0.001648 | 0.002828 | 0.001570 | 0.002702 | 0.001476 | 0.002549 |

2.80 | 0.001591 | 0.002723 | 0.001514 | 0.002598 | 0.001421 | 0.002447 | ||

2.85 | 0.001543 | 0.002631 | 0.001466 | 0.002507 | 0.001373 | 0.002358 | ||

1.7 | 2.75 | 0.001547 | 0.002683 | 0.001474 | 0.002565 | 0.001387 | 0.002422 | |

2.80 | 0.001486 | 0.002572 | 0.001414 | 0.002456 | 0.001327 | 0.002314 | ||

2.85 | 0.001432 | 0.002474 | 0.001361 | 0.002359 | 0.001275 | 0.002219 | ||

1.8 | 2.75 | 0.001474 | 0.002576 | 0.001405 | 0.002464 | 0.001323 | 0.002328 | |

2.80 | 0.001409 | 0.002461 | 0.001341 | 0.002350 | 0.001260 | 0.002217 | ||

2.85 | 0.001353 | 0.002358 | 0.001286 | 0.002249 | 0.001205 | 0.002117 | ||

9 | 1.6 | 2.75 | 0.000821 | 0.001491 | 0.000782 | 0.001424 | 0.000735 | 0.001344 |

2.80 | 0.000789 | 0.001426 | 0.000750 | 0.001361 | 0.000704 | 0.001282 | ||

2.85 | 0.000761 | 0.001371 | 0.000723 | 0.001306 | 0.000677 | 0.001228 | ||

1.7 | 2.75 | 0.000760 | 0.001399 | 0.000724 | 0.001337 | 0.000681 | 0.001262 | |

2.80 | 0.000725 | 0.001331 | 0.000690 | 0.001271 | 0.000647 | 0.001197 | ||

2.85 | 0.000695 | 0.001272 | 0.000660 | 0.001212 | 0.000618 | 0.001140 | ||

1.8 | 2.75 | 0.000716 | 0.001331 | 0.000683 | 0.001273 | 0.000642 | 0.001203 | |

2.80 | 0.000679 | 0.001261 | 0.000646 | 0.001205 | 0.000607 | 0.001136 | ||

2.85 | 0.000647 | 0.001199 | 0.000615 | 0.001144 | 0.000576 | 0.001076 |

${\mathit{\mu}}_{\mathit{H}}$ | ${\mathit{\mu}}_{{\mathit{L}}_{2}}$ | ${\mathit{\mu}}_{{\mathit{L}}_{1}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ | |||
---|---|---|---|---|---|---|---|---|

${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | |||

2.75 | 2.60 | 2.40 | 0.223480 | 0.223851 | 0.223730 | 0.224161 | 0.224073 | 0.224585 |

2.45 | 0.223453 | 0.223827 | 0.223700 | 0.224133 | 0.224038 | 0.224551 | ||

2.50 | 0.223427 | 0.223803 | 0.223671 | 0.224107 | 0.224004 | 0.224519 | ||

2.65 | 2.40 | 0.223458 | 0.223831 | 0.223708 | 0.224141 | 0.224051 | 0.224566 | |

2.45 | 0.223431 | 0.223807 | 0.223678 | 0.224113 | 0.224016 | 0.224532 | ||

2.50 | 0.223405 | 0.223783 | 0.223649 | 0.224087 | 0.223983 | 0.224500 | ||

2.70 | 2.40 | 0.223436 | 0.223812 | 0.223686 | 0.224122 | 0.224030 | 0.224547 | |

2.45 | 0.223409 | 0.223787 | 0.223656 | 0.224094 | 0.223995 | 0.224513 | ||

2.50 | 0.223383 | 0.223764 | 0.223628 | 0.224068 | 0.223962 | 0.224481 | ||

2.80 | 2.60 | 2.40 | 0.223403 | 0.223775 | 0.223653 | 0.224086 | 0.223998 | 0.224514 |

2.45 | 0.223374 | 0.223750 | 0.223622 | 0.224057 | 0.223962 | 0.224479 | ||

2.50 | 0.223347 | 0.223726 | 0.223592 | 0.224030 | 0.223927 | 0.224445 | ||

2.65 | 2.40 | 0.223379 | 0.223754 | 0.223630 | 0.224065 | 0.223975 | 0.224493 | |

2.45 | 0.223351 | 0.223729 | 0.223598 | 0.224037 | 0.223939 | 0.224458 | ||

2.50 | 0.223324 | 0.223705 | 0.223569 | 0.224009 | 0.223904 | 0.224425 | ||

2.70 | 2.40 | 0.223357 | 0.223734 | 0.223608 | 0.224046 | 0.223953 | 0.224474 | |

2.45 | 0.223329 | 0.223709 | 0.223576 | 0.224017 | 0.223917 | 0.224439 | ||

2.50 | 0.223302 | 0.223685 | 0.223547 | 0.223989 | 0.223883 | 0.224406 | ||

2.85 | 2.60 | 2.40 | 0.223327 | 0.223701 | 0.223577 | 0.224013 | 0.223924 | 0.224443 |

2.45 | 0.223297 | 0.223674 | 0.223545 | 0.223983 | 0.223887 | 0.224407 | ||

2.50 | 0.223269 | 0.223649 | 0.223514 | 0.223954 | 0.223851 | 0.224373 | ||

2.65 | 2.40 | 0.223302 | 0.223679 | 0.223553 | 0.223991 | 0.223900 | 0.224422 | |

2.45 | 0.223273 | 0.223653 | 0.223521 | 0.223961 | 0.223863 | 0.224386 | ||

2.50 | 0.223245 | 0.223628 | 0.223490 | 0.223933 | 0.223827 | 0.224351 | ||

2.70 | 2.40 | 0.223279 | 0.223658 | 0.223530 | 0.223970 | 0.223878 | 0.224402 | |

2.45 | 0.223250 | 0.223632 | 0.223498 | 0.223941 | 0.223840 | 0.224366 | ||

2.50 | 0.223222 | 0.223607 | 0.223467 | 0.223912 | 0.223804 | 0.224331 |

**Table 7.**System performances with respect to r, ${\mu}_{H}$, ${\mu}_{{L}_{1}}$, and ${\lambda}_{H}$.

r | ${\mathit{\mu}}_{\mathit{H}}$ | ${\mathit{\mu}}_{{\mathit{L}}_{1}}$ | ${\mathit{E}}_{\mathbf{HPIL}}$ | ${\mathit{E}}_{\mathbf{SB}}$ | ${\mathit{E}}_{\mathbf{SRR}}$ | |||
---|---|---|---|---|---|---|---|---|

${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | |||

0.5 | 2.75 | 2.40 | 0.001919 | 0.002247 | 0.971408 | 0.972786 | 0.015359 | 0.014580 |

2.45 | 0.001921 | 0.002249 | 0.971198 | 0.972595 | 0.015423 | 0.014632 | ||

2.50 | 0.001923 | 0.002251 | 0.970997 | 0.972412 | 0.015484 | 0.014682 | ||

2.80 | 2.40 | 0.001921 | 0.002251 | 0.970795 | 0.972169 | 0.015546 | 0.014750 | |

2.45 | 0.001923 | 0.002253 | 0.970575 | 0.971969 | 0.015615 | 0.014806 | ||

2.50 | 0.001925 | 0.002255 | 0.970364 | 0.971778 | 0.015679 | 0.014859 | ||

2.85 | 2.40 | 0.001922 | 0.002255 | 0.970186 | 0.971555 | 0.015732 | 0.015732 | |

2.45 | 0.001923 | 0.002257 | 0.969956 | 0.971346 | 0.015804 | 0.015804 | ||

2.50 | 0.001925 | 0.002259 | 0.969736 | 0.971146 | 0.015873 | 0.015873 | ||

1 | 2.75 | 2.40 | 0.003858 | 0.004527 | 0.970324 | 0.971718 | 0.015939 | 0.015181 |

2.45 | 0.003863 | 0.004532 | 0.970121 | 0.971534 | 0.015995 | 0.015224 | ||

2.50 | 0.003868 | 0.004536 | 0.969926 | 0.971358 | 0.016048 | 0.015266 | ||

2.80 | 2.40 | 0.003860 | 0.004536 | 0.969673 | 0.971062 | 0.016135 | 0.015358 | |

2.45 | 0.003865 | 0.004540 | 0.969460 | 0.970869 | 0.016196 | 0.015406 | ||

2.50 | 0.003870 | 0.004544 | 0.969255 | 0.970684 | 0.016252 | 0.015451 | ||

2.85 | 2.40 | 0.003860 | 0.004543 | 0.969027 | 0.970408 | 0.016329 | 0.015536 | |

2.45 | 0.003865 | 0.004547 | 0.968804 | 0.970206 | 0.016394 | 0.015588 | ||

2.50 | 0.003870 | 0.004551 | 0.968589 | 0.970012 | 0.016454 | 0.015636 |

N | $\mathit{\gamma}$ | ${\mathit{\mu}}_{\mathit{H}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ | |||
---|---|---|---|---|---|---|---|---|

${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | |||

7 | 0.05 | 2.75 | 1.166812 | 1.294542 | 1.133508 | 1.259561 | 1.092839 | 1.216619 |

2.80 | 1.155714 | 1.281839 | 1.122121 | 1.246556 | 1.081005 | 1.203148 | ||

2.85 | 1.145370 | 1.269916 | 1.111490 | 1.234334 | 1.069927 | 1.190461 | ||

0.1 | 2.75 | 1.176179 | 1.304286 | 1.138123 | 1.264415 | 1.092326 | 1.216121 | |

2.80 | 1.165538 | 1.292048 | 1.127166 | 1.251849 | 1.080899 | 1.203065 | ||

2.85 | 1.155644 | 1.280587 | 1.116958 | 1.240061 | 1.070223 | 1.190789 | ||

0.15 | 2.75 | 1.185531 | 1.314007 | 1.143894 | 1.270412 | 1.094334 | 1.218132 | |

2.80 | 1.175302 | 1.302194 | 1.133339 | 1.258259 | 1.083308 | 1.205488 | ||

2.85 | 1.165815 | 1.291153 | 1.123528 | 1.246881 | 1.073028 | 1.193620 | ||

8 | 0.05 | 2.75 | 1.196611 | 1.333320 | 1.162571 | 1.297436 | 1.120983 | 1.253362 |

2.80 | 1.184390 | 1.319239 | 1.150062 | 1.283055 | 1.108028 | 1.238515 | ||

2.85 | 1.173006 | 1.306032 | 1.138391 | 1.269550 | 1.095910 | 1.224544 | ||

0.1 | 2.75 | 1.207098 | 1.344284 | 1.168216 | 1.303401 | 1.121370 | 1.253814 | |

2.80 | 1.195381 | 1.330726 | 1.156191 | 1.289522 | 1.108882 | 1.239453 | ||

2.85 | 1.184499 | 1.318037 | 1.145001 | 1.276513 | 1.097230 | 1.225963 | ||

0.15 | 2.75 | 1.217523 | 1.355182 | 1.174969 | 1.310463 | 1.124237 | 1.256745 | |

2.80 | 1.206273 | 1.342103 | 1.163403 | 1.297056 | 1.112209 | 1.242855 | ||

2.85 | 1.195845 | 1.329887 | 1.152660 | 1.284515 | 1.101009 | 1.229833 | ||

9 | 0.05 | 2.75 | 1.217496 | 1.361326 | 1.182931 | 1.324789 | 1.140678 | 1.279882 |

2.80 | 1.204370 | 1.346108 | 1.169522 | 1.309276 | 1.126827 | 1.263907 | ||

2.85 | 1.192158 | 1.331848 | 1.157027 | 1.294722 | 1.113888 | 1.248891 | ||

0.1 | 2.75 | 1.228956 | 1.373376 | 1.189474 | 1.331744 | 1.141857 | 1.281196 | |

2.80 | 1.216383 | 1.358733 | 1.176600 | 1.316790 | 1.128530 | 1.265767 | ||

2.85 | 1.204714 | 1.345043 | 1.164633 | 1.302789 | 1.116108 | 1.251293 | ||

0.15 | 2.75 | 1.240316 | 1.385311 | 1.197085 | 1.339751 | 1.145488 | 1.284957 | |

2.80 | 1.228252 | 1.371198 | 1.184714 | 1.325320 | 1.132663 | 1.270051 | ||

2.85 | 1.217078 | 1.358028 | 1.173237 | 1.311835 | 1.120736 | 1.256093 |

${\mathit{\mu}}_{\mathit{H}}$ | ${\mathit{\mu}}_{{\mathit{L}}_{2}}$ | ${\mathit{\mu}}_{{\mathit{L}}_{1}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ | |||
---|---|---|---|---|---|---|---|---|

${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{85}$ | ${\mathbf{\lambda}}_{\mathbf{H}}=\mathbf{0}.\mathbf{95}$ | |||

2.75 | 2.60 | 2.40 | 0.168374 | 0.205167 | 0.172715 | 0.210847 | 0.177406 | 0.217027 |

2.45 | 0.166878 | 0.203129 | 0.171119 | 0.208675 | 0.175696 | 0.214703 | ||

2.50 | 0.165467 | 0.201211 | 0.169613 | 0.206630 | 0.174082 | 0.212513 | ||

2.65 | 2.40 | 0.167100 | 0.203432 | 0.171412 | 0.209068 | 0.176071 | 0.215202 | |

2.45 | 0.165626 | 0.201427 | 0.169838 | 0.206931 | 0.174385 | 0.212914 | ||

2.50 | 0.164236 | 0.199539 | 0.168354 | 0.204918 | 0.172793 | 0.210758 | ||

2.70 | 2.40 | 0.165892 | 0.201789 | 0.170176 | 0.207385 | 0.174805 | 0.213474 | |

2.45 | 0.164439 | 0.199815 | 0.168624 | 0.205280 | 0.173141 | 0.211219 | ||

2.50 | 0.163068 | 0.197957 | 0.167160 | 0.203297 | 0.171571 | 0.209095 | ||

2.80 | 2.60 | 2.40 | 0.162540 | 0.197251 | 0.166733 | 0.202716 | 0.171263 | 0.208663 |

2.45 | 0.161115 | 0.195324 | 0.165212 | 0.200662 | 0.169633 | 0.206463 | ||

2.50 | 0.159772 | 0.193510 | 0.163777 | 0.198726 | 0.168094 | 0.204388 | ||

2.65 | 2.40 | 0.161327 | 0.195611 | 0.165492 | 0.201035 | 0.169992 | 0.206938 | |

2.45 | 0.159923 | 0.193714 | 0.163992 | 0.199012 | 0.168384 | 0.204770 | ||

2.50 | 0.158599 | 0.191929 | 0.162577 | 0.197107 | 0.166866 | 0.202727 | ||

2.70 | 2.40 | 0.160177 | 0.194057 | 0.164315 | 0.199443 | 0.168786 | 0.205303 | |

2.45 | 0.158792 | 0.192190 | 0.162836 | 0.197450 | 0.167199 | 0.203167 | ||

2.50 | 0.157487 | 0.190431 | 0.161440 | 0.195573 | 0.165701 | 0.201154 | ||

2.85 | 2.60 | 2.40 | 0.157094 | 0.189915 | 0.161148 | 0.195181 | 0.165528 | 0.200911 |

2.45 | 0.155734 | 0.188089 | 0.159696 | 0.193232 | 0.163970 | 0.198822 | ||

2.50 | 0.154452 | 0.186369 | 0.158325 | 0.191396 | 0.162500 | 0.196853 | ||

2.65 | 2.40 | 0.155937 | 0.188360 | 0.159964 | 0.193587 | 0.164315 | 0.199275 | |

2.45 | 0.154597 | 0.186563 | 0.158532 | 0.191669 | 0.162779 | 0.197218 | ||

2.50 | 0.153333 | 0.184870 | 0.157181 | 0.189861 | 0.161329 | 0.195278 | ||

2.70 | 2.40 | 0.154839 | 0.186888 | 0.158841 | 0.192078 | 0.163164 | 0.197726 | |

2.45 | 0.153518 | 0.185118 | 0.157428 | 0.190188 | 0.161648 | 0.195698 | ||

2.50 | 0.152272 | 0.183450 | 0.156095 | 0.188406 | 0.160217 | 0.193786 |

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**MDPI and ACS Style**

Jeganathan, K.; Vidhya, S.; Hemavathy, R.; Anbazhagan, N.; Joshi, G.P.; Kang, C.; Seo, C.
Analysis of *M*/*M*/1/*N* Stochastic Queueing—Inventory System with Discretionary Priority Service and Retrial Facility. *Sustainability* **2022**, *14*, 6370.
https://doi.org/10.3390/su14106370

**AMA Style**

Jeganathan K, Vidhya S, Hemavathy R, Anbazhagan N, Joshi GP, Kang C, Seo C.
Analysis of *M*/*M*/1/*N* Stochastic Queueing—Inventory System with Discretionary Priority Service and Retrial Facility. *Sustainability*. 2022; 14(10):6370.
https://doi.org/10.3390/su14106370

**Chicago/Turabian Style**

Jeganathan, K., S. Vidhya, R. Hemavathy, N. Anbazhagan, Gyanendra Prasad Joshi, Chanku Kang, and Changho Seo.
2022. "Analysis of *M*/*M*/1/*N* Stochastic Queueing—Inventory System with Discretionary Priority Service and Retrial Facility" *Sustainability* 14, no. 10: 6370.
https://doi.org/10.3390/su14106370