# Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism

## Abstract

**:**

## 1. Introduction

## 2. Model Description

## 3. Transient Equations for the Queue-Size Distribution

- Event no. 1 (${E}_{1}\left(r\right)$): the first job arrives before or at the moment r but after the completion of the vacation;
- Event no. 2 (${E}_{2}\left(r\right)$): the vacation finishes before or at the moment r and during the vacation (maybe at the last time slot of the vacation duration) the first job enters the system;
- Event no. 3 (${E}_{3}\left(r\right)$): the first job arrives before or at the moment r but the vacation finishes after $r;$
- Event no. 4 (${E}_{4}\left(r\right)$): the first arrival occurs after time $r.$

## 4. System of Equations for PGFs

## 5. Representation for Solution

**Theorem**

**1.**

**Corollary**

**1.**

## 6. Numerical Study

- deterministic: ${b}_{2}=1,\phantom{\rule{5.0pt}{0ex}}{b}_{k}=0,\phantom{\rule{5.0pt}{0ex}}k=1,\phantom{\rule{0.166667em}{0ex}}k\ge 3$;
- finite discrete: ${b}_{1}=0.25,\phantom{\rule{5.0pt}{0ex}}{b}_{2}=0.50,\phantom{\rule{5.0pt}{0ex}}{b}_{3}=0.25,\phantom{\rule{5.0pt}{0ex}}{b}_{k}=0,\phantom{\rule{5.0pt}{0ex}}k\ge 4$;
- geometric : ${b}_{k}={\left(0.5\right)}^{k},\phantom{\rule{5.0pt}{0ex}}k\ge 1$.

- deterministic: ${v}_{2}=1,\phantom{\rule{5.0pt}{0ex}}{v}_{k}=0,\phantom{\rule{5.0pt}{0ex}}k=1,\phantom{\rule{0.166667em}{0ex}}k\ge 3$;
- finite discrete: ${v}_{1}=0.25,\phantom{\rule{5.0pt}{0ex}}{v}_{2}=0.50,\phantom{\rule{5.0pt}{0ex}}{v}_{3}=0.25,\phantom{\rule{5.0pt}{0ex}}{v}_{k}=0,\phantom{\rule{5.0pt}{0ex}}k\ge 4$;
- geometric : ${v}_{k}={\left(0.5\right)}^{k},\phantom{\rule{5.0pt}{0ex}}k\ge 1$.

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Parameter a | Deterministic SV | Finite Discrete SV | Geometric SV |
---|---|---|---|

0.1 | 0.3921 | 0.4052 | 0.4197 |

0.2 | 0.5635 | 0.6037 | 0.6464 |

0.3 | 0.7051 | 0.7742 | 0.8438 |

0.4 | 0.8371 | 0.9253 | 1.0077 |

0.5 | 0.9667 | 1.0560 | 1.1318 |

0.6 | 1.0866 | 1.1583 | 1.2126 |

0.7 | 1.1835 | 1.2277 | 1.2575 |

0.8 | 1.2515 | 1.2711 | 1.2829 |

0.9 | 1.2974 | 1.3021 | 1.3047 |

Parameter a | Deterministic SV | Finite Discrete SV | Geometric SV |
---|---|---|---|

0.1 | 0.3902 | 0.4033 | 0.4179 |

0.2 | 0.5609 | 0.6015 | 0.6448 |

0.3 | 0.6988 | 0.7698 | 0.8417 |

0.4 | 0.8236 | 0.9174 | 1.0061 |

0.5 | 0.9448 | 1.0473 | 1.1356 |

0.6 | 1.0619 | 1.1563 | 1.2295 |

0.7 | 1.1679 | 1.2408 | 1.2915 |

0.8 | 1.2558 | 1.3014 | 1.3297 |

0.9 | 1.3231 | 1.3431 | 1.3542 |

Parameter a | Deterministic SV | Finite Discrete SV | Geometric SV |
---|---|---|---|

0.1 | 0.3830 | 0.3961 | 0.4108 |

0.2 | 0.5509 | 0.5923 | 0.6366 |

0.3 | 0.6810 | 0.7555 | 0.8318 |

0.4 | 0.7931 | 0.8965 | 0.9963 |

0.5 | 0.9001 | 1.0229 | 1.1324 |

0.6 | 1.0088 | 1.1371 | 1.2414 |

0.7 | 1.1206 | 1.2381 | 1.3242 |

0.8 | 1.2225 | 1.3222 | 1.3810 |

0.9 | 1.3365 | 1.3845 | 1.4124 |

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Kempa, W.M. Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism. *Symmetry* **2022**, *14*, 2350.
https://doi.org/10.3390/sym14112350

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Kempa WM. Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism. *Symmetry*. 2022; 14(11):2350.
https://doi.org/10.3390/sym14112350

**Chicago/Turabian Style**

Kempa, Wojciech M. 2022. "Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism" *Symmetry* 14, no. 11: 2350.
https://doi.org/10.3390/sym14112350