# Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Description of the Model

## 3. Equations for Conditional Queue-Size Distribution

## 4. Main Analytical Result

**Theorem**

**1.**

**Proof.**

## 5. Numerical Study

- Poisson arrivals of packets of sizes 100 B;
- CDF of the processing time is a mixture of two exponential distributions (second-order hyper-exponential distribution) and is defined as:$$\begin{array}{cc}\hfill & F\left(t\right)=\alpha \xb7(1-{e}^{-{\mu}_{1}t})+(1-\alpha )\xb7(1-{e}^{-{\mu}_{2}t}),\phantom{\rule{1.em}{0ex}}t>0,\hfill \end{array}$$
- CDF of a single server vacation has 2-Erlang distribution with fixed parameter $\xi ,$ namely:$$\begin{array}{cc}\hfill & G\left(t\right)=1-{e}^{-\xi t}\left(\right)open="("\; close=")">1+\xi t,\phantom{\rule{1.em}{0ex}}t0;\hfill \end{array}$$
- buffer size equals $N=10$ packets (so 1 kB).

#### 5.1. Impact of Service Rate

#### 5.2. Impact of Arrival Rate

#### 5.3. Impact of Initial Buffer State

#### 5.4. Impact of Single Vacation Duration

#### 5.5. Numerical Computations vs. Simulation Results

- The transient probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}$ is calculated for $\lambda =\mu =600$ packets/s (resulting in $\varrho =1$). We take $\xi =1000$ as a parameter of the single vacation duration CDF (corresponding to mean single vacation duration 0.002 s) and the sequence $\left({p}_{k}\right)=\{0.5,0.25,0.25,0,\dots \}$ describing sizes of arriving groups (so the mean group size equals 1.75);
- As in the previous scenario except for the sequence $\left({p}_{k}\right)=\{1,0,0,\dots \}$ (single arrivals).

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Impact of service rate on probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}.$

**Figure 2.**Impact of arrival rate on probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}.$

**Figure 3.**Impact of initial buffer state on probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}$ for different ${n}^{\prime}s.$

**Figure 4.**Impact of single vacation duration on probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}$ for $\lambda =450$ packets/s.

**Figure 5.**Impact of single vacation duration on probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}$ for $\lambda =600$ packets/s.

**Figure 6.**Impact of single vacation duration on probability $\mathbf{P}\left\{X\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}X\left(0\right)=0\}$ for $\lambda =750$ packets/s.

**Figure 7.**Comparison of the $\mathbf{P}\left\{Y\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}Y\left(0\right)=0\}$ probability distribution for the compound Poisson process, obtained: (1) by numerical calculations using the formula (28)—black line on the graph; (2) as a statistical result of a 10,000th random sample—the red squares in the diagram.

**Figure 8.**Comparison of the $\mathbf{P}\left\{Y\right(t)=1\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}Y\left(0\right)=0\}$ probability distribution for the simple Poisson process, obtained: (1) by numerical calculations using the formula (28)—black line on the graph; (2) as a statistical result of a 10,000th random sample—the red squares in the diagram.

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**MDPI and ACS Style**

Kempa, W.M.; Marjasz, R.
Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy. *Entropy* **2021**, *23*, 1410.
https://doi.org/10.3390/e23111410

**AMA Style**

Kempa WM, Marjasz R.
Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy. *Entropy*. 2021; 23(11):1410.
https://doi.org/10.3390/e23111410

**Chicago/Turabian Style**

Kempa, Wojciech M., and Rafał Marjasz.
2021. "Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy" *Entropy* 23, no. 11: 1410.
https://doi.org/10.3390/e23111410