Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks
Abstract
:1. Introduction
2. Model Description
3. Basic Equations for First Buffer Overflow Duration
- ${\mathsf{\Lambda}}_{1}(n):$ the moment r is the arrival time of, at most, the $(N-n-1)$th packet and the next packet enters the system after time i (the buffer does not become saturated before time i);
- ${\mathsf{\Lambda}}_{2}(n):$ the moment r is the arrival time of, at most, the $(N-n-1)$th packet and the next packet enters the system exactly at time i;
- ${\mathsf{\Lambda}}_{3}(n):$ at time r the $(N-n)$th packet arrives, so the buffer overflow period begins at time $r;$
- ${\mathsf{\Lambda}}_{4}(n):$ the first packet (after the opening of the system) arrives exactly at time i;
- ${\mathsf{\Lambda}}_{5}(n):$ the first packet (after the opening of the system) arrives after time $i.$
4. Representation for Solution
5. The Case of Next Buffer Overflows
6. Numerical Study
- -
- the offered traffic load $\varrho $ defined as the quotient of the mean service time and the mean interarrival time;
- -
- the number of jobs n accumulated in the buffer before the starting moment;
- -
- the shape of the service (processing) time distribution;
- -
- the buffer size.
- geometric with fixed parameter $b;$
- deterministic (constant) of duration $B=\mathrm{const};$
- bounded discrete distribution, where the service time takes on finite number of possible values; dealing with the impact of the distribution skewness we analyze separately the following subcases of this type of distribution:
- –
- symmetric;
- –
- with positive skewness (positive asymmetry);
- –
- with negative skewness (negative asymmetry).
6.1. Impact of the Type of Processing Distribution
6.2. Impact of Skewness Type of the Processing Distribution
- symmetric distribution of the form$$\begin{array}{c}{b}_{1}={b}_{2}={b}_{4}={b}_{5}=\frac{1}{8},\phantom{\rule{1.em}{0ex}}{b}_{3}=\frac{1}{2}\hfill \end{array}$$
- distribution with positive skewness (positive asymmetry) of the form$$\begin{array}{c}{b}_{2}=\frac{7}{16},\phantom{\rule{1.em}{0ex}}{b}_{3}=\frac{4}{16},\phantom{\rule{1.em}{0ex}}{b}_{4}=\frac{3}{16},\phantom{\rule{1.em}{0ex}}{b}_{5}=\frac{2}{16}\hfill \end{array}$$
- distribution with negative skewness (negative asymmetry) of the form$$\begin{array}{c}{b}_{1}=\frac{2}{16},\phantom{\rule{1.em}{0ex}}{b}_{2}=\frac{3}{16},\phantom{\rule{1.em}{0ex}}{b}_{3}=\frac{4}{16},\phantom{\rule{1.em}{0ex}}{b}_{4}=\frac{7}{16}\hfill \end{array}$$
6.3. Mean Buffer Overflow Duration in Dependence on Offered Load and Initial Buffer State
6.4. Impact of System Size
7. Conclusions
Funding
Conflicts of Interest
References
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k | Symmetry | Negative Skewness | Positive Skewness |
---|---|---|---|
1 | 0.644089 | 0.409344 | 0.418129 |
2 | 0.361425 | 0.107254 | 0.142688 |
3 | 0.166121 | 4.163336 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 0.030098 |
4 | 0.051831 | 6.938894 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ | 0 |
5 | 0.013312 | 5.551115 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 1.110223 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ |
6 | 3.700743 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 9.251859 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 2.312965 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ |
Buffer State n | $\mathit{\varrho}=0.75$ | $\mathit{\varrho}=1.00$ | $\mathit{\varrho}=1.25$ |
---|---|---|---|
0 | 0.036321 | 0.049755 | 0.038306 |
1 | 0.036321 | 0.049755 | 0.038306 |
2 | 0.037954 | 0.055577 | 0.045982 |
3 | 0.042496 | 0.069353 | 0.062898 |
4 | 0.052282 | 0.094659 | 0.092465 |
5 | 0.071575 | 0.137233 | 0.140520 |
6 | 0.108269 | 0.206258 | 0.216532 |
7 | 0.177884 | 0.318161 | 0.337882 |
8 | 0.313364 | 0.508249 | 0.544020 |
9 | 0.585439 | 0.855341 | 0.925785 |
10 | 1.134407 | 1.511127 | 1.654884 |
System Size N | $\mathit{\varrho}=0.75$ | $\mathit{\varrho}=1.00$ | $\mathit{\varrho}=1.25$ |
---|---|---|---|
2 | 1.682303 | 2.000242 | 2.088127 |
3 | 1.535842 | 1.762685 | 1.818474 |
4 | 1.418714 | 1.639140 | 1.716005 |
5 | 1.327611 | 1.577548 | 1.678921 |
6 | 1.257000 | 1.545704 | 1.664732 |
7 | 1.205492 | 1.528632 | 1.658915 |
8 | 1.171707 | 1.519536 | 1.656483 |
9 | 1.151613 | 1.514796 | 1.655473 |
10 | 1.140419 | 1.512365 | 1.655056 |
11 | 1.134407 | 1.511127 | 1.654884 |
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Kempa, W.M. Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks. Sensors 2020, 20, 5772. https://doi.org/10.3390/s20205772
Kempa WM. Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks. Sensors. 2020; 20(20):5772. https://doi.org/10.3390/s20205772
Chicago/Turabian StyleKempa, Wojciech M. 2020. "Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks" Sensors 20, no. 20: 5772. https://doi.org/10.3390/s20205772