# On Increasing the Accuracy of Modeling Multi-Service Overflow Systems with Erlang-Engset-Pascal Streams

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Scheme of Traffic Overflow under Consideration

## 4. Model of Primary Resources

- $i,j,l$
- indexes for call classes of Erlang, Engset, and Pascal traffic, respectively,
- ${\alpha}_{j,s}^{\mathrm{En}}$
- average traffic intensity of class j for Engset traffic generated by one free source in the resource s,
- ${\alpha}_{l,s}^{\mathrm{Pa}}$
- average traffic intensity of class l for Pascal traffic generated by one free source,
- ${S}_{j,s}^{\mathrm{En}}$
- the number of traffic sources of class j of the Engset type, related to the primary resource s,
- ${S}_{l,s}^{\mathrm{Pa}}$
- the number of traffic sources of class l of the Pascal type, related to the primary resource s,
- ${n}_{j,s}^{\mathrm{En}}$
- the number of traffic sources of class j of the Engset type, serviced in the occupancy state n AU of the primary resource s,
- ${n}_{l,s}^{\mathrm{Pa}}$
- the number of traffic sources of class l of the Pascal type, serviced in the occupancy state n AU of the primary resources s.

## 5. Model of Traffic That Overflows from Primary Resources

#### 5.1. Methods for Determining the Parameters of Overflow Traffic

- –
- Erlang traffic:$${A}_{i,s}^{\mathrm{Er}},\phantom{\rule{0.277778em}{0ex}}{\left[{\sigma}_{i,\Delta s}^{2}\right]}^{\mathrm{Er}}={A}_{i,s}^{\mathrm{Er}},$$
- –
- Engset traffic:$${A}_{j,s}^{\mathrm{En}}={S}_{j,s}^{\mathrm{En}}\frac{{\alpha}_{j,s}^{\mathrm{En}}}{1+{\alpha}_{j,s}^{\mathrm{En}}},\phantom{\rule{0.277778em}{0ex}}{\left[{\sigma}_{j,\Delta s}^{2}\right]}^{\mathrm{En}}={S}_{j,s}^{\mathrm{En}}\frac{{\alpha}_{j,s}^{\mathrm{En}}}{{\left(1+{\alpha}_{j,s}^{\mathrm{En}}\right)}^{2}},$$
- –
- Pascal traffic:$${A}_{l,s}^{\mathrm{Pa}}={S}_{l,s}^{\mathrm{Pa}}\frac{{\alpha}_{l,s}^{\mathrm{Pa}}}{1-{\alpha}_{l,s}^{\mathrm{Pa}}},\phantom{\rule{0.277778em}{0ex}}{\left[{\sigma}_{l,\Delta s}^{2}\right]}^{\mathrm{Pa}}={S}_{l,s}^{\mathrm{Pa}}\frac{{\alpha}_{l,s}^{\mathrm{Pa}}}{{\left(1-{\alpha}_{l,s}^{\mathrm{Pa}}\right)}^{2}}.$$

#### 5.2. Accuracy Analysis of Glabowski-Kmiecik-Stasiak 2018 Method for Determining the Parameters of Overflow Traffic

#### 5.3. New Method for Determining the Parameters of Multi-Service Overflow Systems

## 6. Model of Secondary Resources

- Determination of the occupancy distribution ${\left[{P}_{n}\right]}_{{V}^{s}}$ in the primary resource s, where $1\le s\le r$—Formula (1).
- Determination of the blocking probability ${\left[{E}_{c,s}^{\mathrm{X}}\right]}_{{V}_{s}}$ for traffic streams of all classes in the primary resource s, where $1\le s\le r$, $1\le c\le {m}_{s}$—Formula (4).
- Determination of the capacity ${v}_{c,s}^{\mathrm{Er}}$ for each fictitious primary resource ${s}_{c}^{\mathrm{Er}}$—Formula (13).
- Determination of the capacity ${v}_{c,s}^{\mathrm{En}}$ for each fictitious primary resource ${s}_{c}^{\mathrm{En}}$—Formula (14).
- Determination of the variance ${\left[{\sigma}_{c,\Delta s}^{2}\right]}^{\mathrm{Er}}$ of Erlang traffic offered to the primary resource—Formula (16).
- Determination of the variance ${\left[{\sigma}_{c,\Delta s}^{2}\right]}^{\mathrm{En}}$ of Engset traffic offered to the primary resource—Formula (17).
- Determination of the aggregated peakedness coefficient Z—Formula (31).
- Determination of the occupancy distribution ${\left[{P}_{n}\right]}_{{V}_{0}/Z}$ in the system of secondary resources—Formula (30).
- Determination of the blocking probability ${\left[{E}_{c}\right]}_{{V}_{0}}$ for traffic streams of the classes offered to the system of secondary resources—Formula (32).

## 7. Results of Modeling a Selected Number of Overflow System with Erlang–Engset–Pascal Traffic

`Microsoft.NET`programming platform was chosen to build the simulator. It allows the application to be executed on all devices with Microsoft operating systems. For the development of the simulator the

`C#`language was chosen because of its efficiency and rich library resources. The program interface was designed in the

`WPF`(Windows Presentation Foundation) structure in the

`XAML`language, and its code was separated from the functional code according to the

`MVVM`(model-view-view-model) architectural pattern. Building our own simulator allows us to examine the operation of the models as accurately as possible and allows for high flexibility of system construction.

- m—the number of classes offered to primary resources,
- ${A}_{c,s}^{\mathrm{X}}$—the average intensity of traffic of class c$(c\in m)$ of type X $(\mathrm{X}\in \mathrm{Er},\mathrm{En},\mathrm{Pa})$ offered to primary resource s,
- ${t}_{c}$—demands of calls of class c expressed in AUs,
- ${V}_{s}$—capacity of primary resource s.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Decomposition of the primary resource s into ${m}_{s}$ fictitious primary resources and ${m}_{s}$ equivalent fictitious primary resources.

**Figure 3.**Relative error of the average value of overflow traffic, System 1, Resource 1 ($\alpha $ values specified directly in the figure).

**Figure 4.**Relative error of the average value of overflow traffic, System 1, Resource 2 ($\alpha $ values specified directly in the figure).

**Figure 5.**Relative error of the average value of overflow traffic, System 2, Resource 1 ($\alpha $ values specified directly in the figure).

**Figure 6.**Relative error of the average value of overflow traffic, System 2, Resource 2 ($\alpha $ values specified directly in the figure).

**Figure 7.**Blocking probability of particular traffic classes in the alternative resources of System 1.

**Figure 8.**Blocking probability of particular traffic classes in the alternative resources of System 2.

**Figure 9.**Blocking probability of particular traffic classes in the alternative resources of System 3.

**Figure 10.**Blocking probability of the Pascal traffic class ${t}_{\mathrm{Pa},4}=2$ in the secondary resources in System 1.

**Figure 11.**Blocking probability of the Erlang traffic class ${t}_{\mathrm{Er},1}=7$ in the secondary resources in System 2.

**Figure 12.**Blocking probability of the Pascal traffic class ${t}_{\mathrm{Pa},3}=2$ in the secondary resources in System 2.

**Figure 13.**Blocking probability of the Pascal traffic class ${t}_{\mathrm{Pa},5}=5$ in the secondary resources in System 3.

**Figure 14.**Blocking probability of the Engset traffic class ${t}_{\mathrm{En},2}=2$ in the secondary resources in System 3.

**Figure 15.**Loss probability (call congestion) of three selected traffic classes in the primary resources of System 1.

**Figure 16.**Loss probability (call congestion) of three selected traffic classes in the primary resources of System 2.

**Figure 17.**Loss probability (call congestion) of three selected traffic classes in the primary resources of System 3.

System | Resource | Offered Traffic | Number of Traffic Sources |
---|---|---|---|

1 | ${V}_{1}$ = 120 AUs | ${t}_{\mathrm{Pa},1}$ = 3 AUs | ${S}_{\mathrm{Pa},1}$ = 40 |

${t}_{\mathrm{Pa},2}$ = 4 AUs | ${S}_{\mathrm{Pa},2}$ = 30 | ||

${V}_{2}$ = 90 AUs | ${t}_{\mathrm{Pa},3}$ = 1 AU | ${S}_{\mathrm{Pa},3}$ = 50 | |

${t}_{\mathrm{Pa},4}$ = 2 AUs | ${S}_{\mathrm{Pa},4}$ = 30 | ||

${t}_{\mathrm{Pa},1}$ = 3 AUs | ${S}_{\mathrm{Pa},1}$ = 70 | ||

${V}_{0}$ = 50 AUs | |||

2 | ${V}_{1}$ = 70 AUs | ${t}_{\mathrm{Er},1}$ = 7 AUs | |

${t}_{\mathrm{En},2}$ = 5 AUs | ${S}_{\mathrm{En},2}$ = 50 | ||

${t}_{\mathrm{Pa},3}$ = 2 AUs | ${S}_{\mathrm{Pa},3}$ = 25 | ||

${V}_{2}$ = 100 AUs | ${t}_{\mathrm{Pa},4}$ = 4 AUs | ${S}_{\mathrm{Pa},4}$ = 30 | |

${t}_{\mathrm{Pa},5}$ = 5 AUs | ${S}_{\mathrm{Pa},5}$ = 40 | ||

${V}_{0}$ = 60 AUs |

System | Resource | Offered Traffic | Number of Traffic Sources |
---|---|---|---|

3 | ${V}_{1}$ = 90 AUs | ${t}_{\mathrm{Er},1}$ = 5 AUs | |

${t}_{\mathrm{En},2}$ = 2 AUs | ${S}_{\mathrm{En},2}$ = 80 | ||

${t}_{\mathrm{Pa},3}$ = 3 AUs | ${S}_{\mathrm{Pa},3}$ = 50 | ||

${V}_{2}$ = 110 AUs | ${t}_{\mathrm{Er},4}$ = 4 AUs | ||

${t}_{\mathrm{Pa},5}$ = 5 AUs | ${S}_{\mathrm{Pa},5}$ = 80 | ||

${V}_{0}$ = 40 AUs |

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Głąbowski, M.; Kmiecik, D.; Stasiak, M. On Increasing the Accuracy of Modeling Multi-Service Overflow Systems with Erlang-Engset-Pascal Streams. *Electronics* **2021**, *10*, 508.
https://doi.org/10.3390/electronics10040508

**AMA Style**

Głąbowski M, Kmiecik D, Stasiak M. On Increasing the Accuracy of Modeling Multi-Service Overflow Systems with Erlang-Engset-Pascal Streams. *Electronics*. 2021; 10(4):508.
https://doi.org/10.3390/electronics10040508

**Chicago/Turabian Style**

Głąbowski, Mariusz, Damian Kmiecik, and Maciej Stasiak. 2021. "On Increasing the Accuracy of Modeling Multi-Service Overflow Systems with Erlang-Engset-Pascal Streams" *Electronics* 10, no. 4: 508.
https://doi.org/10.3390/electronics10040508