# Analytical Model of the Connection Handoff in 5G Mobile Networks with Call Admission Control Mechanisms

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## Abstract

**:**

## 1. Introduction

- Firstly, this article presents a generalized model of the limited-availability group that can be used to determine the blocking probability for individual classes of requests offered in 5G systems without CAC mechanisms introduced.
- Secondly, this article describes a model of a limited-availability group with resource reservation mechanisms for blocking probability calculations in 5G systems with reservation mechanisms.
- Then, this article proposes a model of a limited-availability group with threshold mechanisms used for blocking probability calculations in 5G systems.
- Finally, this article presents the algorithm for a blocking probability calculation in the group of cells in 5G systems with CAC mechanisms.

## 2. Limited-Availability Group

#### 2.1. Generalized Model of the Limited-Availability Group with Erlang Traffic Streams

#### 2.2. Structure of Offered Traffic

#### 2.3. Blocking Probability Calculations

- Determination of values of offered traffic ${A}_{s,c}$ based on (3).
- Calculation of the values of the conditional passing coefficients based on (6).
- Determination of state probabilities ${\left[{P}_{n}\right]}_{V}$ using (4).
- Determination of blocking probabilities ${E}_{c}$ for calls to particular traffic classes using (10).

## 3. Limited-Availability Group with CAC Mechanisms

#### 3.1. Resource Reservation Mechanism

- Determination of values of the offered traffic ${A}_{s,c}$ according to (3).
- Calculation of the values of the total conditional passing coefficients based on (15).
- Determination of state probabilities ${\left[{P}_{n}\right]}_{V}$ on the basis of the modified Kaufman–Roberts recursion (16).
- Determination of the blocking probabilities ${E}_{c}$ for calls of particular traffic classes using (17).

#### 3.2. Threshold Mechanism

- Calculation of the values of the total conditional passing coefficients for threshold area u based on (21).
- Determination of state probabilities ${\left[{P}_{n}\right]}_{V}$ on the basis of the modified Kaufman–Roberts recursion (22).
- Determination of the blocking probabilities ${E}_{c}$ for particular traffic class calls using (23).

## 4. Traffic Flows Optimization in 5G Networks

- cell 1 and its neighboring cells: 2, 3, 4, 5, 6, 7;
- cell 2 and its neighboring cells: 3, 1, 7;
- cell 3 and its neighboring cells: 4, 1, 2;
- cell 4 and its neighboring cells: 5, 1, 3;
- cell 5 and its neighboring cells: 6, 1, 4;
- cell 6 and its neighboring cells: 7, 1, 5;
- cell 7 and its neighboring cells: 2, 1, 6;

- In the case of a system without CAC mechanisms or with a reservation mechanism:$${A}_{s,c}^{g}=\frac{{\eta}_{s,c}{\lambda}_{s}^{g}}{{\mu}_{c}},$$
- For a system with the following threshold mechanism:$${A}_{s,c,u}^{g}=\frac{{\eta}_{s,c}{\lambda}_{s}^{g}}{{\mu}_{c,u}},$$

- In the case of a system without CAC mechanisms:$$n{\left[{P}_{n}\right]}_{{V}_{g}}=\sum _{s=1}^{S}\sum _{c=1}^{m}{A}_{s,c}^{g}{\sigma}_{c}^{g}(n-{t}_{i}){t}_{i}{\left[{P}_{n-{t}_{i}}\right]}_{{V}_{g}},$$
- In the case of a system with reservation mechanisms:$$n{\left[{P}_{n}\right]}_{{V}_{g}}=\sum _{s=1}^{S}\sum _{c=1}^{m}{A}_{s,c}^{g}{\sigma}_{c,\mathrm{Tot}}^{g}(n-{t}_{i}){t}_{i}{\left[{P}_{n-{t}_{i}}\right]}_{{V}_{g}},$$
- In the case of a system with threshold mechanisms:$$n{\left[{P}_{n}\right]}_{{V}_{g}}=\sum _{s=1}^{S}\sum _{c=1}^{m}\sum _{u=0}^{{p}_{c}}{A}_{s,c,u}^{g}{\sigma}_{c,u,\mathrm{Tot}}^{g}(n-{t}_{i}){t}_{i}{\left[{P}_{n-{t}_{i}}\right]}_{{V}_{g}}.$$

- In the case of a system without CAC mechanisms:$${E}_{c}^{g}=\sum _{n={V}_{g}-{\sum}_{s=1}^{q}{k}_{s}({t}_{c}-1)}^{{V}_{g}}{\left[{P}_{n}\right]}_{{V}_{g}}[1-{\sigma}_{c}^{q}\left(n\right)].$$
- In the case of a system with reservation mechanisms:$${E}_{c}^{g}=\sum _{n={V}_{g}-{\sum}_{s=1}^{q}{k}_{s}({t}_{c}-1)}^{{V}_{g}}{\left[{P}_{n}\right]}_{{V}_{g}}[1-{\sigma}_{c,\mathrm{Tot}}^{g}\left(n\right)].$$
- In the case of a system with threshold mechanisms:$${E}_{c}^{g}=\sum _{n={V}_{g}-{t}_{c,{p}_{c}}+1}^{{V}_{g}}{\left[{P}_{n}\right]}_{{V}_{g}}(1-{\sigma}_{c}^{g}\left(n\right)).$$

- Method 1:$${E}_{c}=\sqrt[G]{{E}_{c}^{1}\xb7{E}_{c}^{2}\xb7\xb7\xb7{E}_{c}^{G}},$$
- method 2:$${E}_{c}=\sum _{g=1}^{G}{E}_{c}^{g}\xb7{w}_{c}^{g},$$$${w}_{c}^{g}=\frac{{\sum}_{s=1}^{S}{A}_{s,c}^{g}}{{\sum}_{s=1}^{S}{\sum}_{c=1}^{m}{\sum}_{g=1}^{G}{A}_{s,c}^{g}}.$$

- Setting the number of assemblies to $g=1$.
- Increasing the number of assemblies to $g=g+1$.
- Checking the assembly number. If $g\le 7$, go to Step 2.

## 5. Numerical Examples

- Group 1:
- −
- Capacity of particular cells expressed in BBUs: ${f}_{1}=30$, ${f}_{2}=35$, ${f}_{3}=45$, ${f}_{4}=35$, ${f}_{5}=45$, ${f}_{6}=35$, ${f}_{7}=45$;
- −
- Traffic classes: $m=3$, ${t}_{1}=1$ BBU, ${\mu}_{1}^{-1}=1$, ${t}_{2}=4$ BBUs, ${\mu}_{2}^{-1}=1$, ${t}_{3}=8$ BBUs, ${\mu}_{3}^{-1}=1$;
- −
- Sets of traffic sources: $S=2$, ${\mathbb{C}}_{1}=\{1,2,3\}$, ${\eta}_{1,1}=0.4$, ${\eta}_{1,2}=0.3$, ${\eta}_{1,3}=0.3$, ${\mathbb{C}}_{2}=\{1,3\}$, ${\eta}_{2,1}=0.5$, ${\eta}_{2,3}=0.5$;
- −
- Reservation mechanism: $\mathbb{R}=\{1,2\}$, ${Q}_{1}={Q}_{2}=75\%$ (of total system capacity);
- −
- Threshold mechanism: $\mathbb{T}=\left\{3\right\}$, ${p}_{3}=1$, ${Q}_{3,1}=75\%$ (of total system capacity), ${t}_{3,0}={t}_{3}$, ${\mu}_{3,0}^{-1}={\mu}_{3}^{-1}$, ${t}_{3,1}=6$ BBUs, ${\mu}_{3,1}^{-1}=1.33$.

- Group 2:
- −
- Capacity of particular cells expressed in BBUs: ${f}_{1}=55$, ${f}_{2}=45$, ${f}_{3}=55$, ${f}_{4}=45$, ${f}_{5}=55$, ${f}_{6}=45$, ${f}_{7}=55$.
- −
- Traffic classes: $m=4$, ${t}_{1}=1$ BBU, ${\mu}_{1}^{-1}=1$, ${t}_{2}=4$ BBUs, ${\mu}_{2}^{-1}=1$, ${t}_{3}=8$ BBUs, ${\mu}_{3}^{-1}=1$, ${t}_{4}=11$ BBUs, ${\mu}_{4}^{-1}=1$;
- −
- Sets of traffic sources: $S=2$, ${\mathbb{C}}_{1}=\{1,2,3\}$, ${\eta}_{1,1}=0.4$, ${\eta}_{1,2}=0.3$, ${\eta}_{1,3}=0.3$, ${\mathbb{C}}_{2}=\{1,3,4\}$, ${\eta}_{2,1}=0.5$, ${\eta}_{2,3}=0.2$, ${\eta}_{2,4}=0.3$;
- −
- Reservation mechanism: $\mathbb{R}=\{1,2,3\}$, ${Q}_{1}={Q}_{2}={Q}_{3}=75\%$ (of total system capacity);
- −
- Threshold mechanism: $\mathbb{T}=\{3,4\}$, ${p}_{3}=1$, ${p}_{4}=2$, ${Q}_{3,1}={Q}_{4,1}=75\%$, ${Q}_{4,2}=85\%$ (of total system capacity), ${t}_{3,0}={t}_{3}$, ${\mu}_{3,0}^{-1}={\mu}_{3}^{-1}$, ${t}_{3,1}=6$ BBUs, ${\mu}_{3,1}^{-1}=1.33$, ${t}_{4,0}={t}_{4}$, ${\mu}_{4,0}^{-1}={\mu}_{4}^{-1}$, ${t}_{4,1}=9$ BBUs, ${\mu}_{4,1}^{-1}=1$, ${t}_{4,2}=7$ BBUs, ${\mu}_{4,2}^{-1}=1$.

- Group 3:
- −
- Capacity of particular cells expressed in BBUs: ${f}_{1}=80$, ${f}_{2}=100$, ${f}_{3}=80$, ${f}_{4}=100$, ${f}_{5}=80$, ${f}_{6}=100$, ${f}_{7}=80$.
- −
- Traffic classes: $m=3$, ${t}_{1}=1$ BBU, ${\mu}_{1}^{-1}=1$, ${t}_{2}=7$ BBUs, ${\mu}_{2}^{-1}=1$, ${t}_{3}=14$ BBUs, ${\mu}_{3}^{-1}=1$;
- −
- Sets of traffic sources: $S=2$, ${\mathbb{C}}_{1}=\{1,2\}$, ${\eta}_{1,1}=0.6$, ${\eta}_{1,2}=0.4$, ${\mathbb{C}}_{2}=\{2,3\}$, ${\eta}_{2,2}=0.5$, ${\eta}_{2,3}=0.5$;
- −
- Reservation mechanism: $\mathbb{R}=\{1,2\}$, ${Q}_{1}={Q}_{2}=75\%$ (of total system capacity);
- −
- Threshold mechanism: $\mathbb{T}=\left\{3\right\}$, ${p}_{3}=1$, ${Q}_{3,1}=75\%$ (of total system capacity), ${t}_{3,0}={t}_{3}$, ${\mu}_{3,0}^{-1}={\mu}_{3}^{-1}$, ${t}_{3,1}=10$ BBUs, ${\mu}_{3,1}^{-1}=1.4$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BBU | basic bandwidth unit |

CAC | call admission control |

IoT | Internet of Things |

LAG | limited-availability group |

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**Figure 2.**Model of the limited-availability group with reservation mechanisms. Class 1 belongs to the set $\mathbb{R}$.

**Figure 3.**Model of the limited-availability group with the threshold mechanism. Class 1 belongs to the set $\mathbb{T}$.

**Figure 11.**Group 1—method 1; blocking probability in a group of cells with the connection handoff and reservation mechanism.

**Figure 12.**Group 1—method 2; blocking probability in a group of cells with the connection handoff and reservation mechanism.

**Figure 13.**Group 2—method 1; blocking probability in a group of cells with the connection handoff and reservation mechanism.

**Figure 14.**Group 2—method 2; blocking probability in a group of cells with the connection handoff and reservation mechanism.

**Figure 15.**Group 3—method 1; blocking probability in a group of cells with the connection handoff and reservation mechanism.

**Figure 16.**Group 3—method 2; blocking probability in a group of cells with the connection handoff and reservation mechanism.

**Figure 17.**Group 1—method 1; blocking probability in a group of cells with the connection handoff and threshold mechanism.

**Figure 18.**Group 1—method 2; blocking probability in a group of cells with the connection handoff and threshold mechanism.

**Figure 19.**Group 2—method 1; blocking probability in a group of cells with the connection handoff and threshold mechanism.

**Figure 20.**Group 2—method 2; blocking probability in a group of cells with the connection handoff and threshold mechanism.

**Figure 21.**Group 3—method 1; blocking probability in a group of cells with the connection handoff and threshold mechanism.

**Figure 22.**Group 3—method 2; blocking probability in a group of cells with the connection handoff and threshold mechanism.

Simulation | Generated Calls | Lost Calls | ||||
---|---|---|---|---|---|---|

No. | Class 1 | Class 2 | Class 3 | Class 1 | Class 2 | Class 3 |

$a=0.4$ Erl | ||||||

1 | 8,015,163 | 2,002,972 | 1,000,000 | 0 | 4 | 40 |

2 | 8,000,646 | 1,999,606 | 1,000,000 | 0 | 4 | 36 |

3 | 8,003,612 | 2,001,242 | 1,000,000 | 0 | 6 | 23 |

4 | 7,997,956 | 1,998,845 | 1,000,000 | 1 | 6 | 26 |

5 | 8,006,855 | 2,003,639 | 1,000,000 | 1 | 2 | 42 |

$a=0.5$ Erl | ||||||

1 | 8,005,402 | 2,000,097 | 1,000,000 | 3 | 114 | 758 |

2 | 8,002,535 | 2,002,714 | 1,000,000 | 1 | 127 | 807 |

3 | 7,990,866 | 1,998,619 | 1,000,000 | 4 | 112 | 784 |

4 | 8,003,962 | 2,000,647 | 1,000,000 | 4 | 111 | 753 |

5 | 8,000,179 | 2,000,384 | 1,000,000 | 5 | 100 | 662 |

$a=0.6$ Erl | ||||||

1 | 8,016,256 | 2,003,358 | 1,000,000 | 84 | 1414 | 7098 |

2 | 8,001,267 | 2,003,524 | 1,000,000 | 86 | 1477 | 7042 |

3 | 8,001,351 | 1,999,157 | 1,000,000 | 74 | 1394 | 7159 |

4 | 8,001,483 | 1,997,997 | 1,000,000 | 73 | 1516 | 7055 |

5 | 7,997,064 | 1,997,529 | 1,000,000 | 70 | 1419 | 7308 |

$a=0.7$ Erl | ||||||

1 | 8,005,466 | 2,000,655 | 1,000,000 | 668 | 9782 | 35,089 |

2 | 7,989,660 | 2,000,098 | 1,000,000 | 735 | 9672 | 35,231 |

3 | 7,999,643 | 1,998,351 | 1,000,000 | 570 | 9816 | 35,163 |

4 | 7,998,862 | 1,996,429 | 1,000,000 | 797 | 9669 | 35,169 |

5 | 8,011,478 | 2,000,240 | 1,000,000 | 643 | 9626 | 35,155 |

$a=0.8$ Erl | ||||||

1 | 8,016,212 | 2,003,034 | 1,000,000 | 3361 | 37,033 | 103,651 |

2 | 8,000,809 | 1,999,740 | 1,000,000 | 3304 | 37,380 | 104,169 |

3 | 8,007,225 | 2,002,980 | 1,000,000 | 3351 | 36,991 | 103,743 |

4 | 7,989,013 | 1,998,646 | 1,000,000 | 3312 | 36,748 | 103,690 |

5 | 7,990,865 | 1,997,921 | 1,000,000 | 3348 | 37,193 | 104,141 |

$a=0.9$ Erl | ||||||

1 | 8,015,925 | 2,002,674 | 1,000,000 | 10,312 | 91,261 | 208,430 |

2 | 7,996,733 | 2,000,352 | 1,000,000 | 10,123 | 90,897 | 209,014 |

3 | 8,004,606 | 1,998,289 | 1,000,000 | 10,334 | 90,849 | 209,203 |

4 | 7,993,835 | 2,000,331 | 1,000,000 | 10,156 | 90,849 | 209,182 |

5 | 7,993,240 | 1,997,134 | 1,000,000 | 9995 | 90,761 | 209,349 |

$a=1.0$ Erl | ||||||

1 | 8,006,435 | 2,000,442 | 1,000,000 | 22341 | 166982 | 329437 |

2 | 8,007,892 | 2,000,203 | 1,000,000 | 22,911 | 167,469 | 329,535 |

3 | 8,014,370 | 2,003,343 | 1,000,000 | 22,922 | 168,464 | 328,232 |

4 | 7,999,943 | 2,001,715 | 1,000,000 | 22,415 | 166,966 | 329,028 |

5 | 7,996,504 | 1,999,651 | 1,000,000 | 22,775 | 167,821 | 329,304 |

$a=1.1$ Erl | ||||||

1 | 8,018,878 | 2,003,590 | 1,000,000 | 40,424 | 261,294 | 443,488 |

2 | 7,989,703 | 1,997,843 | 1,000,000 | 41,342 | 261,824 | 445,539 |

3 | 8,005,807 | 2,002,445 | 1,000,000 | 41,182 | 260,772 | 443,122 |

4 | 7,996,982 | 2,000,633 | 1,000,000 | 41,035 | 260,411 | 444,261 |

5 | 8,009,529 | 2,001,840 | 1,000,000 | 40,820 | 261,568 | 444,342 |

$a=1.2$ Erl | ||||||

1 | 8,019,782 | 2,003,955 | 1,000,000 | 65,725 | 363,295 | 545,466 |

2 | 8,006,984 | 2,003,003 | 1,000,000 | 65,844 | 363,668 | 547,309 |

3 | 8,008,821 | 2,000,908 | 1,000,000 | 64,933 | 362,008 | 545,762 |

4 | 8,003,306 | 2,001,821 | 1,000,000 | 65,496 | 361,843 | 545,553 |

5 | 8,010,187 | 2,001,188 | 1,000,000 | 65,035 | 363,342 | 546,434 |

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**MDPI and ACS Style**

Głąbowski, M.; Sobieraj, M.; Stasiak, M.
Analytical Model of the Connection Handoff in 5G Mobile Networks with Call Admission Control Mechanisms. *Sensors* **2024**, *24*, 697.
https://doi.org/10.3390/s24020697

**AMA Style**

Głąbowski M, Sobieraj M, Stasiak M.
Analytical Model of the Connection Handoff in 5G Mobile Networks with Call Admission Control Mechanisms. *Sensors*. 2024; 24(2):697.
https://doi.org/10.3390/s24020697

**Chicago/Turabian Style**

Głąbowski, Mariusz, Maciej Sobieraj, and Maciej Stasiak.
2024. "Analytical Model of the Connection Handoff in 5G Mobile Networks with Call Admission Control Mechanisms" *Sensors* 24, no. 2: 697.
https://doi.org/10.3390/s24020697