# Optimal Allocation of Virtual Machines in Cloud Computing

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## Abstract

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## 1. Introduction

- Global optimality guarantee: Compared with heuristic methods, the proposed approach transforms the virtual machine placement problem into a mixed-integer linear programming problem and is thus guaranteed to reach a global optimum.
- Computational efficiency enhancement: Compared with the existing deterministic method [5], the proposed method adds appropriate constraints to reduce the number of feasible solutions for enhancing computational efficiency.

## 2. Literature Review

- Heuristic Bin Packing: The VM placement problem is formulated as vector bin packing. VMs are considered to be small items that are tightly packed into the minimum number of bins, each considered a PM. Several heuristic methods are developed to approximate the optimal solution to this packing problem.
- Biology-based optimization: Several bio-inspired optimization techniques such as the ant colony optimization method, the self-adaptive particle swarm optimization method, and genetic algorithms are applied to pack VMs into the smallest number of PMs, given the current workload.
- Linear programming: The VM placement problem is constructed as a linear programming problem which considers a number of constraints derived from practical applications. LP-relaxation-based methods are developed to solve the formulated model.
- Constraint programming: Van et al. [11] have presented a resource management framework, which includes a dynamic utility-based VM provisioning manager and a dynamic VM placement manager, to obtain a suitable VM-PM mapping. Both management tasks are regarded as constraint satisfaction problems. More practical aspects can be taken into consideration by extending the constraints in these problems.
- Stochastic integer programming: Because the future demand of VM for providing network services is uncertain, the stochastic integer programming technique is used to predict a suitable VM-PM mapping.
- Simulated annealing optimization: Liao et al. [12] have proposed a dynamic runtime mapping framework that adopts a simulated annealing optimization algorithm to map VMs onto a small set of PMs in order to minimize power consumption without significant system degradation.

- N: set of data centers;
- K: set of virtual machines;
- U: set of users;
- ${a}_{i}$: capacity in number of VMs that data center i can host;
- ${B}_{ij}$: bandwidth between data centers i and j;
- ${L}_{ij}$: latency between data centers i and j;
- ${C}_{ij}$: cost of transferring a unit of data between data centers i and j;
- ${b}_{vw}$: required bandwidth between VMs v and w;
- ${l}_{vw}$: required latency between VMs v and w;
- ${t}_{vu}$: required latency between user u and VM v;
- d(u): the data center which hosts user u;
- ${c}_{iv}$: cost of allocating VM v in data center i;
- $z$: scaling cost factor.

## 3. Proposed Method

## 4. Numerical Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Experimental results of the LMVMP (Linear Model for the VM Placement) model [5] and the proposed method.

Case No. | Instance | Objective | CPU Time (Seconds) | |
---|---|---|---|---|

LMVMP | Proposed Method | |||

1 | 005_015_007_070 | 25,844.02 | 0.73 | 0.66 |

2 | 005_015_007_090 | 23,557.30 | 7.27 | 3.25 |

3 | 005_015_015_070 | 10,904.78 | 0.37 | 0.39 |

4 | 005_015_015_090 | 24,354.96 | 1.15 | 0.69 |

5 | 005_015_022_070 | 14,163.60 | 0.37 | 0.41 |

6 | 005_015_022_090 | 32,318.02 | 1.81 | 1.53 |

7 | 005_020_010_070 | 38,572.62 | 33.24 | 41.43 |

8 | 005_020_010_090 | 64,710.80 | 16.07 | 12.86 |

9 | 005_020_020_070 | 55,288.76 | 25.65 | 31.81 |

10 | 005_020_020_090 | 57,574.90 | 0.83 | 0.70 |

11 | 005_020_030_070 | 28,433.34 | 2.42 | 1.68 |

12 | 005_020_030_090 | 66,088.70 | 0.73 | 0.70 |

13 | 005_025_012_070 | 43,300.76 | 154.39 | 151.15 |

14 | 005_025_012_090 | 10,0865.02 | N/A * | 6877.55 |

15 | 005_025_025_070 | 42,890.40 | 10.75 | 13.96 |

16 | 005_025_025_090 | 103.791.96 | N/A * | 8021.09 |

17 | 005_025_037_070 | 97,335.12 | 16.24 | 27.81 |

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**MDPI and ACS Style**

Lin, M.-H.; Tsai, J.-F.; Hu, Y.-C.; Su, T.-H.
Optimal Allocation of Virtual Machines in Cloud Computing. *Symmetry* **2018**, *10*, 756.
https://doi.org/10.3390/sym10120756

**AMA Style**

Lin M-H, Tsai J-F, Hu Y-C, Su T-H.
Optimal Allocation of Virtual Machines in Cloud Computing. *Symmetry*. 2018; 10(12):756.
https://doi.org/10.3390/sym10120756

**Chicago/Turabian Style**

Lin, Ming-Hua, Jung-Fa Tsai, Yi-Chung Hu, and Tzu-Hsuan Su.
2018. "Optimal Allocation of Virtual Machines in Cloud Computing" *Symmetry* 10, no. 12: 756.
https://doi.org/10.3390/sym10120756