### Abstract

Mean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of mean-field models for continuous-time Markov chains (CTMC), and focuses on mean-field models that are represented as finite-dimensional dynamical systems with a unique equilibrium point. By applying Stein's method and the perturbation theory, the paper shows that under some mild conditions, if the mean-field model is glob- Ally asymptotically stable and locally exponentially stable, the mean square difference between the stationary distribution of the stochastic system with size M and the equilibrium point of the corresponding mean-field system is O(1/M). The result of this paper establishes a general theorem for establishing the convergence and the approximation error (i.e., the rate of convergence) of a large class of CTMCs to their mean-field limit by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. Two applications of mean-field models in data center networks are presented to demonstrate the novelty of our results.

Original language | English (US) |
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Title of host publication | SIGMETRICS/ Performance 2016 - Proceedings of the SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Science |

Publisher | Association for Computing Machinery, Inc |

Pages | 285-297 |

Number of pages | 13 |

ISBN (Electronic) | 9781450342667 |

DOIs | |

State | Published - Jun 14 2016 |

Event | 13th Joint International Conference on Measurement and Modeling of Computer Systems, ACM SIGMETRICS / IFIP Performance 2016 - Antibes Juan-les-Pins, France Duration: Jun 14 2016 → Jun 18 2016 |

### Other

Other | 13th Joint International Conference on Measurement and Modeling of Computer Systems, ACM SIGMETRICS / IFIP Performance 2016 |
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Country | France |

City | Antibes Juan-les-Pins |

Period | 6/14/16 → 6/18/16 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Networks and Communications
- Computational Theory and Mathematics
- Hardware and Architecture

### Cite this

*SIGMETRICS/ Performance 2016 - Proceedings of the SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Science*(pp. 285-297). Association for Computing Machinery, Inc. https://doi.org/10.1145/2896377.2901463

**On the approximation error of mean-field models.** / Ying, Lei.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*SIGMETRICS/ Performance 2016 - Proceedings of the SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Science.*Association for Computing Machinery, Inc, pp. 285-297, 13th Joint International Conference on Measurement and Modeling of Computer Systems, ACM SIGMETRICS / IFIP Performance 2016, Antibes Juan-les-Pins, France, 6/14/16. https://doi.org/10.1145/2896377.2901463

}

TY - GEN

T1 - On the approximation error of mean-field models

AU - Ying, Lei

PY - 2016/6/14

Y1 - 2016/6/14

N2 - Mean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of mean-field models for continuous-time Markov chains (CTMC), and focuses on mean-field models that are represented as finite-dimensional dynamical systems with a unique equilibrium point. By applying Stein's method and the perturbation theory, the paper shows that under some mild conditions, if the mean-field model is glob- Ally asymptotically stable and locally exponentially stable, the mean square difference between the stationary distribution of the stochastic system with size M and the equilibrium point of the corresponding mean-field system is O(1/M). The result of this paper establishes a general theorem for establishing the convergence and the approximation error (i.e., the rate of convergence) of a large class of CTMCs to their mean-field limit by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. Two applications of mean-field models in data center networks are presented to demonstrate the novelty of our results.

AB - Mean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of mean-field models for continuous-time Markov chains (CTMC), and focuses on mean-field models that are represented as finite-dimensional dynamical systems with a unique equilibrium point. By applying Stein's method and the perturbation theory, the paper shows that under some mild conditions, if the mean-field model is glob- Ally asymptotically stable and locally exponentially stable, the mean square difference between the stationary distribution of the stochastic system with size M and the equilibrium point of the corresponding mean-field system is O(1/M). The result of this paper establishes a general theorem for establishing the convergence and the approximation error (i.e., the rate of convergence) of a large class of CTMCs to their mean-field limit by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. Two applications of mean-field models in data center networks are presented to demonstrate the novelty of our results.

UR - http://www.scopus.com/inward/record.url?scp=84978640617&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978640617&partnerID=8YFLogxK

U2 - 10.1145/2896377.2901463

DO - 10.1145/2896377.2901463

M3 - Conference contribution

AN - SCOPUS:84978640617

SP - 285

EP - 297

BT - SIGMETRICS/ Performance 2016 - Proceedings of the SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Science

PB - Association for Computing Machinery, Inc

ER -