### Abstract

A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r^{-1/2}) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

Original language | English (US) |
---|---|

Pages (from-to) | 251-269 |

Number of pages | 19 |

Journal | Advances in Applied Probability |

Volume | 47 |

Issue number | 1 |

State | Published - Mar 1 2015 |

Externally published | Yes |

### Fingerprint

### Keywords

- Diffusion limit
- Fluid limit
- Limit interchange
- Many-server model
- Priority discipline
- Tightness of invariant distribution

### ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics

### Cite this

**Diffusion-scale tightness of invariant distributions of a large-scale flexible service system.** / Stolyar, A. L.

Research output: Contribution to journal › Article

*Advances in Applied Probability*, vol. 47, no. 1, pp. 251-269.

}

TY - JOUR

T1 - Diffusion-scale tightness of invariant distributions of a large-scale flexible service system

AU - Stolyar, A. L.

PY - 2015/3/1

Y1 - 2015/3/1

N2 - A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r-1/2) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

AB - A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r-1/2) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

KW - Diffusion limit

KW - Fluid limit

KW - Limit interchange

KW - Many-server model

KW - Priority discipline

KW - Tightness of invariant distribution

UR - http://www.scopus.com/inward/record.url?scp=84940381924&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940381924&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84940381924

VL - 47

SP - 251

EP - 269

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 1

ER -