## Abstract

We consider general large-scale service systems with multiple customer classes and multiple server (agent) pools, mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing 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 natural (load balancing) routing/scheduling rule, Longest-Queue Freest-Server (LQFS-LB), in the many-server asymptotic regime, such that the exogenous arrival rates of the customer classes, as well as the number of agents in each pool, grow to infinity in proportion to some scaling parameter r. Equilibrium point of the system under LQBS-LB is the desired operating point, with server pool loads minimized and perfectly balanced. Our main results are as follows. (a)We show that, quite surprisingly (given the tree assumption), for certain parameter ranges, the fluid limit of the system may be unstable in the vicinity of the equilibrium point; such instability may occur if the activity graph is not "too small." (b) Using (a), we demonstrate that the sequence of stationary distributions of diffusion-scaled processes [measuring O(√r) deviations from the equilibrium point] may be nontight, and in fact may escape to infinity. (c) In one special case of interest, however, we show that the sequence of stationary distributions of diffusionscaled processes is tight, and the limit of stationary distributions is the stationary distribution of the limiting diffusion process.

Original language | English (US) |
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Pages (from-to) | 2099-2138 |

Number of pages | 40 |

Journal | Annals of Applied Probability |

Volume | 23 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2013 |

Externally published | Yes |

## Keywords

- Diffusion limit
- Fluid limit
- Instability
- Load balancing
- Many server models
- Tightness of invariant distributions

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty