### Abstract

A class of stochastic processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson [Bramson M (1998) State space collapse with applications to heavy-traffic limits for multiclass queueing networks. Queueing Systems 30:89-148] and Williams [Williams RJ (1998b) Diffusion approximations for open multiclass queueuing networks: Sufficient conditions involving state space collapse. Queueing Systems 30:27-88] laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre- and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates interesting connections between stability of the pre- and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Original language | English (US) |
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Pages (from-to) | 121-162 |

Number of pages | 42 |

Journal | Mathematics of Operations Research |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2014 |

### Keywords

- Heavy traffic
- Multiclass
- Network
- Queue-ratio
- Steady-state

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research