# Mathematics Colloquia and Seminars

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### Fluid and Diffusion Approximations for a Heavily Loaded Processor Sharing Queue

**Mathematical Physics & Probability**

Speaker: | Ruth Williams, UC San Diego |

Location: | 693 Kerr |

Start time: | Tue, Feb 26 2002, 3:10PM |

This talk will concern a heavily loaded single server queue operating under a processor sharing service discipline. This system is naturally modeled using a measure-valued process that keeps track of the residual service times of all jobs currently in the system. The fluid (law of large numbers) limit for this model will be identified as the unique solution of a nonlinear integral equation. The large time asymptotics of this fluid limit will be described, including rates of convergence to an invariant state. The latter play a key role in establishing a certain "state space collapse" that features in establishing a diffusion limit for the heavily loaded processor sharing queue. Indeed, the diffusion limit of the measure-valued process is obtained via a lifting map from a one-dimensional reflected Brownian motion. This talk is based on joint work with Christian Gromoll and Amber Puha.

Maury Bramson is the host.