**Organiser**: David McDonald and André Dabrowski,

Department of Mathematics and Statistics,University of Ottawa

**Format:** 2 days: Invited Talks.

### Invited Lecturers

**There will be 5 hour long talks on the state of the art of the
following subjects all closely related to rare events in networks:**

**Robert Foley**, Georgia Institute of Technology

*Doob's h-transform technique, Markov Additive processes and rare
events in networks*
**Peter Glynn**, Stanford University

*Rare Event Simulation*
**Irina Ignatiouk-Robert**, l'Université de Cergy-Pontoise

*Large deviations for Markov jump processes with boundaries *
**Phil Pollett**, University of Queensland

*Ratio limit theorems, quasi stationary distributions, the Yaglom
limit*
**Marcel Neuts**(tentative):

*The Matrix Geometric method *

**AA, Mogulski**, Novosibirsk, Russia

**Masakiyo Miyazawa**, Science University of Tokyo, Japan

Sean Meyn, University of Illinois, Urbana

**Tomasz Rolski**, Mathematical Institute Wroclaw

**Hui Wang **, Brown

### Invited participants:

**Atahiru Alfa,** University of Manitoba

**Doug Down**, McMaster University

**Winfried Grassman**, University of Saskachewan

**David Stanford**, University of Western Ontario

**Yiqiang Zhao**, Carleton University

### Workshop Description

The subject of queueing networks finds applications in the description
of the flow of packets in the internet or in the flow of jobs through
the many work stations in a modern factory. The study of congestion
and control in such networks drawns on diverse areas of probability.
One aspect is the calculation of rare event probabilities like the
probability a particular node in a network overflows. Large deviation
theory must be adapted to the network context because there are natural
boundaries where the queues become empty. Large deviation theory along
boundaries has been the subject of research from some time.

At the heart large deviation theory one uses the classical change
of measure or h-transform technique due to Doob. In special cases
the h-transform technique can even yield sharp asymptotics and a description
of the network when the large deviation occurs. Sharp asymptotics
are related to strong ratio limit theorems and the study of harmonic
functions, quasi-stationary distributions and the Yaglom limit. The
latter may be interpreted as the limiting conditional distribution
of a Markov chain given the chain has not yet been killed by the nth
step. This in turn is related to the literature on random walks conditioned
to stay positive. In some sense the Yaglom limit is a more practical
and fundamental concept than the traditional equilibrium limit since,
as Keynes said, “in the long run we are all dead”. In other
words most equilibrium states we may identify are more likely to be
quasi-stationary distributions.

There is also recent work on determining the steady state and hence
the large deviation probabilities using the Matrix Geometric method
with infinite phase.

The aim of the workshop is to connect all these different threads.

### Summer School (June 27th to June 30th)

The Rare Events Summer School will be held in room STE C0136 starting
9am on Monday June 27th.

This room is in the SITE (School of Information Technology and Engineering)
building at the

extreme south end of the University of Ottawa Campus.

** (This building is located on the South of the Campus).See: http://www.uottawa.ca/map/#).**

For emergencies on arrival please call David McDonald at (613) 236
5893.

Professors Foley, McDonald, and Rolski will give a series of lectures
on this theme. We also hope to have a couple of guest lecturers. The
aim is to prepare the participants for the following rare events workshop
and the Informs meeting.

There was some financial assistance for local expenses for qualified
graduate students.

First round deadline is May 1 with notification of funding by May
9th.

We have arranged space in residence at the University of Ottawa for
students attending the summer school, right in the city center.

Note: Classes will be cut short on Friday July 1st due to Canada Day
celebrations.

Contact rare_events@fields.utoronto.ca
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