Reflecting Brownian Motions and Queueing Networks

                  Vien Nguyen and Ruth Williams

                       Tentative Schedule 

Lecture 1: Overview 
Some outline of what will be covered in the
lectures, with some general motivation and a simple example 
of the heavy traffic approximation, e.g., single class tandem queue, without
proof -- this would be good to get people interested
and also to give them a feel for where this is all going.

Lecture 2: SRBM -- existence and uniqueness.
One dimensional construction, pathwise theorems (Harrison-Reiman)
(state), weak existence and uniqueness (Taylor-Williams) --
state, motivate, maybe describe construction, maybe state Bernard-El
Kharroubi oscillation estimate (a key estimate).

Mention Dai-Williams result for non-simple polyhedrons (no
details).

Lecture 3: SRBM -- recurrence, stationary distributions.
Recurrence:
Dupuis-Williams theorem and application via linear Lyapunov function
to show positive recurrence  when R=I-P' (single class case) and
R^{-1} theta < 0.
Stationary distributions: BAR (application of stochastic calculus)
State necessary and sufficient conditions for product form solutions
(Harrison-Williams result).
Mention existence of numerical procedure for finding stationary
distribution from BAR. 


Lecture 4:  HT Limit for a multiclass FIFO STATION   
Introduce notation here which can be used consistently in
the following lectures.  Use a workload formulation.

Heavy traffic limit of multiclass GI/GI/1, FIFO queue,
after the style of Peterson -- do this rigorously
with scaling explicitly laid out
      Principle 1:  The snapshot principle
      Principle 2:  State space collapse 
      Principle 3:  The uniform mixing principle 
  
It seems that this 
is a good way to get the students warmed up to HT approximation because, (i), 
all the results can be proved rigorously and (ii) this analysis contains many 
of the important insights into the behavior of queues under heavy traffic 
conditions.

Lecture 5: HT Limit for a (single class FIFO) NETWORK
  HT limit of open generalized Jackson network -- this can still be 
  done pretty rigorously, as we can rely on the path-to-path mapping
  Use workload formulation

Lecture 6: Performance analysis of a make-to-order manufacturing job
shop 
  By this we mean the "QNET" method for analyzing a multiclass FIFO
  open
  network (Vien's paper with Mike).  Some of the important ideas in this
  lecture:  how to "guess" at the limiting result using heavy traffic
  principles; the usual "enhancements" of heavy traffic results 
  for better performance measures; show "insightful" numerical examples.

Lecture 7:  Some surprising examples:  unstable networks and unconventional 
limit theorems
HT limits: Dai-Wang, Dai-Nguyen examples
Stability: Lu-Kumar/Rybko-Stolyar examples, mention results of
  Bramson. Mention fluid models for determining stability: Dai,
  etc.
HT limit theorems: Harrison-Williams unconventional heavy traffic limit theorem

Lecture 8: Other networks and applications:  
Performance analysis of make-to-stock and 
hybrid make-to-order/make-to-stock production systems
  -- Vien's work on mixed networks; this has a nice structure
  and provides some good insights into behavior of networks and
  implications of heavy traffic theory


Throughout one can mention extensions and open
problems in passing.
It would be good to mention optimization as an additional
problem which has not been covered here.