        The Law Second Class Customers Obey:
     A result for queues and particle systems
               Balaji Prabhakar
             Stanford University

The celebrated Burke's Theorem says that the law of the departure
process from a ./M/1 queue is Poisson when the arrival process is
Poisson.  We consider a ./M/1 queue with two classes of customers: 
second class customers yield absolute priority to first class customers. 
We ask: What law is invariant for the second class customers.  This 
classicial question turns out to have a nice, simple answer, via a 
connection with the Exclusion Process.

The $n$-type TASEP (Totally Asymmetric Simple Exclusion Process) 
has $n$ classes of particle, and particles of class $i$ cannot 
exclude particles of class $j<i$.  There is a well-known correspondence 
between this process and a multi-class exponential-server queue in 
which class $i$ customers yield priority to customers of class $j<i$.

The invariant distribution for the $n$-type TASEP has recently been 
determined by Angel.  Ferrari and Martin exploit the queueing connection 
and obtain the fixed points for the multi-class queue.  

We obtain a simple alternate proof using Weber's result on the 
interchangeability of exponential-server queues.  

Joint work with James Martin, Oxford University.




