Tassiulas and Ephremides established that Max Weight (MW) algorithm with
weight equal to the linear function of queue-size is throughput optimal 
for a large class of constrained scheduling problems. Various researchers 
established that the MW algorithm with weight equal to the queue-size to 
power x, for any positive x is also throughput optimal. However, the algorithms
with different weight seem to differ in performance in terms of average queue-size.
Specifically, in the context of switch scheduling Keslassy and McKeown had 
conjectured that as x decreases the average queue-size induced by MW algorithm,
with weight equal queue-size to power x, decreases.

In our previous work, we provided intuitive justification of this conjecture 
by means of state space collapse space of algorithm under heavy traffic limit.
In this talk, we will present a rigorous justification by means of critical 
fluid models for a class of scheduling problems. We will explain results in 
the context of switch scheduling and wireless scheduling.