We revisit smoothing networks, which are made up of balancers and wires. 
Tokens arrive arbitrarily on w input wires and propagate asynchronously 
through the network; each token gets service on the output wire it 
arrives at. The smoothness is the maximum discrepancy among the numbers 
of tokens arriving at the w output wires. We assume that balancers are 
oriented independently and uniformly at random. We present a collection 
of lower and upper bounds on smoothness, which are to some extent 
surprising:

- The smoothness of a cube-connected-cycles network is log log w + 
Theta(1) (with high probability). This bound improves vastly over the 
previously known upper bound of O(sqrt{log w}) from Herlihy and Tirthapura.

-Most significantly, the smoothness of the cascade of two 
cube-connected-cycles networks is no more than 17 (with high 
probability). This is the first known randomized network with so small 
depth 2 log w and so good smoothness.
 
-There is no randomized 1-smoothing network of width w and depth d that 
achieves 1-smoothness with probability better than d/(w-1). In view of 
the deterministic 1-smoothing network by Klugerman and Plaxton, this 
result implies the first separation between deterministic and randomized 
smoothing networks, which demonstrates an unexpected limitation of 
randomization: it can get to constant smoothness very easily, but after 
that, the progress to 1-smoothing is very limited.