We study a stream of traffic or message as it is transferred over an A
TM connection consisting of burst reducing servers. A message is model
ed as a deterministic fluid flow, and an ATM node is modeled as a serv
er which allocates bandwidth to messages. A message's burstiness curve
b(mu) is the buffer size needed to prevent cell loss if it is served
at rate mu. A server is burst reducing if its output message is always
less bursty than the input message. Two popular bandwidth allocation
schemes - the fixed rate and the leaky bucket server - are shown to be
burst reducing. We also present a new class of burst reducing servers
, the affine servers. We derive buffer requirements along a multi-hop
connection and the final fluid flow reaching the destination as a mess
age goes through a sequence of burst reducing servers. Finally, we sug
gest an approach to defining service quality.