This paper addresses the following questions related to buffer management s
chemes for high speed integrated services networks: (i) given the pattern o
f cell arrivals from different classes of traffic, can buffer control signi
ficantly influence the effect of cell loss, and (ii) what are the "best" po
licies for selecting cells for transmission from buffers in the network nod
es as well as for rejecting cells when the buffers are full. The basic appr
oach to answering these questions is to impute a cost of losing cells which
could depend on the class of application, and to minimize this cost over t
he finite or infinite time horizons. The policies we derive using this cost
minimization approach are best in the sense that they minimize linear cost
functions of cell losses. at each instant of time during the system's oper
ation. We also show how to construct policies that minimize general cost fu
nctions of cell loss rates.