On the Dependence of the Queue Tail Distribution on Multiple Time Scales of ATM Multiplexers

For a family of arrival processes A ; > 0, we give suucient conditions under which the queue length distribution satisses lim !0 lim x!1 ? log IPPQ x]=x= ; for some > 0 called the asymptotic decay rate. Using this result we show, under strict stability conditions , that the asymptotic decay rate of an ATM network multiplexer does not depend on the slow time scale statistics. However, the rate at which the queue length distribution decreases for small buuer sizes can be much larger than the asymptotic decay rate. This implies that an equivalent bandwidth admission control policy (based only on) may signiicantly un-derutilize the system resources. We illustrate this discrepancy with two examples. We also show that the graph of log IPPQ > x] may have a polygonal shape approximation, i.e., it consists of more than one exponential. We argue that this situation arises whenever the dynamics of the arrival process to the multiplexer \spreads" over multiple time scales.