We consider the steady state distribution of the end-to-end delay of a tagged flow in queueing networks where the queues have self-similar cross traffic. We assume that such cross traffic at each queue, say queue i, is modeled by fractional Brownian Motion (FBM) with Hurst parameter Hi (1/2 <= Hi < 1), and is independent of other queues.The arrival process of the tagged flow is renewal. The virtual service time of the tagged flow is self similar due to the cross traffic. Two types of queueing networks are considered.

We show that the end-to-end delay of the tagged flow in a tandem queueing network, and more generally in a tree network, is completely dominated by one of the queues. The dominant queue is the one with the maximal Hurst parameter. If several queues have the same maximal Hurst parameter, then we have to compare the ratio 1/s * (1- r)^H to determine the dominant queue, where r is the load of the queue.

In the case that the tagged flow is controlled through a window based congestion control mechanism, the end-to-end delay is still asymptotically Weibullian with the same shape parameter. We provide upper and lower bounds on the constant that determines the scale parameter of the corresponding Weibull distribution.

This is joint work with Marc LeLarge and Zhen Liu.