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The purpose of this paper is to provide a simple evaluation formula for the asymptotic tail probability of the end-to-end backlog in a packet network with a wide range of traffic flows including fractional Brownian motions (fBms). In the previous paper, the authors proposed the concept of tractable effective bandwidth (tEBW). It is the effective bandwidths (EBWs) of one type and can carry out the end-to-end evaluation into a single node evaluation at the bottleneck node. Though almost all of the known traffic flow models have tEBWs, but fBms don't. In this paper we discuss the end-to-end evaluation under mixed traffic flows with tEBWs and fBm variances to include fBms. We show that, by suitably evaluating the input traffics (both forwarding traffic and the cross traffics), the end-to-end evaluation can be reduced into a single node evaluation, and obtain a simple asymptotic evaluation formula. The formula depends on the evaluation of the input traffics, but it is independent of the number of nodes as the case for tEBWs only.

Asymptotic End-to-end Backlog Evaluation in a Packet Network with a Wide Range of Traffic Flows Including Fractional Brownian Motions