mBm-Based Scalings of Traffic Propagated in Internet

Scaling phenomena of the Internet traffic gain people's interests, ranging from computer scientists to statisticians. There are two types of scales. One is small-time scaling and the other large-time one. Tools to separately describe them are desired in computer communications, such as performance analysis of network systems. Conventional tools, such as the standard fractional Brownian motion (fBm), or its increment process, or the standard multifractional fBm (mBm) indexed by the local Holder function 𝐻(𝑡) may not be enough for this purpose. In this paper, we propose to describe the local scaling of traffic by using 𝐷(𝑡) on a point-by-point basis and to measure the large-time scaling of traffic by using E[𝐻(𝑡)] on an interval-by-interval basis, where E implies the expectation operator. Since E[𝐻(𝑡)] is a constant within an observation interval while 𝐷(𝑡) is random in general, they are uncorrelated with each other. Thus, our proposed method can be used to separately characterize the small-time scaling phenomenon and the large one of traffic, providing a new tool to investigate the scaling phenomena of traffic.