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In this paper, we investigate time-correlation of the connection request process of Web browsing applications and other Internet applications. Evidence of self-similarity in the Internet traffic has been pointed out in several papers, but mainly with reference to the volume of traffic, to the packet arrival process, or to the connection arrival process of aggregated traffic. In our study, instead, we focus on the process of connection requests coming from a single client and study whether asymptotic self-similarity is evident even when there is low client activity, the observation window is short, or data is partial. The analysis is performed on publicly available traffic traces that include both Wide Area and Campus Network traffic. To identify time correlations, we use the novel, unbiased estimator of the power-law exponent based on the Modified Allan Variance (MAVAR). Our results show that self-similarity is evident in Web traffic and Domain Name requests, provided that the client is active for more than a few connections. This study is valuable for researchers interested in the modeling of packet traffic sources or in the monitoring of network activity.