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Instant messaging (IM) has become increasingly popular due to its social functionality. Despite its popularity and large user base, little has been done to the analysis and characteristic of the IM traffic, especially at the chat session level. In this paper, we analyze the traffic of two popular instant messaging systems, Yahoo messenger and MSN/Windows Live Messenger. We mainly consider the distribution of chat session level traffic between end users, and find the distribution in many cases follows a power law, as shown in recent work. This power law finding was previously used to support the hypothesis that chat interval time has a power law tail. We further show the scale property by V-T plot and R/S estimation. That is, the interval time is characterized by a strong self-similarity for larger time scales. And the packet arrival times are independent explaining the weak correlation of the data. Our analysis sheds light on instant messaging system design and optimization and provides a scientific basis for instant messaging traffic generation.