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Network analysis is the process of identifying and remediating the processes and systems within a network, including performance, connectivity and security. The process is performed through the use of tools developed for monitoring and analyzing network activity. Network problems that involve finding an optimal way of doing something are studied under the name combinatorial optimization. Examples include network flow, shortest path problem, transport problem, transshipment problem, location problem, matching problem, assignment problem, packing problem, routing problem, Critical Path Analysis and PERT (Program Evaluation & Review Technique).
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But for free , check out http://www.sysinternals.com/
Also, most likely the reason your server times out is because of network traffic.
So, if you know how to read packets, then check out WireShark