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We describe a measurement-analytic approach for estimating the overflow probability, an important measure of the quality of service (QoS), at a given multiplexing point in the network. A multiplexing point in the network could be a multiplexer or an output port of a switch or router where resources such as bandwidth and buffers are shared. Our approach impinges on using the notion of the dominant time scale (DTS), which corresponds to the most probable time scale over which overflow occurs. The DTS provides us with a measurement window for the statistics of the traffic, but is in fact itself defined in terms of the statistics of the traffic over all time. This, in essence, results in a chicken-and-egg type of unresolved problem. For the DTS to be useful for on-line measurements, we need to be able to break this chicken-and-egg cycle, and to estimate the DTS with only a bounded window of time over which the statistics of the traffic are to be measured. We present a stopping criterion to successfully break this cycle and find a bound on the DTS. Thus, the result has significant implications for network measurements. Our approach is quite different from other works in the literature that require off-line measurements of the entire trace of the traffic. In our case, we need to measure only the statistics of the traffic up to a bound on the DTS. We also investigate the characteristics of this upper bound on the DTS, and provide numerical results to illustrate the utility of our measurement analytic approach.