Any autonomicsystem must implement mechanisms to automatically capture the most significant information about the internal state and also adapt the monitoring system to internal and external conditions. We refer to these activities as self-inspection and we consider them inthe context ofInternet-based services that are subject to workloads characterized by burst arrivals and heavy-tailed distributions. The large majority ofthemechanisms driving these systems must take fast decisions on the basis of past and/or present load conditions ofthesystem resources. In this context, self-inspection requires an adequate representation ofthe load behavior ofthesystem resources that makes it possible to perform good actions under soft real-time constraints. In this paper, we show through a large set of experiments the need of basing load analyses and decisions on linear and non-linear models, such as the exponential moving average and the 90-percentile models. All the considered models are applied to a multi-tier Web-basedsystem that is instrumented with suitable self-inspectionmechanisms at operating system level. However, the results can be extended to other Internet-based contexts where thesystems are characterized by similar workload and resource behaviors.