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We present a new technique for predicting the resource demand requirements of services implemented by multi-tier systems. Accurate demand estimates are essential to ensure the efficient provisioning of services in an increasingly service-oriented world. The demand estimation technique proposed in this paper has several advantages compared with regression-based demand estimation techniques, which many practitioners employ today. In contrast to regression, it does not suffer from the problem of multicollinearity, it provides more reliable aggregate resource demand and confidence interval predictions, and it offers a measurement-based validation test. The technique can be used to support system sizing and capacity planning exercises, costing and pricing exercises, and to predict the impact of changes to a service upon different service customers.