The performance analysis of a network link is a well-studied problem. However, the most interesting issue for a service provider is to evaluate the end-to-end quality of service (QoS). The evaluation of the end-to-end QoS (e.g. loss probability or delay) depends on the traffic statistic which is constantly modified as the traffic traverse the network, making its analysis a very difficult problem. In this work we use a simplified framework known as fictitious network analysis that allows us to estimate on-line the end-to-end loss ratio from input traffic traces statistics. We prove that the defined estimator is consistent and that a Central Limit Theorem is verified. Based on these estimations an admission control mechanism can be implemented. More precisely, we propose a simply method to estimate the control admission region, i.e. which are the flows that can be accepted in the network that verifies that its end-to-end loss ratio is smaller than a given threshold. While decisions based on the fictitious network analysis are safe, it may lead to network resources under-utilization (it generally overestimates the QoS parameters). In this work we establish sufficient conditions to assure that results obtained by means of the fictitious network coincide with real ones (there is no overestimation). We present first the conditions in the one-link case and extend them to the multilink case, necessary to evaluate the end-to-end loss ratio. When different results are obtained we define a method to find a bound for the overestimation. We also present numerical examples to compare the performance obtained in the real and the fictitious network, validating our main results.