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Grid networks have captured a lot of attention in recent years because of their potential to generate new applications thanks to network, computing and storage resources virtualization. Quality of service (QoS) is a key issue for grid services provisioning. Current grid services are provided on multi service networks such as the Internet. Thus, QoS architectures originally developed for the Internet such as DiffServ (DS) have been tested in grid environment. Since grid network services based on Internet networks will be developed in the next years, we propose in this paper to investigate the potentialities of an innovative Internet QoS architecture known as flow-aware networking (FAN). FAN appears as a promising alternative to DS for QoS provisioning in IP networks. DS proceeds to traffic differentiation and QoS provisioning through IP packet marking whereas FAN consists in implicit IP flow differentiation and a flow-based admission control. A grid session may be seen as a succession of parallel TCP flows with voluminous data transfers. In this paper, we compare by means of computer simulations the performance of FAN and DS architectures under grid environment. Two metrics are adopted for that purpose: the average transit delay and the average goodput of a grid session in an IP access router.