In this paper, a comprehensive and new closed form analytical approach to calculate the saturated throughput and delay of a differentiated p-persistent CSMA protocol with a multirate capability is presented for the first time. The p-persistent analysis can be used to model accurately 802.11e EDCA for WLAN applications. The analysis allows 802.11e to be characterised in the case of multirate operation without undue numerical complexity unlike state-of-the-art analytical approaches.