In this paper we compute the uplink capacity of power-control CDMA mobile networks with an idealized power control, that contain best-effort type applpications, i.e. applications whose transmission rate can be controlled. An arriving best-effort call is assumed to have a fixed amount of traffic to send, so the transmission rate assigned to it determines the duration of the call. We allow for multi-services (so that mobile stations have different quality of service requirements). Unlike some previous published work where soft blocking was considered (and the system was thus allowed to operate beyond capacity), we assume that a call admission mechanism is implemented in order to prevent a new call to arrive when the system is already saturated. This guarantees the quality of service of ongoing calls.