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In this paper we compute the uplink capacity of power-control CDMA mobile networks with an idealized power control, that contain best-effort type applications, 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. Our first result is that slowing the transmission rates in the case of a single cell with homogeneous quality of service characteristics increases capacity. This suggests that there is a limit capacity that can be approached when slowing down the transmission rates. We identify this limit and show that it has the following property: as long as the arrival rate of information is below some level, blocking probability can become arbitrarily small by sufficiently slowing down the transmission rates. We then extend the results to the general heterogeneous and multi-cell case.