This paper considers the scenario of a targeted drug delivery system, which consists of deploying a number of biological nanomachines close to a biological target (e.g., a tumor), able to deliver drug molecules in the diseased area. Suitably located transmitters are designed to release a continuous flow of drug molecules in the surrounding environment, where they diffuse and reach the target. These molecules are received when they chemically react with compliant receptors deployed on the receiver surface. In these conditions, if the release rate is relatively high and the drug absorption time is significant, congestion may happen, essentially at the receiver site. This phenomenon limits the drug absorption rate and makes the signal transmission ineffective, with an undesired diffusion of drug molecules elsewhere in the body. The original contribution of this paper consists of a theoretical analysis of the causes of congestion in diffusion-based molecular communications. For this purpose, it is proposed a reception model consisting of a set of pure loss queuing systems. The proposed model exhibits an excellent agreement with the results of a simulation campaign made by using the Biological and Nano-Scale communication simulator version 2 (BiNS2), a well-known simulator for molecular communications, whose reliability has been assessed through in vitro experiments. The obtained results can be used in rate control algorithms to optimally determine the optimal release rate of molecules in drug delivery applications.