In this paper we analyze the M/G/1 processor sharing queue with heavy tailed services and with impatient customers. It is assumed that impatience depends on the value of the service required. We prove that a reduced service rate (RSR) approximation holds for estimating the sojourn time of a customer in the system, when the queue capacity is finite or infinite. This allows us to evaluate the reneging probability of customers with very large service times. We then use these results to investigate the impact of admission control on a link of a packet network. Admission control simply consists of limiting the number of simultaneous connections. It turns out that there is a real benefit for the efficiency of the system to perform admission control: It globally increases the fraction of customers, who complete their service (i.e. without being impatient). Finally, we investigate the fairness of the system and propose a criterion to assess the capacity of the system so as to allow the completion of very large service times.