Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level K. A basic performance measure of this system is the probability PK that a customer gets rejected in steady state. This paper presents asymptotic expansions for PK as K ... If the service time B is light-tailed and interarrival times are exponential, it is shown that the loss probability has an exponential tail. The proof of this result heavily relies on results on the two-sided exit problem for Levy processes with no positive jumps. For heavy-tailed (subexponential) service times and generally distributed inter-arrival times, the loss probability is shown to be asymptotically equivalent to the trivial lower bound P(B > K)