Loss rates in the single-server queue with complete rejection

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 $P_K$ that a customer gets rejected in steady state. This paper presents asymptotic expansions for $P_K$ as $K\to \infty$ . If the service time $B$ is light‐tailed and inter‐arrival 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 Lévy 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 $P(B>K)$ .