Loss rates in the single-server queue with complete rejection

Abstract

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 $$K\rightarrow \infty $$ . 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 $$P(B>K)$$ .