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 $$K$$K. A basic performance measure of this system is the probability $$P_K$$PK that a customer gets rejected in steady state. This paper presents asymptotic expansions for $$P_K$$PK as $$K\rightarrow \infty $$K→∞. If the service time $$B$$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 $$P(B>K)$$P(B>K).

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