Let {Xt} be a Lévy process which is reflected at 0 and K> 0. The reflected process {V K t} is constructed as V K t = V K 0 + Xt + L0 t − LK t where {L0 t} and {LK t} are the local times at 0 and K, respectively. We consider the loss rate ℓK, defined by ℓK = EπK LK1, where EπK is the expectation under the stationary measure πK. The main result of the paper is the identification of ℓK in terms of πK and the characteristic triplet of {Xt}. We also derive asymptotics of ℓK as K → ∞ when EX1 < 0 and the Lévy measure of {Xt} is light-tailed.

We consider M/G/1 queues with workload-dependent arrival rate, service speed, and restricted accessibility. The admittance of customers typically depends on the amount of work found upon arrival in addition to its own service requirement. Typical examples are the finite dam, systems with customer impatience and queues regulated by the complete rejection discipline. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet. First, we compare the steady-state distribution of the workload in two finite-buffer models, in which the ratio of arrival and service speed is equal. Second, we find an explicit expression for the cycle maximum in an M/G/1 queue with workload-dependent arrival and service rate. And third, we derive a formal solution for the steady-state workload density in case of restricted accessibility. The proportionality relation between some finite and infinite-buffer queues is extended. Level crossings and Volterra integral equations play a key role in our approach.

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