Abstract
We consider the lower boundary crossing problem for the difference of two independent compound Poisson processes. This problem arises in the busy period analysis of single-server queueing models with work removals. The Laplace transform of the crossing time is derived as the unique solution of an integral equation and is shown to be given by a Neumann series. In the case of ±1 jumps, corresponding to queues with deterministic service times and work removals, we obtain explicit results and an approximation useful for numerical purposes. We also treat upper boundaries and two-sided stopping times, which allows to derive the conditional distribution of the maximum workload up to time t, given the busy period is longer than t.
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Perry, D., Stadje, W. & Zacks, S. Boundary Crossing for the Difference of Two Ordinary or Compound Poisson Processes. Annals of Operations Research 113, 119–132 (2002). https://doi.org/10.1023/A:1020957827834
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DOI: https://doi.org/10.1023/A:1020957827834