We consider the steady-state distribution of the sojourn time of a job entering an M/GI/1 queue with the foreground–background scheduling policy in heavy traffic. The growth rate of its mean as well as the limiting distribution are derived under broad conditions. Assumptions commonly used in extreme value theory play a key role in both the analysis and the results.

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Stochastic Systems
Bridging Probabilistic and Competitive Analysis of Scheduling Policies , Rare events: Asymptotics, Algorithms, Applications
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Kamphorst, B., & Zwart, B. (2020). Heavy-traffic analysis of sojourn time under the foreground–background scheduling policy. Stochastic Systems, 10(1), 1–28. doi:10.1287/stsy.2019.0036