In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, we derive recursive equations for the steady-state distributions of the virtual waiting times in M/G/1 vacation systems with a general vacation time and two vacation rules.
"A note on M/G/1 vacation systems with sojourn time limits." J. Appl. Probab. 49 (4) 1194 - 1199, December 2012. https://doi.org/10.1239/jap/1354716667