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A Path Construction for the Virtual Waiting Time of an M/G/1 Queue
Author(s) -
Hooghiemstra G.
Publication year - 1987
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1987.tb01209.x
Subject(s) - queue , statement (logic) , path (computing) , mathematics , distribution (mathematics) , state (computer science) , work (physics) , computer science , discrete mathematics , combinatorics , algorithm , physics , computer network , mathematical analysis , political science , law , thermodynamics
Consider the M/G/1 queue, the finite dam M/G/1 with capacity T, and the impatient customer M/G/1 model, where customers become lost customers if their waiting time exceeds τ. In this note we prove that for all three models and each xe(0, r) the distribution of the number of downcrossings of the virtual waiting time process with level x during a busy cycle is identical. This implies the weaker statement that on [0, T) the distribution functions of the steady state distributions of the amount of unprocessed work (virtual waiting time) are proportional. A number of applications is given.