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For the general G/G/1 processor sharing (PS) system a sample path result for the sojourn times in a busy period is proved, which yields a relation between the sojourn times under PS and FCFS discipline. In particular, the result provides a formula for the mean sojourn time in G/D/1-PS in terms of the mean sojourn time in the corresponding G/D/1-FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of the formula provide the basis for a two-moment approximation of the mean sojourn time in G/GI/1-PS in terms of a related G/D/1-FCFS.
The paper deals with the workload and busy period for the M/GI/1
system under FCFS discipline, where the customers may become
impatient during their waiting for service with generally
distributed maximal waiting times and also during their service
with generally distributed maximal service times depending
on the time waited for service. This general impatience mechanism,
originally introduced by Kovalenko (1961) and considered
by Daley (1965), too, covers the special cases of impatience
on waiting times as well as impatience on sojourn times,
for which Boxma et al. (2010), (2011) gave new results
and outlined special cases recently. Our unified approach
bases on the vector process of workload and busy time.
Explicit representations for the LSTs of workload and
busy period are given in case of phase-type distributed
impatience.