  Four different fluid model formulations have been recently developed for
$G_t/GI/N+GI$ queues, including a two-parameter fluid model in Whitt (2006) by
tracking elapsed service and patience times of each customer, a measure-valued
fluid model in Kang and Ramanan (2010) and its extension in Zu{\~n}iga (2014)
by tracking elapsed service and patience times of each customer, and a
measure-valued fluid model in Zhang (2013) by tracking residual service and
patience times of each customer. We show that the two fluid models tracking
elapsed times (Whitt's and Kang and Ramanan's fluid models) are equivalent
formulations for the same $G_t/GI/N+GI$ queue, whereas Zu{\~n}iga's fluid model
and Zhang's fluid model are not entirely equivalent under general initial
conditions. We then identify necessary and sufficient conditions under which
Zu{\~n}iga's fluid model and Zhang's fluid model can be derived from each other
for the same system, in which certain measure-valued fluid processes tracking
residual service and patience times of each customer derived from Kang-Ramanan
and Zu{\~n}iga's fluid models play an important role.The equivalence properties
discovered provide important implications for the understanding of the recent
development for non-Markovian many-server queues.

Probability