  In this paper, we study large multi-server loss models under power-of-$d$
routing scheme when service time distributions are general with finite mean.
Previous works have addressed the exponential service time case when the number
of servers goes to infinity giving rise to a mean field model. The fixed point
of limiting mean field equations (MFE) was shown to be insensitive to the
service time distribution through simulation. Showing insensitivity to general
service time distributions has remained an open problem. Obtaining the MFE in
this case poses a challenge due to the resulting Markov description of the
system being in positive orthant as opposed to a finite chain in the
exponential case. In this paper, we first obtain the MFE and then show that the
MFE has a unique fixed point that coincides with the fixed point in the
exponential case thus establishing insensitivity. The approach is via a
measure-valued Markov process representation and the martingale problem to
establish the mean-field limit. The techniques can be applied to other queueing
models.
