Busy-probability comparison between the true system and mean-field approximations

Let

$$be below the stability threshold$$

. Denote by $(q>0)$, $^{\mathrm{MF1}}(q>0)$, and $^{\mathrm{MF2}}(q>0)$ the stationary busy probabilities of the original system and the Mean-Field 1 and Mean-Field 2 approximations, respectively.

Mean-field busy-probability comparison. For $<$,

$$^{\mathrm{MF2}}(q>0)\leq (q>0)\leq ^{\mathrm{MF1}}(q>0).$$

Thus Mean-Field 1 overestimates the stationary busy probability, whereas Mean-Field 2 underestimates it. The comparison quantifies the relative congestion predicted by the two approximations, but the supplied text does not establish whether the claim is proved or remains conjectural.