What is growth of $\psi_q(m) = \min \{ p: m ∣ (q^p−1) \}$ for fixed $q$?

Question

I have to estimate the computational complexity of some algorithm that does $\psi_q(m)$ iterations. Assume that all inputs $m$ are coprime to $q$.

So I need to know what growth the $\psi(m)$ has.