Recent study suggests that the streaming instability, one of the leading mechanisms for driving the formation of planetesimals, may not be as efficient as previously thought. Under some disc conditions, the growth timescale of the instability can be longer than the disc lifetime when multiple dust species are considered. To further explore this finding, we use both linear analysis and direct numerical simulations with gas fluid and dust particles to mutually validate and study the unstable modes of the instability in more detail. We extend the previously studied parameter space by one order of magnitude in both the range of the dust-size distribution $[T_{s,min}, T_{s,max}]$ and the total solid-to-gas mass ratio $varepsilon$ and introduce a third dimension with the slope $q$ of the size distribution. We find that the fast-growth regime and the slow-growth regime are distinctly separated in the $varepsilon$-$T_{s,max}$ space, while this boundary is not appreciably sensitive to $q$ or $T_{s,min}$. With a wide range of dust sizes present in the disc (e.g. $T_{s,min}lesssim10^{-3}$), the growth rate in the slow-growth regime decreases as more dust species are considered. With a narrow range of dust sizes (e.g. $T_{s,max}/T_{s,min}=5$), on the other hand, the growth rate in most of the $varepsilon$-$T_{s,max}$ space is converged with increasing dust species, but the fast and the slow growth regimes remain clearly separated. Moreover, it is not necessary that the largest dust species dominate the growth of the unstable modes, and the smaller dust species can affect the growth rate in a complicated way. In any case, we find that the fast-growth regime is bounded by $varepsilongtrsim 1$ or $T_{s,max}gtrsim 1$, which may represent the favourable conditions for planetesimal formation.
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