  The modulational instability of dust-ion-acoustic (DIA) mode and associated
rogue waves in a three component dusty plasma system (containing inertial warm
ion and negatively charged dust fluids along with inertialess $q$-distributed
electrons) has been theoretically investigated. A nonlinear Schr\"{o}dinger
equation (NLSE) has been derived by employing the reductive perturbation
method. It is observed that the dusty plasma system under consideration
supports the fast and slow DIA modes, and that the dispersion and nonlinear
coefficients of the NLSE determine the parametric regimes not only for the
modulationally stable and unstable fast DIA mode, but also for the formation of
the DIA rogue waves. The parametric regimes for the modulational instability of
the fast DIA mode, and the criterion for the formation of the DIA rogue waves
have been found to be significantly modified by the effects of the relevant
plasma parameters, particularly, mass and charge state of ion and dust species,
number density of the plasma species, and non-extensive parameter $q$, etc. It
is found that the modulationally stable parametric regime decreases (increases)
with the increase in the value of positive (negative) $q$. The numerical
analysis has also shown that the nonlinearity as well as the amplitude and
width of the rogue waves increases (decreases) with the mass of positive ion
(negative dust grains) while decreases (increases) with the charge state of the
positive ion (negative dust). The applications of our present work in space
(viz., Earth ionosphere, magnetosphere, molecular clouds, interstellar medium,
cometary tails, and planetary rings, etc.) and laboratory plasmas have been
pinpointed.
