  We study the polarized quark hadronization in a Monte Carlo (MC) framework
based on the recent extension of the quark-jet framework, where a
self-consistent treatment of the quark polarization transfer in a sequential
hadronization picture has been presented. Here, we first adopt this approach
for MC simulations of hadronization process with finite number of produced
hadrons, expressing the relevant probabilities in terms of the eight leading
twist quark-to-quark transverse momentum dependent (TMD) splitting functions
(SFs) for elementary $q \to q'+h$ transition. We present explicit expressions
for the unpolarized and Collins fragmentation functions (FFs) of unpolarized
hadrons emitted at rank two. Further, we demonstrate that all the current
spectator-type model calculations of the leading twist quark-to-quark TMD SFs
violate the positivity constraints, and propose quark model based ansatz for
these input functions that circumvents the problem. We validate our MC
framework by explicitly proving the absence of unphysical azimuthal modulations
of the computed polarized FFs, and by precisely reproducing the earlier derived
explicit results for rank two pions. Finally, we present the full results for
pion unpolarized and Collins FFs, as well as the corresponding analyzing powers
from high statistics MC simulations with a large number of produced hadrons for
two different model input elementary SFs. The results for both sets of input
functions exhibit the same general features of opposite signed Collins function
for favored and unfavored channels at large $z$, and at the same time
demonstrate the flexibility of the quark-jet framework by producing
significantly different dependences of the results at mid- to low-$z$ for the
two model inputs.
