  We extend collisional quantum thermometry schemes to allow for stochasticity
in the waiting time between successive collisions. We establish that
introducing randomness through a suitable waiting time distribution, the
Weibull distribution, allows to significantly extend the parameter range for
which an advantage over the thermal Fisher information is attained. These
results are explicitly demonstrated for dephasing interactions and also hold
for partial swap interactions. Furthermore, we show that the optimal
measurements can be performed locally, thus implying that genuine quantum
correlations do not play a role in achieving this advantage. We explicitly
confirm this by examining the correlation properties for the deterministic
collisional model.
