I tried to use small signal model and introduce a measuring voltage between source and drain, but I can not get a solvable equation of Vds/Id or V_measure/I_measure.
You get a non-linear relation Vds=f(Ids), you need to specify either Vds or Ids to calculate a small signal rds = dVds/dIds or large signal Rds = Vds/Ids. In any case, they don't depend on gate resistor R.
You get a non-linear relation Vds=f(Ids), you need to specify either Vds or Ids to calculate a small signal rds = dVds/dIds or large signal Rds = Vds/Ids. In any case, they don't depend on gate resistor R.