Pull-up and Pull-down ratios of NMOS

Pull-up and Pull-down ratio: 

It is defined as the ratio between the resistance offered by pull-up transistor, Rp.u. (depletion mode NMOS) and pull-down transistor, Rp.d. (Enhancement mode transistor) i.e. Rp.u./Rp.d. 

To determine Rp.u./Rp.d. of NMOS inverter driven by another NMOS inverter  

The basic arrangement of NMOS inverter driven by another NMOS inverter is shown in figure below:
 

 Here,

Vout = Vin and set Vinv = 0.5VDD. At this point, both the NMOS inverters are in saturation region

From the MOS Device design equations, we know that, the drain-source current (Ids) under saturation region is given as,

But, for depletion mode transistor (Normally-ON pull-up transistor), Vgs = 0, therefore,

Similarly, for enhancement mode transistor (pull-down transistor), Vgs = Vinv, therefore,

Where,
Wpu = width of pull-up transistor
Lpu = length of pull-up transistor
Wpd = width of pull-down transistor
Lpd = length of pull-down transistor

Since, both depletion and enhancement mode transistors are connected in series to make an NMOS inverter, the Ids shall be same for both, therefore,