In the small signal model of a MOSFET, the current from the drain to the source (for an n channel MOSFET) is a function of the gate to source voltage only.
But, if a positive voltage was not applied at the drain, the current wouldn't flow in the first place.
So, why is the drain to source voltage not considered?
A small-signal model is only valid around a certain operating point. Your particular model makes the assumption that the DC bias voltage on the drain is such that the model is valid, and the AC drive voltage at the gate is small enough that the gm approximation works.
As an example, the Id vs VGS characteristic for a particular FET is shown below with Vds=15V. Graphic from here. If your gate is biased at the point on the diagram where the slope of the curve is gm, then those would be the conditions around which your small-signal model would be valid.
For different DC conditions you may have different values of gm and output resistance.
You can Google all kinds of detailed explanations of how to derive a small-signal model and the assumptions made.
Here's one example: Small-signal derivation Here's another: MOSFET Small Signal Modelling
Maybe after reviewing what you can find with Google you could post a new question with any specific doubts you have about what you find.
