Writeback formula doubt

Question

In writeback cache, formula for write is given as

$T_{write} = H \times T_{cache} + (1-H) \times (T_{cache} + T_{memory\_block} + T_{write\_back}) \\ \\ \text{ where } T_{write\_back} = x \times T_{memory\_block}, \text{ where } x \text{ is the fraction of dirty blocks}$

Here in case of write miss, why are we not adding the time to update the word? Incase of write miss, the word is updated in the block and then that block is brought back to cache.(As mentioned here http://web.cs.iastate.edu/~prabhu/Tutorial/CACHE/interac.html)

So shouldn't the formula be

$T_{write} = H \times T_{cache} + (1-H) \times (T_{cache} +T_{Mem(Update word)} + T_{memory\_block} + T_{write\_back}) \\ \text{ where } T_{write\_back} = x \times T_{memory\_block}, \text{ where } x \text{ is the fraction of dirty blocks}$

$T_{cache}$ = hierarchical cache time.

$T_{Mem(Update word)}$ = Update the word in main memory.

$T_{memory\_block}$= Bring the block containing updated word to cache .

Please help me .