Audio transfer function modelling with pink noise + signal
your generated pink noise is just a signal.  it's not noise because you know what it is.  


you can toss whatever broadbanded signal you want into your linear, time-invariant system.  call that "$x[n]$".
you can measure the output of that LTI system, call that "$y[n]$", given the known input.
you can FFT both the input and the output.  call those FFT results "$X[k]$" and "$Y[k]$", respectively.
if the input was broadbanded, $X[k] \ne 0$ for all $k$.  so you can divide $Y[k]$ with $X[k]$.
the result: $$ H[k]=\frac{Y[k]}{X[k]} $$ is your system response.  this is called the "Dual-channel FFT" method of measuring system response. if there is a delay, you will see a linear trend to the unwrapped angle of $H[k]$. the negative of the slope of that linear trend is proportional to the delay.


you can also use cross-correlation between the input and output to discover the prominent delay of the system.  the definition of that is

$$ R_{xy}[m] = \sum_n x[n] y[n+m] $$

i'm deliberately playing fast-and-loose with the limits of the summation.  the value of $m$ that maximizes $R_{xy}[m]$ is a good candidate for the delay of the system.