  We study the statistical properties of the complex generalization of Wigner
time delay $\tau_\text{W}$ for sub-unitary wave chaotic scattering systems. We
first demonstrate theoretically that the mean value of the
$\text{Re}[\tau_\text{W}]$ distribution function for a system with uniform
absorption strength $\eta$ is equal to the fraction of scattering matrix poles
with imaginary parts exceeding $\eta$. The theory is tested experimentally with
an ensemble of microwave graphs with either one or two scattering channels, and
showing broken time-reversal invariance and variable uniform attenuation. The
experimental results are in excellent agreement with the developed theory. The
tails of the distributions of both real and imaginary time delay are measured
and are also found to agree with theory. The results are applicable to any
practical realization of a wave chaotic scattering system in the
short-wavelength limit.
