  The expected event rate of lensed gravitational wave sources scales with the
merger rate at redshift $z\geq 1$, where the optical depth for lensing is high.
It is commonly assumed that the merger rate of the astrophysical compact
objects is closely connected with the star formation rate, which peaks around
redshift $z\sim 2$. However, a major source of uncertainty is the delay time
between the formation and merger of compact objects. We explore the impact of
delay time on the lensing event rate. We show that as the delay time increases,
the peak of the merger rate of gravitational wave sources gets deferred to a
lower redshift. This leads to a reduction in the event rate of the lensed
events which are detectable by the gravitational wave detectors. We show that
for a delay time of around $10$ Gyr or larger, the lensed event rate can be
less than one per year for the design sensitivity of LIGO/Virgo. We also
estimate the merger rate for lensed sub-threshold for different delay time
scenarios, finding that for larger delay times the number of lensed
sub-threshold events is reduced, whereas for small-delay time models they are
significantly more frequent. This analysis shows for the first time that
lensing is a complementary probe to explore different formation channels of
binary systems by exploiting the lensing event rate from the well-detected
events and sub-threshold events which are measurable using the network of
gravitational wave detectors.
