  An information engine based on a two level system in contact with a thermal
reservoir is studied analytically. The model incorporates delay time between
the measurement of the state of the system and the feedback. The engine
efficiency and work extracted per cycle are studied as a function of delay time
and energy spacing between the two levels. It is found that the range of delay
time over which one can extract work from the information engine increases with
temperature. For delay times comparable to the relaxation time, efficiency and
work per cycle are both maximum when $k_B T \approx 2 U_0$, the energy
difference between the levels. Generalized Jarzynski equality and the
generalized integral fluctuation theorem are explicitly verified for the model.
The results from the model are compared with the simulation results for a
feedback engine based on a particle moving in a $1D$ square potential. The
variation of efficiency, work per cycle and efficacy with the delay time is
compared using relaxation time in the two state model as the fitting parameter
and leads to good fit.
