An alternative scheme is described to estimate the layer 1 LAXPC 20 background for faint sources where the source contribution to the 50-80 keV count rate is less than 0.25 counts/sec (15 milli-crabs or $6 \times 10^{-11}$ ergs/s/cm$^2$). We consider 12 blank sky observations and based on their 50-80 keV count rate in 100 second time-bins, generate four template spectra which are then used to estimate the background spectrum and lightcurve for a given faint source observation. The variance of the estimated background subtracted spectra for the 12 blank sky observations is taken as the energy dependent systematic uncertainty which will dominate over the statistical one for exposures longer than 5 ksecs. The estimated 100 second time bin background lightcurve in the 4-20 keV band with a 3\% systematic error matches with the blank sky ones. The 4-20 keV spectrum can be constrained for a source with flux $\gtrapprox 1$ milli-crab. Fractional r.m.s variability of 10\% can be determined for a $\sim 5$ milli-crab source lightcurve binned at 100 seconds. To illustrate the scheme, the lightcurves, and spectra of three different blank sky observations, three AGN sources (Mrk 0926, Mrk 110, NGC 4593), and LMC X-1 are shown.