  There have now been three supernova-associated gamma-ray bursts (GRBs) at
redshift z < 0.17, namely 980425, 030329, and 031203, but the nearby and
under-luminous GRBs 980425 and 031203 are distinctly different from the
`classical' or standard GRBs. It has been suggested that they could be
classical GRBs observed away from their jet axes, or they might belong to a
population of under-energetic GRBs. Recent radio observations of the afterglow
of GRB 980425 suggest that different engines may be responsible for the
observed diversity of cosmic explosions. Given this assumption, a crude
constraint on a luminosity function for faint GRBs with a mean luminosity
similar to that of GRB 980425 and an upper limit on the rate density of
980425-type events, we simulate the redshift distribution of under-luminous
GRBs assuming BATSE and Swift sensitivities. A local rate density of about 0.6%
of the local supernova Type Ib/c rate yields simulated probabilities for
under-luminous events to occur at rates comparable to the BATSE GRB
low-redshift distribution. In this scenario the probability of BATSE/HETE
detecting at least one GRB at z<0.05 is 0.78 over 4.5 years, a result that is
comparable with observation. Swift has the potential to detect 1--5
under-luminous GRBs during one year of observation.
