We study the mass of quasar-hosting dark matter halos at z $sim$ 6 and further constrain the fraction of dark matter halos hosting an active quasar $f_{on}$ and the quasar beaming angle $i_{rm max}$ using observations of CII lines in the literature. We make assumptions that (1) more massive halos host brighter quasars, (2) a fraction of the halos host active quasars with a certain beaming angle, (3) cold gas in galaxies has rotational velocity $V_{rm circ}=alpha V_{rm max}$, and that (4) quasars point randomly on the sky. We find that for a choice of specific $alpha gtrsim 1$, the most likely solution has $f_{rm on} < 0.01$, corresponding to a small duty cycle of quasar activity. However, if we marginalize over $alpha$, for some choices of a prior a second solution with $f_{rm on}=1$ appears. Overall, our the constraints are highly sensitive to $alpha$ and hence inconclusive. Stronger constraints on $f_{rm on}$ can be made if we better understand the dynamics of cold gas in these galaxies.
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