  We derive scaling laws for the signal-to-noise ratio of the optimal
cross-correlation statistic, and show that the large power-law increase of the
signal-to-noise ratio as a function of the the observation time $T$ that is
usually assumed holds only at early times. After enough time has elapsed,
pulsar timing arrays enter a new regime where the signal to noise only scales
as $\sqrt{T}$. In addition, in this regime the quality of the pulsar timing
data and the cadence become relatively un-important. This occurs because the
lowest frequencies of the pulsar timing residuals become gravitational-wave
dominated. Pulsar timing arrays enter this regime more quickly than one might
naively suspect. For T=10 yr observations and typical stochastic background
amplitudes, pulsars with residual RMSs of less than about $1\,\mu$s are already
in that regime. The best strategy to increase the detectability of the
background in this regime is to increase the number of pulsars in the array. We
also perform realistic simulations of the NANOGrav pulsar timing array, which
through an aggressive pulsar survey campaign adds new millisecond pulsars
regularly to its array, and show that a detection is possible within a decade,
and could occur as early as 2016.
