  A millisecond pulsar is a neutron star that has been substantially spun up by
accretion from a binary companion. A previously unrecognized factor governing
the spin evolution of such pulsars is the crucial effect of non-steady or
transient accretion. We numerically compute the evolution of accreting neutron
stars through a series of outburst and quiescent phases considering the drastic
variation of the accretion rate and the standard disk-magnetosphere
interaction. We find that, for the same long-term average accretion rate, X-ray
transients can spin up pulsars to rates several times higher than can
persistent accretors, even when the spin down due to electromagnetic radiation
during quiescence is included. We also compute an analytical expression for the
equilibrium spin frequency in transients, by taking spin equilibrium to mean
that no net angular momentum is transferred to the neutron star in each
outburst cycle. We find that the equilibrium spin rate for transients, which
depends on the peak accretion rate during outbursts, can be much higher than
that for persistent sources. This explains our numerical finding. This finding
implies that any meaningful study of neutron star spin and magnetic field
distributions requires the inclusion of the transient accretion effect, since
most accreting neutron star sources are transients. Our finding also implies
the existence of a submillisecond pulsar population, which is not observed.
This may point to the need for a competing spin-down mechanism for the
fastest-rotating accreting pulsars, such as gravitational radiation.
