We study the effects of the accretion of normal matter on to strange-m
atter pulsar models. It is assumed that, because of the high strangene
ss barrier, normal matter is inert in contact with Q(alpha) matter. Fo
r this reason, normal matter accretion is able to form a thick outer l
ayer with densities far above neutron drip. Accretion can make the sup
erfluid quark-alpha Q(alpha) layer disappear by solidification but, fo
r the same reason, a superfluid neutron layer is formed. The fractiona
l moment of inertia of the latter is large enough to fulfil the vortex
-creep glitch model requirements. The high rotational stability of mil
lisecond pulsars is discussed in the framework of the strange-matter h
ypothesis. If the validity of the vortex-creep model is assumed, it is
shown that, for the case of accreted strange-pulsar models, this stab
ility cannot be interpreted as due to the solidification of the superf
luid Q(alpha) layer. Nevertheless,`standard' homogeneous strange stars
remain as interesting candidates for the internal composition of thes
e objects. The main conclusions of this work should be unaltered if Q(
alpha) is not preferred to strange matter at low pressures, but some o
ther strange complex plays a similar role.