  Observation of fast radio bursts (FRBs) are rising very quickly with the
advent of specialised instruments and surveys, and it has recently been shown
that some of them repeat quasi-periodically. In particular, evidence of a
$P=16.35$ day period has been reported for FRB 180916.J0158+65. We seek an
explanation within the frame of our orbiting asteroid model, whereby FRBs are
produced in the plasma wake of asteroids immersed in the wind of a pulsar or a
magnetar. We used the data reported by the CHIME/FRB collaboration in order to
infer the orbital characteristics of asteroid swarms, and performed parametric
studies to explore the possible characteristics of the pulsar, its wind, and of
the asteroids, under the constraint that the latter remain dynamically and
thermally stable. We found a plausible configuration in which a young pulsar is
orbited by a main $\sim 10^{-3}M_\odot$ companion with a period $3P = 49$d,
three times longer than the apparent periodicity $P$. Asteroids responsible for
FRBs are located in three dynamical swarms near the L3, L4 and L5 Lagrange
points, in a 2:3 orbital resonance akin to the Hildas class of asteroids in the
Solar system. In addition, asteroids could be present in the Trojan swarms at
the L4 and L5 Lagrange points. Together these swarms form a carousel that
explains the apparent $P$ periodicity and dispersion. We estimated that the
presence of at least a few thousand asteroids, of size $\sim20$km, is necessary
to produce the observed burst rate. We show how radius-to-frequency mapping in
the wind and small perturbations by turbulence can suffice to explain
downward-drifting sub-pulses, micro-structures, and narrow spectral occupancy.
