  We investigate the phenomenon of particle creation of the massless scalar
field in the model of spacetime in which, depending on the model's parameter
$\aout$, the chronology horizon could be formed. The model represents a
two-dimensional curved spacetime with the topology $R^1 \times S^1$ which is
asymptotically flat in the past and in the future. The spacetime is globally
hyperbolic and has no causal pathologies if $\aout<1$, and closed timelike
curves appear in the spacetime if $\aout\ge 1$. We obtain the spectrum of
created particles in the case $\aout<1$. In the limit $\aout \to 1$ this
spectrum gives the number of particles created into mode $n$ near the
chronology horizon. The main result we have obtained is that the number of
scalar particles created into each mode as well as the full number of particles
remain finite at the moment of forming of the chronology horizon.
