= Solution
Assign each arriving car independently to a uniformly chosen lot. The <Poisson thinning theorem> gives \b[three independent Poisson processes, each of rate $\lambda/3$]. For an interval of length $t$, their joint <probability generating function> is
$$
\exp\!\left[\lambda t\left(\frac{z_a+z_b+z_c}{3}-1\right)\right]
=\prod_{i=a,b,c}\exp\left[\frac{\lambda t}{3}(z_i-1)\right].
$$
The factorization proves independence of the counts in that interval; independence over disjoint intervals proves independence of the processes.
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