Traffic lights, clumping and QBDs

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. \ We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics algebraically for $2\leq\ell\leq3$ and numerically for $\ell\geq4$.

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