Traffic lights, clumping and QBDs

In discrete time, $\mathrm{\ell }$-blocks of red lights are separated by $\mathrm{\ell }$-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 $2\le \mathrm{\ell }\le 3$ and numerically for $\mathrm{\ell }\ge 4$.