Devise a two$-$way bus service scheme on the $($regional$)$ route $\{ABC$:$CBA\}$ that will have regular departures from $A,$ but buses continuing on from $B$ to $C$ will depart less frequently. Some buses will be turned back at $B$ and will return to A without continuing on to $C.$ Passengers from $A$ to $C$ should not transfer at $B$ but they may be delayed at $B.$

If the target service frequencies on $AB$ are $12$ buses $\frac{?}{hr}$ in each direction, and on $BC,8$ buses $\frac{?}{hr},$ and if the trip times from $AB,BA,CB,$ and $B$A are each $30$ minutes, then depict a set of bus trajectories that will meet these requirements with regular headways on all legs. You need to determine how much to delay each bus at $B,$ and if a bus just came from $A,$ whether to turn it around at B$.$

What size of bus fleet is required to accommodate your scheme?