A camel with a less strict diet

A camel can carry $B$ bananas on its back. It can have $2$ bananas at a time in its stomach. For each mile the camel walks, the amount of bananas in its stomach decreases $1$. As soon as the amount of bananas in the camel's stomach is at most $1$, it can eat a new banana. When the camel's stomach is empty, the camel must eat a new banana (in order to be able to continue its itinerary). Let there be a stock of $N$ bananas at the border of the desert. How far can the camel penetrate into the desert, starting at this point? (Of course it can form new stocks with transported bananas.) The case $B=1$ is solved completely. The round trip variant is solved for $B=1$ as well. For $B=2$, the round trip variant is solved for $N$ which are a power of $2$ and $N \le 8$, and estimated up to $1/(N-1)$ miles for general $N$.