Say I have a gas station where cars arrive to refuel. I know that if the waiting times between arrivals is exponentially distributed with mean $1/\lambda$ time-units, then I can model the number of cars arriving in $(0,t]$ as a Poisson process $N_t$ with mean rate $\lambda$ per time unit - which is Markov.
However, now I want to model the amount of fuel contained in the gas station - that is, for every unit time period (say every hour) I check to see how much fuel was taken by the cars that came to refuel. Would making the quantity of the hourly fuel consumption exponentially distributed make this a Markov process?