Suppose we simulate the following system:
In the beginning there are $N$ robots and a large reserve of fuel: $T$ units of fuel in total. Each robot takes some amount of fuel to start out. They consume fuel at a constant rate: 1 unit / second - regardless of what they are doing. If a robot runs out of fuel, it loses (dies in a sense) and it doesn't affect the system anymore.
The robots can access the reserve and they can take some amount of fuel for themselves. The amount of elapsed time between refuels follows an exponential distribution with rate $\lambda$. The amount of fuel the robots take at any given occasion follows a uniform distribution ~ $U(0, F)$.
If there is only 1 robot remaining then it gets all the remaining fuel. For the sake of simplicity, let's assume that taking/transfering fuel happens pretty much instantly. Let's also assume that the robots can store and use any amount of fuel.
No additional fuel or robots are added during the simulation, no parameters are changed, only the initial values are given: $T, N, \lambda, F$.
Is it possible to state anything about the distribution of the termination time in terms of these parameters?
The main motivation was to figure out for how long would it take to live up some sort of resource that's nonrenewable and is necessary for life.
I've tried to define a function for the remaining fuel based on the time points when robots die. Let $d_k$ denote the times of deaths where $k \in [1;n-1]$ and $0 \le d_1 \le d_2 \le ... \le d_{n-1}$. Following $r(t)$ shows the remaining fuel in the system at time point $t$:
$r(t)=\begin{cases}T-Nt & 0 \le t \le d_1\\ T-(N-1)t - d_1 & d_1 \lt t \le d_2 \\ ...\\ T-2t - d_1 - ... - d_{n-2} & d_{n-2} \lt t \le d_{n-1} \\ T-t - d_1 - ... - d_{n-2} - d_{n-1} & d_{n-1} \lt t \le T - d_1 - ... - d_{n-1} \\ \end{cases}$
However, this didn't really bring me closer to anything. Here is a desmos graph for $N=5$ https://www.desmos.com/calculator/fshxexlgx5
