Successive times between completed handling of requests are described by a distribution with p.d.f.
$$f(x) = \frac{25}{12} e^{-5x/2} + \frac{1}{18}e^{-x/3}$$
It may be assumed that successive interdeparture times of handled requests are independent.
How could one propose a stochastic model for the process of completed requests?
What you have there is a mixture of two exponential distributions.
Write the density as $$f(x) = \frac{5}{6}\frac{5}{2}\mathrm e^{-5x/2} + \frac{1}{6}\frac{1}{3}\mathrm e^{-x/3}.$$
This describes a random variable that is $Exp(5/2)$ with probability $5/6$, and $Exp(1/3)$ with probability $1/6$.
Edit after comment
You can model the departure times as a Markov chain. Let $t_n$ be the $n$th departure chain and $\{\Delta_n\}$ be the sequence of interdeparture times -- an iid sequence of random variables with density $f(x)$.
Then, the departure times can be descibed as the Markov chain $$t_{n+1} = t_n +\Delta_n.$$