Imagine a scenario where customers arrive in some queue according to a Poisson process with rate parameter $\lambda_{arr}$, and where the process of responding to the customers has a kind of "hysteresis" s.t. no agents are assigned to customers until at least one is detected. Here, this "detection" occurs at a rate $\lambda_{det}$ per customer. Once at least a single customer is detected, immediately all of the customers are assigned an agent that serially and sequentially handles their complaints with a "per-complaint time" given by an exponentially distributed parameter $\mu$. No further customers are admitted to the queue while this takes place. Finally, once all customers have been dealt with, the process resets.
The twist is that, the longer a customer waits in the queue prior to being assigned an agent, he'll generate new complaints that need to be dealt with by the agent at a rate $\lambda_{com}$. Thus, it's not simply the case that we have an $M/M/\infty$ queuing process once at least a single customer is detected. Let me stress that no further complaints will be thought of after a customer is assigned an agent.
Provided the above scenario, what fraction of the time is the queue empty? Can we derive a probability distribution for number of customers in the queue at a random time point?
(Update) I suppose we can decompose the above queuing scenario into a cycle of three sequential/successive stages $(...$ $\to 3 \to 1 \to 2 \to 3 \to 1 \to 2 \to$ $...)$:
(1) Time until first customer arrives in queue. The duration of this stage can be characterized by a simple exponential decay function with rate parameter $\lambda_{arr}$.
(2) Time between the first customer entering the queue and until at least one customer in the queue is detected.
(3) Time to empty the queue.
GOTO (1).
It would be really neat to be able to explicitly describe the probability distribution for the duration of events (2) and (3).