This is a question about intuition in understanding the fluid limit queuing system.

Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit service rate. Assume the arrival rate of the system is $\lambda N$, where $\lambda<1$ is a constant. People come to the system and select one server to queue there. Let $k_i$ be the number of servers with $i$ people in line. So $\pi_i=\frac{k_i}{N}$ is the fraction of servers with $i$ people in line.

My first question is about understanding the situation when $N\rightarrow\infty$. I don't know if it is called the "fluid limit". Imagining the system as described above starts with empty, then every unit of time, roughly $\lambda N dt$ amount of people come in. We assume that each person chooses the shortest queue, i.e., the empty queue. To make life even simpler, for this moment, assume not service process, i.e., people coming in, and select a server who never start to work and just standing there. Then there are two ways of understanding the system:

A) the amount $\lambda N dt$ of people is uniformly assigned to each server, so that each server will have people, but not a whole people, it is only some fraction of people (e.g., $\lambda<1$ people)

B) only $\lambda$ fraction of the servers will have 1 people and the remaining $1-\lambda$ fraction of servers will have 0 people.

Which intuition is correct? There is a big difference between A and B. If A is correct, then this means, there is no queue of integer length. If we imagining people coming in like "fluid", then it makes sense. However, if A is correct, then it will implies that $\pi_1=\frac{k_1}{N}$ converges to 0, since there is no queue of integer length. If B is correct, then this means, there are only queues of integer length, which sounds more intuitive. And it will implies $\pi^1=\frac{k_1}{N}$ converges to $\lambda t$.

The things that are puzzling me the most is that, I think intuition B is very clearly in my setting, i.e., queue should increase or decrease with integer size, e.g., if there are empty queue, then people join that empty queue and hence we have queue of length one, if all queue are of length one, then people join queue of length one and hence create queue of length two, etc. If there is a service process, e.g., each server has identical service rate $\mu=1$, then as $N\rightarrow\infty$, then it should be clear that $\frac{d\pi_1(t)}{dt}=-\pi_1(t)+\lambda$. I am wondering where does intuition A come from? Is this because, in the "fluid model", people usually talking about the process $Q(t)$, which is the number of people in the queue, then they do scaling $\frac{1}{N}Q(t)$ and call this process $\bar{Q}(t)$, then because in the original $Q(t)$ process, every time when people come or leave, the process jump up or down by 1 unit, hence, after the scaling $\frac{1}{N}$, the scaled process $\bar{Q}(t)$ has a continuous state, so that's why they say queue (or people) become "fluid" such that "queues" are no longer has integer size?

So in my problem, should I call it "Fluid model"? Is there a clear definition of the terminology of "Fluid model"?