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 people choose 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 are 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 $\lambda t$.