Understanding the Fluid limit model in Queueing Theory

Question

I am self-studying the fluid limit model in queuing theory.

The basic setting is that: let $Q(t)$ be the total number of customers in the system. Consider the simplest case of M/M/1 queuing system. Usually in papers, they accelerate time by $n$ and scale the state by $1/n$. Then the new process $X(t)=Q(nt)/n$ will converge to the solution of an ODE. My question is, how should we interpret the process $X(t)$? Any physical relationship between this $X(t)$ to the original process $Q(t)$? I think our ultimate goal is still trying to study the process $Q(t)$, now we do this transformation and obtain $X(t)$, which should be a much simpler process, but how does this $X(t)$ help us to understand $Q(t)$?

Answer 1

You can interpret it similarly to a sample mean: measuring the height of 100 randomly selected people and taking the average will give you a good idea of the average height in the sample population (the Law of Large Numbers is the formal mathematical meaning of a "good idea" here).

A queuing process is a random process (analogous to the random variable of height in the example). The fluid limit is the "average" process. That is, it a deterministic process defined by the ODE which represents the average sample path of the random queuing process. The time acceleration is similar to taking a large sample and taking average, in the sense that you have more arrivals/departures at every unit of time but then divide them by the same speed up rate to compute the average. This is known as a Functional Law of Large Numbers (FLLN).

The above explanation is intuitive, but all of the arguments can be made formal, as is done in many Queueing Theory books. I hope it helps.

Answer 2

An intuitive way of thinking about the fluid limit is to think of it as how the process would look like if we "zoom out" and view the process from a large distance (dividing space by $n$) and at large timescale (multiplying time by $n$). This way, stochastic fluctuations of order $\sqrt n$ would be negligible and will not be seen from this "zoomed out" view. A good reference for a rigorous treatment of fluid limits with intuitive explanation can be found here.