Product form Solution of Jackson Network

Question

Consider a Jackson network with nodes $\{i:1\leq i\leq n\}$ which have the arrival rates $\{\lambda_i\}_{i=1}^n$ from outside and service rates at each node $\{\mu\}_{i=1}^n$. Define $\rho_i=\frac{\lambda_i}{\mu_i}\forall\ i$ Then we know from queueing theory that the steady-state distribution of the number in the system is given by $$\pi(k_1,k_2,\cdots, k_n)=\prod_{i=1}^n \rho_i^{k_i}(1-\rho_i) $$ which takes the product form meaning that "as if" the distribution in the different queues in the steady-state are independent, but really they are not. My question is, does anybody know, what is the physical significance of this remarkably simple result?

Answer 1

It's just independent for a special instant of time. Of course the behaviour of the Queues over a span of time can not be independent!